Explanation:
Okay, so I need to find the value of k such that (3^k)^6 equals 3^6. Let me think about how to approach this.

First, I remember that when you have an exponent raised to another exponent, like (a^b)^c, you can multiply the exponents. So, (a^b)^c = a^(b*c). Is that right? Let me check with a simple example. If I take (2^3)^2, that should be 2^(3*2) = 2^6, which is 64. And if I calculate it step by step, (2^3) is 8, and 8 squared is 64. Yep, that works. So the rule seems correct.

Applying that to the problem here: (3^k)^6 would be 3^(k*6). So the equation becomes 3^(6k) = 3^6.

Now, since the bases are the same (both are base 3), the exponents must be equal for the equation to hold true. That is, if a^m = a^n, then m = n, provided that a is not 0 or 1. Since 3 is neither 0 nor 1, this rule applies here.

Therefore, setting the exponents equal: 6k = 6.

To solve for k, I can divide both sides of the equation by 6.

So, 6k / 6 = 6 / 6, which simplifies to k = 1.

Wait, let me verify that. If k is 1, then (3^1)^6 = 3^6. Which is 3^6 = 3^6. That's definitely true.

But let me think again if there could be other solutions. For example, sometimes with exponents, especially if dealing with negative numbers or fractions, there might be multiple solutions. But in this case, since the base is 3, which is positive and not equal to 1, the function 3^x is one-to-one. That means each output corresponds to exactly one input. Therefore, there's only one solution for k, which is 1.

Alternatively, if I didn't remember the exponent rules, I could try plugging in numbers. Let's see. If k is 1, then (3^1)^6 = 3^6. If k is 0, then (3^0)^6 = 1^6 = 1, which is not equal to 3^6. If k is 2, then (3^2)^6 = 9^6, which is way bigger than 3^6. So k=1 is the only one that works.

Another way to think about it: 3^6 is 729. Let me compute (3^k)^6 and see when it equals 729.

If k=1: 3^1 = 3, then 3^6 = 729. Perfect.

If k=0: 3^0 = 1, 1^6 = 1 ≠ 729.

k=2: 3^2 = 9, 9^6 = 531441, which is way larger.

k=1/2: 3^(1/2) is sqrt(3), then (sqrt(3))^6 = (3^(1/2))^6 = 3^(3) = 27 ≠ 729.

So yeah, only k=1 works.

Is there a possibility of a negative k? Let's try k=-1. Then (3^-1)^6 = (1/3)^6 = 1/729 ≠ 729. So no.

Therefore, k must be 1.

Wait, but let me check if there's another way to interpret the original equation. Maybe the problem is written as 3^k^6 = 3^6? But no, the original problem is written as (3^k)^6 = 3^6. The parentheses make it clear that it's (3^k) raised to the 6th power. If it were 3^(k^6), that would be different, but the parentheses indicate that the exponent 6 applies to the entire 3^k.

So, no ambiguity here.

Alternatively, if someone misreads the problem as 3^(k^6) = 3^6, then the solution would be different. In that case, we would have k^6 = 6, so k = 6^(1/6). But since the original problem has parentheses around 3^k, it's definitely (3^k)^6.

Therefore, my conclusion is that k=1.

But just to make sure, let me go through the exponent rules again.

The rule is (a^b)^c = a^(b*c). So, (3^k)^6 = 3^(k*6). Therefore, 3^(6k) = 3^6. Since the bases are equal, exponents must be equal. Therefore, 6k = 6 => k=1.

Yes, that's straightforward.

Another thought: maybe the problem is designed to test understanding of exponents, so maybe there's a trick? But I don't see any trick here. It's a straightforward application of exponent rules.

Alternatively, if someone forgets the exponent rule, they might try to expand (3^k)^6. Let's see, 3^k multiplied by itself 6 times. So, 3^k * 3^k * 3^k * 3^k * 3^k * 3^k. Which is 3^(k + k + k + k + k + k) = 3^(6k). So same result.

Therefore, 3^(6k) = 3^6 implies 6k = 6, so k=1.

Therefore, I think the answer is definitely 1.

Just to recap:

Original equation: (3^k)^6 = 3^6

Apply exponent rule: 3^(6k) = 3^6

Set exponents equal: 6k = 6

Solve for k: k = 6 / 6 = 1

Check solution: Plug back in, (3^1)^6 = 3^6, which is true.

Therefore, k=1 is the correct solution.

**Final Answer**
The value of $ k $ is \boxed{1}.

To solve the equation:

$$
(3^k)^6 = 3^6
$$

we begin by applying the **exponentiation rule** for powers of powers:

$$
(a^b)^c = a^{b \cdot c}
$$

Applying this rule to the left-hand side of the equation:

$$
(3^k)^6 = 3^{k \cdot 6} = 3^{6k}
$$

So the equation becomes:

$$
3^{6k} = 3^6
$$

---

Since the **bases are the same** (both are 3), and the function $ 3^x $ is **one-to-one** (i.e., it is strictly increasing and injective), the **exponents must be equal** for the equality to hold. Therefore, we equate the exponents:

$$
6k = 6
$$

Now, solve for $ k $:

$$
k = \frac{6}{6} = 1
$$

---

### Final Verification:

Substitute $ k = 1 $ back into the original expression:

$$
(3^1)^6 = 3^6
$$

$$
3^6 = 3^6
$$

This confirms the solution is correct.

---

### Final Answer:

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of k such that (3^k)^6 equals 3^6. Let me think about how to approach this.

First, I remember that when you have an exponent raised to another exponent, like (a^b)^c, you can multiply the exponents. So, (a^b)^c = a^(b*c). Is that right? Let me check with a simple example. If I take (2^3)^2, that should be 2^(3*2) = 2^6, which is 64. And if I calculate it step by step, (2^3) is 8, and 8 squared is 64. Yep, that works. So the rule seems correct.

Applying that to the problem here: (3^k)^6 would be 3^(k*6). So the equation becomes 3^(6k) = 3^6.

Now, since the bases are the same (both are base 3), the exponents must be equal for the equation to hold true. That is, if a^m = a^n, then m = n, provided that a is not 0 or 1. Since 3 is neither 0 nor 1, this rule applies here.

Therefore, setting the exponents equal: 6k = 6.

To solve for k, I can divide both sides of the equation by 6.

So, 6k / 6 = 6 / 6, which simplifies to k = 1.

Wait, let me verify that. If k is 1, then (3^1)^6 = 3^6. Which is 3^6 = 3^6. That's definitely true.

But let me think again if there could be other solutions. For example, sometimes with exponents, especially if dealing with negative numbers or fractions, there might be multiple solutions. But in this case, since the base is 3, which is positive and not equal to 1, the function 3^x is one-to-one. That means each output corresponds to exactly one input. Therefore, there's only one solution for k, which is 1.

Alternatively, if I didn't remember the exponent rules, I could try plugging in numbers. Let's see. If k is 1, then (3^1)^6 = 3^6. If k is 0, then (3^0)^6 = 1^6 = 1, which is not equal to 3^6. If k is 2, then (3^2)^6 = 9^6, which is way bigger than 3^6. So k=1 is the only one that works.

Another way to think about it: 3^6 is 729. Let me compute (3^k)^6 and see when it equals 729.

If k=1: 3^1 = 3, then 3^6 = 729. Perfect.

If k=0: 3^0 = 1, 1^6 = 1 ≠ 729.

k=2: 3^2 = 9, 9^6 = 531441, which is way larger.

k=1/2: 3^(1/2) is sqrt(3), then (sqrt(3))^6 = (3^(1/2))^6 = 3^(3) = 27 ≠ 729.

So yeah, only k=1 works.

Is there a possibility of a negative k? Let's try k=-1. Then (3^-1)^6 = (1/3)^6 = 1/729 ≠ 729. So no.

Therefore, k must be 1.

Wait, but let me check if there's another way to interpret the original equation. Maybe the problem is written as 3^k^6 = 3^6? But no, the original problem is written as (3^k)^6 = 3^6. The parentheses make it clear that it's (3^k) raised to the 6th power. If it were 3^(k^6), that would be different, but the parentheses indicate that the exponent 6 applies to the entire 3^k.

So, no ambiguity here.

Alternatively, if someone misreads the problem as 3^(k^6) = 3^6, then the solution would be different. In that case, we would have k^6 = 6, so k = 6^(1/6). But since the original problem has parentheses around 3^k, it's definitely (3^k)^6.

Therefore, my conclusion is that k=1.

But just to make sure, let me go through the exponent rules again.

The rule is (a^b)^c = a^(b*c). So, (3^k)^6 = 3^(k*6). Therefore, 3^(6k) = 3^6. Since the bases are equal, exponents must be equal. Therefore, 6k = 6 => k=1.

Yes, that's straightforward.

Another thought: maybe the problem is designed to test understanding of exponents, so maybe there's a trick? But I don't see any trick here. It's a straightforward application of exponent rules.

Alternatively, if someone forgets the exponent rule, they might try to expand (3^k)^6. Let's see, 3^k multiplied by itself 6 times. So, 3^k * 3^k * 3^k * 3^k * 3^k * 3^k. Which is 3^(k + k + k + k + k + k) = 3^(6k). So same result.

Therefore, 3^(6k) = 3^6 implies 6k = 6, so k=1.

Therefore, I think the answer is definitely 1.

Just to recap:

Original equation: (3^k)^6 = 3^6

Apply exponent rule: 3^(6k) = 3^6

Set exponents equal: 6k = 6

Solve for k: k = 6 / 6 = 1

Check solution: Plug back in, (3^1)^6 = 3^6, which is true.

Therefore, k=1 is the correct solution.

**Final Answer**
The value of $ k $ is \boxed{1}.

To solve the equation:

$$
(3^k)^6 = 3^6
$$

we begin by applying the **exponentiation rule** for powers of powers:

$$
(a^b)^c = a^{b \cdot c}
$$

Applying this rule to the left-hand side of the equation:

$$
(3^k)^6 = 3^{k \cdot 6} = 3^{6k}
$$

So the equation becomes:

$$
3^{6k} = 3^6
$$

---

Since the **bases are the same** (both are 3), and the function $ 3^x $ is **one-to-one** (i.e., it is strictly increasing and injective), the **exponents must be equal** for the equality to hold. Therefore, we equate the exponents:

$$
6k = 6
$$

Now, solve for $ k $:

$$
k = \frac{6}{6} = 1
$$

---

### Final Verification:

Substitute $ k = 1 $ back into the original expression:

$$
(3^1)^6 = 3^6
$$

$$
3^6 = 3^6
$$

This confirms the solution is correct.

---

### Final Answer:

$$
\boxed{1}
$$

Solution: Following exponent laws, ${(3^k)}^6=3^{6k}$.  Because $3^{6k}=3^6$, we have $6k=6$, which, dividing by 6, solves to $k=\boxed{1}$.


Explanation:
Okay, so Hillary has eleven coins, and they're all either dimes or nickels. The total value of these coins is 75 cents. I need to figure out how many nickels she has. Let me think about how to approach this.

First, let me recall what dimes and nickels are worth. A dime is 10 cents, and a nickel is 5 cents. So, if I can figure out how many of each coin there are, I can add up their values to get 75 cents. Since there are eleven coins in total, I can set up some equations to solve this.

Let me denote the number of dimes as D and the number of nickels as N. According to the problem, the total number of coins is 11. So, that gives me my first equation:

D + N = 11

That's straightforward. Now, the total value of the coins is 75 cents. Since each dime is 10 cents and each nickel is 5 cents, the second equation would be:

10D + 5N = 75

So now I have a system of two equations with two variables. I can solve this system using substitution or elimination. Let me try substitution because it seems straightforward.

From the first equation, I can express D in terms of N:

D = 11 - N

Now, substitute this expression for D into the second equation:

10(11 - N) + 5N = 75

Let me compute that step by step. First, multiply out the 10:

10*11 - 10*N + 5N = 75

Which is:

110 - 10N + 5N = 75

Combine like terms:

110 - 5N = 75

Now, subtract 110 from both sides:

-5N = 75 - 110

Calculating the right side:

75 - 110 = -35

So:

-5N = -35

Now, divide both sides by -5:

N = (-35)/(-5) = 7

So, N = 7. That means Hillary has 7 nickels. Let me check if this makes sense.

If there are 7 nickels, then the number of dimes would be 11 - 7 = 4. Let me check the total value:

7 nickels * 5 cents = 35 cents

4 dimes * 10 cents = 40 cents

Total value: 35 + 40 = 75 cents. Perfect, that matches the given total. So, the answer is 7 nickels.

Wait, let me just make sure I didn't make any arithmetic errors. Let me go through the steps again.

Starting with D + N = 11 and 10D + 5N = 75.

Substituting D = 11 - N into the second equation:

10*(11 - N) + 5N = 75

110 - 10N + 5N = 75

110 - 5N = 75

Subtract 110: -5N = 75 - 110 = -35

Divide by -5: N = 7. Yep, that's correct. And checking the numbers again, 7 nickels and 4 dimes add up to 11 coins and 75 cents. Seems solid.

Alternatively, maybe I can try solving it another way to confirm. Let's try elimination.

Original equations:

1) D + N = 11

2) 10D + 5N = 75

If I multiply the first equation by 5, I get:

5D + 5N = 55

Then subtract that from the second equation:

(10D + 5N) - (5D + 5N) = 75 - 55

Which simplifies to:

5D = 20

Therefore, D = 4. Then substituting back into D + N = 11, we get N = 7. Same result. So that confirms it again.

Another way to think about it: Let's assume all coins are nickels. Then total value would be 11 * 5 = 55 cents. But we need 75 cents, which is 20 cents more. Each time we replace a nickel with a dime, we add 5 cents (since a dime is 10 cents vs. 5 cents). So, how many replacements do we need to get an additional 20 cents?

Number of replacements = 20 / 5 = 4. Therefore, there are 4 dimes and 11 - 4 = 7 nickels. Yep, same answer. So that's another way to think about it, using the concept of replacing coins.

So all methods lead to 7 nickels. Therefore, I can be confident that the answer is 7.

**Final Answer**
Hillary has \boxed{7} nickels.

To determine how many nickels Hillary has, we begin by analyzing the information provided:

- Hillary has **11 coins** in total.
- All coins are either **dimes (10 cents)** or **nickels (5 cents)**.
- The **total value** of the coins is **75 cents**.

---

### Step 1: Define Variables

Let:
- $ D $ = number of dimes
- $ N $ = number of nickels

From the problem, we can write two equations:

1. **Total number of coins:**
   $$
   D + N = 11
   $$

2. **Total value of coins:**
   $$
   10D + 5N = 75
   $$

---

### Step 2: Solve the System of Equations

From the first equation, solve for $ D $ in terms of $ N $:

$$
D = 11 - N
$$

Substitute this into the second equation:

$$
10(11 - N) + 5N = 75
$$

Simplify:

$$
110 - 10N + 5N = 75
$$
$$
110 - 5N = 75
$$

Subtract 110 from both sides:

$$
-5N = -35
$$

Divide both sides by -5:

$$
N = 7
$$

---

### Step 3: Verify the Solution

If $ N = 7 $, then $ D = 11 - 7 = 4 $.

Check the total value:

- 7 nickels = $ 7 \times 5 = 35 $ cents
- 4 dimes = $ 4 \times 10 = 40 $ cents
- Total = $ 35 + 40 = 75 $ cents ✅

This matches the given total value, confirming the solution is correct.

---

### Final Answer

$$
\boxed{7}
$$

[Assistant]
Explanation:
Okay, so Hillary has eleven coins, and they're all either dimes or nickels. The total value of these coins is 75 cents. I need to figure out how many nickels she has. Let me think about how to approach this.

First, let me recall what dimes and nickels are worth. A dime is 10 cents, and a nickel is 5 cents. So, if I can figure out how many of each coin there are, I can add up their values to get 75 cents. Since there are eleven coins in total, I can set up some equations to solve this.

Let me denote the number of dimes as D and the number of nickels as N. According to the problem, the total number of coins is 11. So, that gives me my first equation:

D + N = 11

That's straightforward. Now, the total value of the coins is 75 cents. Since each dime is 10 cents and each nickel is 5 cents, the second equation would be:

10D + 5N = 75

So now I have a system of two equations with two variables. I can solve this system using substitution or elimination. Let me try substitution because it seems straightforward.

From the first equation, I can express D in terms of N:

D = 11 - N

Now, substitute this expression for D into the second equation:

10(11 - N) + 5N = 75

Let me compute that step by step. First, multiply out the 10:

10*11 - 10*N + 5N = 75

Which is:

110 - 10N + 5N = 75

Combine like terms:

110 - 5N = 75

Now, subtract 110 from both sides:

-5N = 75 - 110

Calculating the right side:

75 - 110 = -35

So:

-5N = -35

Now, divide both sides by -5:

N = (-35)/(-5) = 7

So, N = 7. That means Hillary has 7 nickels. Let me check if this makes sense.

If there are 7 nickels, then the number of dimes would be 11 - 7 = 4. Let me check the total value:

7 nickels * 5 cents = 35 cents

4 dimes * 10 cents = 40 cents

Total value: 35 + 40 = 75 cents. Perfect, that matches the given total. So, the answer is 7 nickels.

Wait, let me just make sure I didn't make any arithmetic errors. Let me go through the steps again.

Starting with D + N = 11 and 10D + 5N = 75.

Substituting D = 11 - N into the second equation:

10*(11 - N) + 5N = 75

110 - 10N + 5N = 75

110 - 5N = 75

Subtract 110: -5N = 75 - 110 = -35

Divide by -5: N = 7. Yep, that's correct. And checking the numbers again, 7 nickels and 4 dimes add up to 11 coins and 75 cents. Seems solid.

Alternatively, maybe I can try solving it another way to confirm. Let's try elimination.

Original equations:

1) D + N = 11

2) 10D + 5N = 75

If I multiply the first equation by 5, I get:

5D + 5N = 55

Then subtract that from the second equation:

(10D + 5N) - (5D + 5N) = 75 - 55

Which simplifies to:

5D = 20

Therefore, D = 4. Then substituting back into D + N = 11, we get N = 7. Same result. So that confirms it again.

Another way to think about it: Let's assume all coins are nickels. Then total value would be 11 * 5 = 55 cents. But we need 75 cents, which is 20 cents more. Each time we replace a nickel with a dime, we add 5 cents (since a dime is 10 cents vs. 5 cents). So, how many replacements do we need to get an additional 20 cents?

Number of replacements = 20 / 5 = 4. Therefore, there are 4 dimes and 11 - 4 = 7 nickels. Yep, same answer. So that's another way to think about it, using the concept of replacing coins.

So all methods lead to 7 nickels. Therefore, I can be confident that the answer is 7.

**Final Answer**
Hillary has \boxed{7} nickels.

To determine how many nickels Hillary has, we begin by analyzing the information provided:

- Hillary has **11 coins** in total.
- All coins are either **dimes (10 cents)** or **nickels (5 cents)**.
- The **total value** of the coins is **75 cents**.

---

### Step 1: Define Variables

Let:
- $ D $ = number of dimes
- $ N $ = number of nickels

From the problem, we can write two equations:

1. **Total number of coins:**
   $$
   D + N = 11
   $$

2. **Total value of coins:**
   $$
   10D + 5N = 75
   $$

---

### Step 2: Solve the System of Equations

From the first equation, solve for $ D $ in terms of $ N $:

$$
D = 11 - N
$$

Substitute this into the second equation:

$$
10(11 - N) + 5N = 75
$$

Simplify:

$$
110 - 10N + 5N = 75
$$
$$
110 - 5N = 75
$$

Subtract 110 from both sides:

$$
-5N = -35
$$

Divide both sides by -5:

$$
N = 7
$$

---

### Step 3: Verify the Solution

If $ N = 7 $, then $ D = 11 - 7 = 4 $.

Check the total value:

- 7 nickels = $ 7 \times 5 = 35 $ cents
- 4 dimes = $ 4 \times 10 = 40 $ cents
- Total = $ 35 + 40 = 75 $ cents ✅

This matches the given total value, confirming the solution is correct.

---

### Final Answer

$$
\boxed{7}
$$

Solution:
Let the number of dimes Hillary has be $d$ and the number of nickels she has be $n$. We have the two equations \begin{align*}
d+n&=11\\
10d+5n&=75
\end{align*} (The last equation is in terms of cents.) To make the second equation nicer, we divide both sides by 5 to get $2d+n=15$. From the first given equation, we have $d=11-n$. Substituting this into the simplified second given equation to eliminate $d$, we get $2(11-n)+n=15\Rightarrow n=7$. Thus, Hillary has $\boxed{7}$ nickels.


Explanation:
Okay, so I need to solve this equation: 441 plus 2 times 21 times 19 plus 361 equals x. Let me write that down first to make sure I have it right.

So, the equation is 441 + 2*(21)*(19) + 361 = x. Alright, let me break this down step by step.

First, I should probably handle the multiplication parts before doing the addition. The equation has three terms: 441, 2*(21)*(19), and 361. So, I need to calculate the middle term first.

Let me start with the middle term: 2*(21)*(19). Hmm, multiplication is associative, so I can do 2*21 first and then multiply by 19, or maybe 21*19 first and then multiply by 2. Let me see which is easier.

Calculating 2*21 is straightforward. 2*21 is 42. Then, multiplying that by 19. So, 42*19. Hmm, how do I compute that? Maybe I can break it down.

I know that 42*20 would be 840, right? Because 42*10 is 420, so times 2 is 840. But since it's 42*19, that's 42*(20 - 1) = 42*20 - 42*1 = 840 - 42 = 798. So, 42*19 is 798.

Alternatively, if I did 21*19 first, that might be another way. Let me check that. 21*19. Hmm, 20*19 is 380, and then 1*19 is 19, so 380 + 19 = 399. Then, multiplying by 2 gives 399*2 = 798. Yep, same result. So that middle term is 798.

Okay, so now the equation becomes 441 + 798 + 361 = x. Now I need to add those three numbers together. Let me do that step by step.

First, let me add 441 and 798. Let me think. 441 + 798. Maybe I can break it down. 400 + 700 is 1100, 40 + 90 is 130, and 1 + 8 is 9. So adding those together: 1100 + 130 is 1230, plus 9 is 1239. Wait, let me check that again.

Alternatively, maybe adding 441 + 798. Let me do it vertically:

  441
+798
------
Let me add the units place: 1 + 8 = 9. Then the tens place: 4 + 9 = 13. So write down 3 and carry over 1. Then the hundreds place: 4 + 7 = 11, plus the carried over 1 is 12. So the total is 1239. Yes, that matches. So 441 + 798 = 1239.

Now, add that to 361. So 1239 + 361. Let me do that. Again, maybe breaking it down. 1239 + 300 is 1539, then +60 is 1599, then +1 is 1600. Wait, that seems too easy. Let me check again.

Alternatively, adding vertically:

 1239
+ 361
-------
Starting from the right: 9 + 1 = 10. Write down 0, carry over 1.

Next, 3 + 6 = 9, plus the carried over 1 is 10. Write down 0, carry over 1.

Next, 2 + 3 = 5, plus the carried over 1 is 6.

Then, 1 + 0 = 1.

So the total is 1600. Wait, so 1239 + 361 = 1600? Let me check with another method.

Alternatively, 1239 + 361. Let me think of 1239 + 361 as 1239 + 300 + 60 + 1.

1239 + 300 = 1539.

1539 + 60 = 1599.

1599 + 1 = 1600. Yes, that's correct. So the total is 1600.

Therefore, x equals 1600.

Wait, but let me check if I did all steps correctly. Let me verify each part again.

Original problem: 441 + 2*(21)*(19) + 361.

First, 2*21*19.

2*21 is 42. 42*19.

As before, 42*19. Let me compute 42*19 again.

Another way: 19*40 = 760, 19*2 = 38. So 760 + 38 = 798. Yes, that's correct.

So 441 + 798 + 361.

Adding 441 + 798:

441 + 798. Let me think of 441 + 798.

If I take 441 + 798, maybe think of 441 + 800 - 2 = 1241 - 2 = 1239. Yes, that's another way.

Then 1239 + 361.

Alternatively, 1239 + 361. Let me think of 1239 + 361.

If I add 1239 + 361, maybe think of 1239 + 361 = (1200 + 39) + (300 + 61) = 1200 + 300 + 39 + 61 = 1500 + 100 = 1600. Oh, that's a nice way. Because 39 + 61 is 100.

So that's another way to see it. So that's 1500 + 100 = 1600.

So that seems to confirm it again.

Alternatively, maybe there's a smarter way to do this problem? Let me think.

Looking at the original numbers: 441, 361. Wait, 441 is 21 squared, right? 21*21 is 441. And 361 is 19 squared, because 19*19 is 361.

Oh! So the original expression is 21² + 2*(21)*(19) + 19².

Wait, that looks like a perfect square formula. Because (a + b)² = a² + 2ab + b².

So if a is 21 and b is 19, then (21 + 19)² = 21² + 2*21*19 + 19².

So that would be 40², since 21 + 19 is 40.

So 40 squared is 1600.

Oh! So that's a much quicker way. Instead of calculating each term separately, recognizing the pattern of a perfect square.

So that's a good check. So if I recognize that 441 is 21², 361 is 19², and the middle term is 2*21*19, then the entire expression is (21 + 19)² = 40² = 1600.

So that's a much more efficient way.

But since I didn't notice that initially, I did the step-by-step calculation and arrived at the same answer.

So either way, the answer is 1600.

Therefore, x is 1600.

Let me just confirm once again.

21 + 19 is 40. 40 squared is 1600.

Yes.

Alternatively, if I didn't notice the perfect square, but just did the arithmetic step by step, I still arrived at 1600.

So that gives me confidence that the answer is correct.

Therefore, x equals 1600.

**Final Answer**
The value of $ x $ is \boxed{1600}.

To solve the expression:

$$
441 + 2(21)(19) + 361 = x
$$

we can approach this in two ways: either by computing each term step-by-step or by recognizing a pattern that simplifies the calculation.

---

### **Step-by-Step Calculation**

1. **Compute the middle term:**

   $$
   2 \times 21 \times 19
   $$

   First, compute $ 2 \times 21 = 42 $, then multiply by 19:

   $$
   42 \times 19 = 798
   $$

2. **Now substitute back into the original expression:**

   $$
   441 + 798 + 361
   $$

3. **Add the numbers:**

   - $ 441 + 798 = 1239 $
   - $ 1239 + 361 = 1600 $

So, the value of $ x $ is:

$$
x = 1600
$$

---

### **Alternative Method: Recognizing a Perfect Square**

Observe that:

- $ 441 = 21^2 $
- $ 361 = 19^2 $
- $ 2 \times 21 \times 19 $ is the middle term of the expansion of $ (a + b)^2 $

This matches the identity:

$$
(a + b)^2 = a^2 + 2ab + b^2
$$

So, the expression becomes:

$$
(21 + 19)^2 = 40^2 = 1600
$$

---

### **Final Answer**

$$
\boxed{1600}
$$

[Assistant]
Explanation:
Okay, so I need to solve this equation: 441 plus 2 times 21 times 19 plus 361 equals x. Let me write that down first to make sure I have it right.

So, the equation is 441 + 2*(21)*(19) + 361 = x. Alright, let me break this down step by step.

First, I should probably handle the multiplication parts before doing the addition. The equation has three terms: 441, 2*(21)*(19), and 361. So, I need to calculate the middle term first.

Let me start with the middle term: 2*(21)*(19). Hmm, multiplication is associative, so I can do 2*21 first and then multiply by 19, or maybe 21*19 first and then multiply by 2. Let me see which is easier.

Calculating 2*21 is straightforward. 2*21 is 42. Then, multiplying that by 19. So, 42*19. Hmm, how do I compute that? Maybe I can break it down.

I know that 42*20 would be 840, right? Because 42*10 is 420, so times 2 is 840. But since it's 42*19, that's 42*(20 - 1) = 42*20 - 42*1 = 840 - 42 = 798. So, 42*19 is 798.

Alternatively, if I did 21*19 first, that might be another way. Let me check that. 21*19. Hmm, 20*19 is 380, and then 1*19 is 19, so 380 + 19 = 399. Then, multiplying by 2 gives 399*2 = 798. Yep, same result. So that middle term is 798.

Okay, so now the equation becomes 441 + 798 + 361 = x. Now I need to add those three numbers together. Let me do that step by step.

First, let me add 441 and 798. Let me think. 441 + 798. Maybe I can break it down. 400 + 700 is 1100, 40 + 90 is 130, and 1 + 8 is 9. So adding those together: 1100 + 130 is 1230, plus 9 is 1239. Wait, let me check that again.

Alternatively, maybe adding 441 + 798. Let me do it vertically:

  441
+798
------
Let me add the units place: 1 + 8 = 9. Then the tens place: 4 + 9 = 13. So write down 3 and carry over 1. Then the hundreds place: 4 + 7 = 11, plus the carried over 1 is 12. So the total is 1239. Yes, that matches. So 441 + 798 = 1239.

Now, add that to 361. So 1239 + 361. Let me do that. Again, maybe breaking it down. 1239 + 300 is 1539, then +60 is 1599, then +1 is 1600. Wait, that seems too easy. Let me check again.

Alternatively, adding vertically:

 1239
+ 361
-------
Starting from the right: 9 + 1 = 10. Write down 0, carry over 1.

Next, 3 + 6 = 9, plus the carried over 1 is 10. Write down 0, carry over 1.

Next, 2 + 3 = 5, plus the carried over 1 is 6.

Then, 1 + 0 = 1.

So the total is 1600. Wait, so 1239 + 361 = 1600? Let me check with another method.

Alternatively, 1239 + 361. Let me think of 1239 + 361 as 1239 + 300 + 60 + 1.

1239 + 300 = 1539.

1539 + 60 = 1599.

1599 + 1 = 1600. Yes, that's correct. So the total is 1600.

Therefore, x equals 1600.

Wait, but let me check if I did all steps correctly. Let me verify each part again.

Original problem: 441 + 2*(21)*(19) + 361.

First, 2*21*19.

2*21 is 42. 42*19.

As before, 42*19. Let me compute 42*19 again.

Another way: 19*40 = 760, 19*2 = 38. So 760 + 38 = 798. Yes, that's correct.

So 441 + 798 + 361.

Adding 441 + 798:

441 + 798. Let me think of 441 + 798.

If I take 441 + 798, maybe think of 441 + 800 - 2 = 1241 - 2 = 1239. Yes, that's another way.

Then 1239 + 361.

Alternatively, 1239 + 361. Let me think of 1239 + 361.

If I add 1239 + 361, maybe think of 1239 + 361 = (1200 + 39) + (300 + 61) = 1200 + 300 + 39 + 61 = 1500 + 100 = 1600. Oh, that's a nice way. Because 39 + 61 is 100.

So that's another way to see it. So that's 1500 + 100 = 1600.

So that seems to confirm it again.

Alternatively, maybe there's a smarter way to do this problem? Let me think.

Looking at the original numbers: 441, 361. Wait, 441 is 21 squared, right? 21*21 is 441. And 361 is 19 squared, because 19*19 is 361.

Oh! So the original expression is 21² + 2*(21)*(19) + 19².

Wait, that looks like a perfect square formula. Because (a + b)² = a² + 2ab + b².

So if a is 21 and b is 19, then (21 + 19)² = 21² + 2*21*19 + 19².

So that would be 40², since 21 + 19 is 40.

So 40 squared is 1600.

Oh! So that's a much quicker way. Instead of calculating each term separately, recognizing the pattern of a perfect square.

So that's a good check. So if I recognize that 441 is 21², 361 is 19², and the middle term is 2*21*19, then the entire expression is (21 + 19)² = 40² = 1600.

So that's a much more efficient way.

But since I didn't notice that initially, I did the step-by-step calculation and arrived at the same answer.

So either way, the answer is 1600.

Therefore, x is 1600.

Let me just confirm once again.

21 + 19 is 40. 40 squared is 1600.

Yes.

Alternatively, if I didn't notice the perfect square, but just did the arithmetic step by step, I still arrived at 1600.

So that gives me confidence that the answer is correct.

Therefore, x equals 1600.

**Final Answer**
The value of $ x $ is \boxed{1600}.

To solve the expression:

$$
441 + 2(21)(19) + 361 = x
$$

we can approach this in two ways: either by computing each term step-by-step or by recognizing a pattern that simplifies the calculation.

---

### **Step-by-Step Calculation**

1. **Compute the middle term:**

   $$
   2 \times 21 \times 19
   $$

   First, compute $ 2 \times 21 = 42 $, then multiply by 19:

   $$
   42 \times 19 = 798
   $$

2. **Now substitute back into the original expression:**

   $$
   441 + 798 + 361
   $$

3. **Add the numbers:**

   - $ 441 + 798 = 1239 $
   - $ 1239 + 361 = 1600 $

So, the value of $ x $ is:

$$
x = 1600
$$

---

### **Alternative Method: Recognizing a Perfect Square**

Observe that:

- $ 441 = 21^2 $
- $ 361 = 19^2 $
- $ 2 \times 21 \times 19 $ is the middle term of the expansion of $ (a + b)^2 $

This matches the identity:

$$
(a + b)^2 = a^2 + 2ab + b^2
$$

So, the expression becomes:

$$
(21 + 19)^2 = 40^2 = 1600
$$

---

### **Final Answer**

$$
\boxed{1600}
$$

Solution: We note that $361=19^2$ and $441=21^2$, so $x=21^2+2(21)(19)+19^2$. This is just the binomial expansion of $(21+19)^2=40^2=\boxed{1600}$.


Explanation:
Okay, so I need to find the value of x + y + z given that xy = 24, xz = 48, and yz = 72. All variables are positive. Hmm, let me think about how to approach this.

First, I remember that when you have multiple equations with products of variables, sometimes multiplying all the equations together can help. Let me try that. If I multiply xy, xz, and yz together, that would be (xy)(xz)(yz). Let me compute that:

(xy)(xz)(yz) = x^2 y^2 z^2.

On the other hand, multiplying the right-hand sides of the equations: 24 * 48 * 72.

So, x^2 y^2 z^2 = 24 * 48 * 72.

Therefore, if I take the square root of both sides, I get xyz = sqrt(24 * 48 * 72).

But maybe I can compute that. Let me calculate the right-hand side first. Let me compute 24 * 48 first. 24*48... Well, 24*40 is 960, and 24*8 is 192, so total is 960 + 192 = 1152. Then multiply that by 72. Hmm, 1152 * 72. That seems big. Maybe there's a smarter way to factor these numbers?

Alternatively, maybe I can factor each of the numbers into primes. Let me try that.

24 factors into 2^3 * 3^1.

48 is 2^4 * 3^1.

72 is 2^3 * 3^2.

So multiplying them all together:

24 * 48 * 72 = (2^3 * 3^1) * (2^4 * 3^1) * (2^3 * 3^2)

Adding exponents for like bases:

For 2: 3 + 4 + 3 = 10

For 3: 1 + 1 + 2 = 4

So total is 2^10 * 3^4.

Therefore, sqrt(2^10 * 3^4) = 2^(5) * 3^(2) = 32 * 9 = 288.

So xyz = 288.

Okay, so now I know that xyz = 288.

But how does that help me find x, y, z individually? Well, if I can find each variable by dividing by the product of the other two.

For example, since xy = 24, then z = (xyz)/(xy) = 288 / 24 = 12.

Similarly, x = (xyz)/(yz) = 288 / 72 = 4.

And y = (xyz)/(xz) = 288 / 48 = 6.

Wait, so x is 4, y is 6, z is 12? Let me check if these values satisfy the original equations.

Check xy: 4*6 = 24. Correct.

Check xz: 4*12 = 48. Correct.

Check yz: 6*12 = 72. Correct.

So that works. Then x + y + z = 4 + 6 + 12 = 22.

Wait, so is the answer 22? Let me just make sure I didn't make any calculation errors.

Let me recheck the multiplication steps.

Original equations: xy=24, xz=48, yz=72.

Multiplying all three: (xy)(xz)(yz) = 24*48*72.

Which is x^2 y^2 z^2 = 24*48*72.

Taking square root: xyz = sqrt(24*48*72).

Then factoring each number:

24 = 2^3 * 3

48 = 2^4 * 3

72 = 2^3 * 3^2

Multiply them: 2^(3+4+3) * 3^(1+1+2) = 2^10 * 3^4.

Square root is 2^5 * 3^2 = 32 * 9 = 288.

Yes, that's correct.

Then xyz = 288.

So z = 288 / (xy) = 288 / 24 = 12.

Similarly, x = 288 / (yz) = 288 / 72 = 4.

y = 288 / (xz) = 288 / 48 = 6.

Adding them up: 4 + 6 + 12 = 22.

Seems correct.

Alternatively, maybe I can solve the equations step by step without multiplying all three. Let me try another approach to verify.

From the first equation, xy = 24. Let me solve for y: y = 24/x.

From the second equation, xz = 48. Solve for z: z = 48/x.

From the third equation, yz = 72. Substitute y and z from above: (24/x)*(48/x) = 72.

Compute that: (24*48)/x² = 72.

So 24*48 = 72x².

Calculate 24*48: 24*48. Let me compute 24*48 again. 24*40=960, 24*8=192, total 960+192=1152.

So 1152 = 72x².

Divide both sides by 72: 1152 / 72 = x².

Calculate 1152 / 72. Let me divide numerator and denominator by 8: 144 / 9 = 16.

Wait, 72*16 = 1152? Let me check: 70*16=1120, 2*16=32, so 1120+32=1152. Yes.

So x² = 16, so x = 4 (since x is positive).

Then, from earlier, y = 24/x = 24/4 = 6.

z = 48/x = 48/4 = 12.

Same result. So x + y + z = 4 + 6 + 12 = 22.

So that's consistent.

Alternatively, maybe I can use ratios. Let me see.

From xy = 24 and xz = 48, if I divide the second equation by the first, I get (xz)/(xy) = 48/24 => z/y = 2 => z = 2y.

Similarly, from yz = 72, if z = 2y, then y*(2y) = 72 => 2y² = 72 => y² = 36 => y = 6 (positive).

Then z = 2y = 12. Then from xy = 24, x = 24 / y = 24 / 6 = 4.

Same result again.

So x = 4, y = 6, z = 12. Sum is 22.

Therefore, regardless of the method, I get the same answer.

I think that's solid. So the answer is 22.

**Final Answer**
\boxed{22}

To solve the problem, we are given the following equations involving three positive variables $ x $, $ y $, and $ z $:

$$
xy = 24, \quad xz = 48, \quad yz = 72
$$

Our goal is to find the value of $ x + y + z $.

---

### Step 1: Multiply All Three Equations

Multiplying the left-hand sides and the right-hand sides of the equations:

$$
(xy)(xz)(yz) = 24 \cdot 48 \cdot 72
$$

The left-hand side becomes:

$$
x^2 y^2 z^2
$$

So:

$$
x^2 y^2 z^2 = 24 \cdot 48 \cdot 72
$$

Taking the square root of both sides:

$$
xyz = \sqrt{24 \cdot 48 \cdot 72}
$$

---

### Step 2: Simplify the Product

We simplify the product on the right-hand side by factoring each number into its prime components:

- $ 24 = 2^3 \cdot 3 $
- $ 48 = 2^4 \cdot 3 $
- $ 72 = 2^3 \cdot 3^2 $

Multiplying them:

$$
24 \cdot 48 \cdot 72 = (2^3 \cdot 3) \cdot (2^4 \cdot 3) \cdot (2^3 \cdot 3^2) = 2^{3+4+3} \cdot 3^{1+1+2} = 2^{10} \cdot 3^4
$$

Taking the square root:

$$
xyz = \sqrt{2^{10} \cdot 3^4} = 2^5 \cdot 3^2 = 32 \cdot 9 = 288
$$

So:

$$
xyz = 288
$$

---

### Step 3: Solve for Each Variable

We can now find each variable by dividing the product $ xyz $ by the product of the other two variables:

- $ x = \frac{xyz}{yz} = \frac{288}{72} = 4 $
- $ y = \frac{xyz}{xz} = \frac{288}{48} = 6 $
- $ z = \frac{xyz}{xy} = \frac{288}{24} = 12 $

---

### Step 4: Compute the Sum

$$
x + y + z = 4 + 6 + 12 = \boxed{22}
$$

---

### Final Answer

$$
\boxed{22}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of x + y + z given that xy = 24, xz = 48, and yz = 72. All variables are positive. Hmm, let me think about how to approach this.

First, I remember that when you have multiple equations with products of variables, sometimes multiplying all the equations together can help. Let me try that. If I multiply xy, xz, and yz together, that would be (xy)(xz)(yz). Let me compute that:

(xy)(xz)(yz) = x^2 y^2 z^2.

On the other hand, multiplying the right-hand sides of the equations: 24 * 48 * 72.

So, x^2 y^2 z^2 = 24 * 48 * 72.

Therefore, if I take the square root of both sides, I get xyz = sqrt(24 * 48 * 72).

But maybe I can compute that. Let me calculate the right-hand side first. Let me compute 24 * 48 first. 24*48... Well, 24*40 is 960, and 24*8 is 192, so total is 960 + 192 = 1152. Then multiply that by 72. Hmm, 1152 * 72. That seems big. Maybe there's a smarter way to factor these numbers?

Alternatively, maybe I can factor each of the numbers into primes. Let me try that.

24 factors into 2^3 * 3^1.

48 is 2^4 * 3^1.

72 is 2^3 * 3^2.

So multiplying them all together:

24 * 48 * 72 = (2^3 * 3^1) * (2^4 * 3^1) * (2^3 * 3^2)

Adding exponents for like bases:

For 2: 3 + 4 + 3 = 10

For 3: 1 + 1 + 2 = 4

So total is 2^10 * 3^4.

Therefore, sqrt(2^10 * 3^4) = 2^(5) * 3^(2) = 32 * 9 = 288.

So xyz = 288.

Okay, so now I know that xyz = 288.

But how does that help me find x, y, z individually? Well, if I can find each variable by dividing by the product of the other two.

For example, since xy = 24, then z = (xyz)/(xy) = 288 / 24 = 12.

Similarly, x = (xyz)/(yz) = 288 / 72 = 4.

And y = (xyz)/(xz) = 288 / 48 = 6.

Wait, so x is 4, y is 6, z is 12? Let me check if these values satisfy the original equations.

Check xy: 4*6 = 24. Correct.

Check xz: 4*12 = 48. Correct.

Check yz: 6*12 = 72. Correct.

So that works. Then x + y + z = 4 + 6 + 12 = 22.

Wait, so is the answer 22? Let me just make sure I didn't make any calculation errors.

Let me recheck the multiplication steps.

Original equations: xy=24, xz=48, yz=72.

Multiplying all three: (xy)(xz)(yz) = 24*48*72.

Which is x^2 y^2 z^2 = 24*48*72.

Taking square root: xyz = sqrt(24*48*72).

Then factoring each number:

24 = 2^3 * 3

48 = 2^4 * 3

72 = 2^3 * 3^2

Multiply them: 2^(3+4+3) * 3^(1+1+2) = 2^10 * 3^4.

Square root is 2^5 * 3^2 = 32 * 9 = 288.

Yes, that's correct.

Then xyz = 288.

So z = 288 / (xy) = 288 / 24 = 12.

Similarly, x = 288 / (yz) = 288 / 72 = 4.

y = 288 / (xz) = 288 / 48 = 6.

Adding them up: 4 + 6 + 12 = 22.

Seems correct.

Alternatively, maybe I can solve the equations step by step without multiplying all three. Let me try another approach to verify.

From the first equation, xy = 24. Let me solve for y: y = 24/x.

From the second equation, xz = 48. Solve for z: z = 48/x.

From the third equation, yz = 72. Substitute y and z from above: (24/x)*(48/x) = 72.

Compute that: (24*48)/x² = 72.

So 24*48 = 72x².

Calculate 24*48: 24*48. Let me compute 24*48 again. 24*40=960, 24*8=192, total 960+192=1152.

So 1152 = 72x².

Divide both sides by 72: 1152 / 72 = x².

Calculate 1152 / 72. Let me divide numerator and denominator by 8: 144 / 9 = 16.

Wait, 72*16 = 1152? Let me check: 70*16=1120, 2*16=32, so 1120+32=1152. Yes.

So x² = 16, so x = 4 (since x is positive).

Then, from earlier, y = 24/x = 24/4 = 6.

z = 48/x = 48/4 = 12.

Same result. So x + y + z = 4 + 6 + 12 = 22.

So that's consistent.

Alternatively, maybe I can use ratios. Let me see.

From xy = 24 and xz = 48, if I divide the second equation by the first, I get (xz)/(xy) = 48/24 => z/y = 2 => z = 2y.

Similarly, from yz = 72, if z = 2y, then y*(2y) = 72 => 2y² = 72 => y² = 36 => y = 6 (positive).

Then z = 2y = 12. Then from xy = 24, x = 24 / y = 24 / 6 = 4.

Same result again.

So x = 4, y = 6, z = 12. Sum is 22.

Therefore, regardless of the method, I get the same answer.

I think that's solid. So the answer is 22.

**Final Answer**
\boxed{22}

To solve the problem, we are given the following equations involving three positive variables $ x $, $ y $, and $ z $:

$$
xy = 24, \quad xz = 48, \quad yz = 72
$$

Our goal is to find the value of $ x + y + z $.

---

### Step 1: Multiply All Three Equations

Multiplying the left-hand sides and the right-hand sides of the equations:

$$
(xy)(xz)(yz) = 24 \cdot 48 \cdot 72
$$

The left-hand side becomes:

$$
x^2 y^2 z^2
$$

So:

$$
x^2 y^2 z^2 = 24 \cdot 48 \cdot 72
$$

Taking the square root of both sides:

$$
xyz = \sqrt{24 \cdot 48 \cdot 72}
$$

---

### Step 2: Simplify the Product

We simplify the product on the right-hand side by factoring each number into its prime components:

- $ 24 = 2^3 \cdot 3 $
- $ 48 = 2^4 \cdot 3 $
- $ 72 = 2^3 \cdot 3^2 $

Multiplying them:

$$
24 \cdot 48 \cdot 72 = (2^3 \cdot 3) \cdot (2^4 \cdot 3) \cdot (2^3 \cdot 3^2) = 2^{3+4+3} \cdot 3^{1+1+2} = 2^{10} \cdot 3^4
$$

Taking the square root:

$$
xyz = \sqrt{2^{10} \cdot 3^4} = 2^5 \cdot 3^2 = 32 \cdot 9 = 288
$$

So:

$$
xyz = 288
$$

---

### Step 3: Solve for Each Variable

We can now find each variable by dividing the product $ xyz $ by the product of the other two variables:

- $ x = \frac{xyz}{yz} = \frac{288}{72} = 4 $
- $ y = \frac{xyz}{xz} = \frac{288}{48} = 6 $
- $ z = \frac{xyz}{xy} = \frac{288}{24} = 12 $

---

### Step 4: Compute the Sum

$$
x + y + z = 4 + 6 + 12 = \boxed{22}
$$

---

### Final Answer

$$
\boxed{22}
$$

Solution:
Multiplying all three equations, we get $x^2 y^2 z^2 = 82944.$  Since $x,$$y,$ and $z$ are positive, $xyz = \sqrt{82944} = 288.$  Then
\begin{align*}
x &= \frac{xyz}{yz} = \frac{288}{72} = 4, \\
y &= \frac{xyz}{xz} = \frac{288}{48} = 6, \\
z &= \frac{xyz}{xy} = \frac{288}{24} = 12.
\end{align*}Hence, $x + y + z = \boxed{22}.$


Explanation:
Okay, so I need to compute the product of the number 5 plus the square root of 3 and its radical conjugate. Hmm, let me think. First, what exactly is a radical conjugate? I remember that for expressions involving radicals, the radical conjugate is formed by changing the sign between the terms. So if I have something like a + b√c, its radical conjugate would be a - b√c, right? Let me confirm that. Yeah, I think that's correct. For example, the radical conjugate of 2 + 3√5 would be 2 - 3√5. So applying that to the given problem, the number is 5 + √3, so its radical conjugate should be 5 - √3.

Alright, so the problem is asking for the product of these two numbers: (5 + √3) multiplied by (5 - √3). Let me write that down:

(5 + √3)(5 - √3)

Hmm, this looks like a difference of squares situation. Remember, (a + b)(a - b) equals a² - b². Let me check if that applies here. Here, a would be 5 and b would be √3. So applying the formula, it should be 5² minus (√3)². Let me compute that.

First, 5 squared is 25. Then, (√3) squared is just 3, because the square and the square root cancel each other out. So subtracting those, 25 - 3 equals 22. Therefore, the product should be 22.

Wait, let me verify that by actually expanding the multiplication step by step, just to make sure I didn't make a mistake. Let's do the distributive property, also known as the FOIL method for binomials.

First, multiply the First terms: 5 * 5 = 25.

Then, the Outer terms: 5 * (-√3) = -5√3.

Next, the Inner terms: √3 * 5 = 5√3.

Finally, the Last terms: √3 * (-√3) = - (√3)^2 = -3.

Now, adding all those together: 25 - 5√3 + 5√3 - 3.

Wait, the -5√3 and +5√3 should cancel each other out, right? Because they are like terms. So combining those, they sum to zero. Then we have 25 - 3, which is 22. Yep, that matches the earlier result.

So both methods give me 22. That seems straightforward. But let me think again if there's any chance I misapplied the concept of radical conjugate. For instance, is there a different way to define radical conjugate? Maybe in some contexts, but I think in the context of quadratic expressions, the radical conjugate is indeed changing the sign of the radical part.

Alternatively, if the original number was something like a + b√c, then the conjugate is a - b√c. So in this case, since the original number is 5 + √3, which can be thought of as 5 + 1√3, so the conjugate is 5 - 1√3, which is 5 - √3. So that's correct.

Another way to think about this is that multiplying a number by its conjugate is a way to rationalize the denominator if you have a fraction with a radical in the denominator. For example, if you have 1/(5 + √3), multiplying numerator and denominator by the conjugate 5 - √3 would rationalize the denominator. But in this case, we aren't dealing with a fraction, just the product of the number and its conjugate.

But regardless, the product is a rational number, which is 22. That makes sense because when you multiply a number by its conjugate, you eliminate the radical. So in this case, the radical √3 is eliminated, resulting in a rational number.

Let me check with actual decimal approximations to see if the product is indeed 22. Let me compute 5 + √3 numerically. The square root of 3 is approximately 1.732. So 5 + 1.732 is approximately 6.732. Then the radical conjugate is 5 - √3, which is approximately 5 - 1.732 = 3.268. Multiplying these two approximate numbers: 6.732 * 3.268. Let me compute that.

First, approximate 6.732 * 3.268. Let me do 6.732 * 3 = 20.196, and 6.732 * 0.268. Let me compute 6.732 * 0.2 = 1.3464, 6.732 * 0.06 = 0.40392, and 6.732 * 0.008 = 0.053856. Adding those together: 1.3464 + 0.40392 = 1.75032 + 0.053856 ≈ 1.804176. Then total product is approximately 20.196 + 1.804176 ≈ 22.000176. Wow, that's really close to 22. So that numerical approximation confirms that the exact value is indeed 22.

Therefore, I can be confident that the product of 5 + √3 and its radical conjugate 5 - √3 is 22.

Just to recap, the key steps were:

1. Identify the radical conjugate of 5 + √3 as 5 - √3.
2. Recognize that multiplying these two binomials is a difference of squares.
3. Apply the formula (a + b)(a - b) = a² - b².
4. Calculate 5² - (√3)² = 25 - 3 = 22.
5. Verify by expanding the multiplication step-by-step, confirming that the radical terms cancel out.
6. Cross-check with decimal approximations to ensure the result is accurate.

All steps lead to the same conclusion, so I think that's solid. I don't see any mistakes in my reasoning. It's always good to check with multiple methods, especially when dealing with radicals, to make sure that the operations are performed correctly.

Another thing to note is that this concept is useful in various areas of algebra, such as simplifying expressions, rationalizing denominators, and solving quadratic equations. For instance, if you have a quadratic equation with irrational roots, the product of the roots can be found using the formula related to the coefficients, and in some cases, that product is exactly the product of the number and its conjugate.

In this problem, even though it's straightforward, it's a good exercise to practice working with radicals and their conjugates. It reinforces the understanding of how to manipulate expressions involving square roots and how to simplify them effectively.

So, in summary, the product is 22, and all methods of calculation confirm this result. I think that's thorough enough. No steps seem to be missing, and all checks align. Therefore, I can confidently say the answer is 22.

**Final Answer**
The product of the number $5+\sqrt{3}$ and its radical conjugate is \boxed{22}.

To compute the product of the number $ 5 + \sqrt{3} $ and its radical conjugate, we begin by identifying the radical conjugate of the expression.

---

### Step 1: Identify the Radical Conjugate

The **radical conjugate** of an expression of the form $ a + b\sqrt{c} $ is $ a - b\sqrt{c} $. In this case, the given number is:

$$
5 + \sqrt{3}
$$

So, its radical conjugate is:

$$
5 - \sqrt{3}
$$

---

### Step 2: Multiply the Expression by Its Conjugate

We now compute the product:

$$
(5 + \sqrt{3})(5 - \sqrt{3})
$$

This is a classic **difference of squares** expression, which follows the identity:

$$
(a + b)(a - b) = a^2 - b^2
$$

Here, $ a = 5 $ and $ b = \sqrt{3} $, so:

$$
(5 + \sqrt{3})(5 - \sqrt{3}) = 5^2 - (\sqrt{3})^2
$$

---

### Step 3: Simplify the Expression

Now compute each term:

- $ 5^2 = 25 $
- $ (\sqrt{3})^2 = 3 $

So the product becomes:

$$
25 - 3 = 22
$$

---

### Step 4: Verification via Expansion

To double-check, we can expand the product using the distributive property (FOIL method):

$$
(5 + \sqrt{3})(5 - \sqrt{3}) = 5 \cdot 5 + 5 \cdot (-\sqrt{3}) + \sqrt{3} \cdot 5 + \sqrt{3} \cdot (-\sqrt{3})
$$

$$
= 25 - 5\sqrt{3} + 5\sqrt{3} - 3
$$

The middle terms $ -5\sqrt{3} $ and $ +5\sqrt{3} $ cancel each other, leaving:

$$
25 - 3 = 22
$$

---

### Final Answer

The product of the number $ 5 + \sqrt{3} $ and its radical conjugate is:

$$
\boxed{22}
$$

[Assistant]
Explanation:
Okay, so I need to compute the product of the number 5 plus the square root of 3 and its radical conjugate. Hmm, let me think. First, what exactly is a radical conjugate? I remember that for expressions involving radicals, the radical conjugate is formed by changing the sign between the terms. So if I have something like a + b√c, its radical conjugate would be a - b√c, right? Let me confirm that. Yeah, I think that's correct. For example, the radical conjugate of 2 + 3√5 would be 2 - 3√5. So applying that to the given problem, the number is 5 + √3, so its radical conjugate should be 5 - √3.

Alright, so the problem is asking for the product of these two numbers: (5 + √3) multiplied by (5 - √3). Let me write that down:

(5 + √3)(5 - √3)

Hmm, this looks like a difference of squares situation. Remember, (a + b)(a - b) equals a² - b². Let me check if that applies here. Here, a would be 5 and b would be √3. So applying the formula, it should be 5² minus (√3)². Let me compute that.

First, 5 squared is 25. Then, (√3) squared is just 3, because the square and the square root cancel each other out. So subtracting those, 25 - 3 equals 22. Therefore, the product should be 22.

Wait, let me verify that by actually expanding the multiplication step by step, just to make sure I didn't make a mistake. Let's do the distributive property, also known as the FOIL method for binomials.

First, multiply the First terms: 5 * 5 = 25.

Then, the Outer terms: 5 * (-√3) = -5√3.

Next, the Inner terms: √3 * 5 = 5√3.

Finally, the Last terms: √3 * (-√3) = - (√3)^2 = -3.

Now, adding all those together: 25 - 5√3 + 5√3 - 3.

Wait, the -5√3 and +5√3 should cancel each other out, right? Because they are like terms. So combining those, they sum to zero. Then we have 25 - 3, which is 22. Yep, that matches the earlier result.

So both methods give me 22. That seems straightforward. But let me think again if there's any chance I misapplied the concept of radical conjugate. For instance, is there a different way to define radical conjugate? Maybe in some contexts, but I think in the context of quadratic expressions, the radical conjugate is indeed changing the sign of the radical part.

Alternatively, if the original number was something like a + b√c, then the conjugate is a - b√c. So in this case, since the original number is 5 + √3, which can be thought of as 5 + 1√3, so the conjugate is 5 - 1√3, which is 5 - √3. So that's correct.

Another way to think about this is that multiplying a number by its conjugate is a way to rationalize the denominator if you have a fraction with a radical in the denominator. For example, if you have 1/(5 + √3), multiplying numerator and denominator by the conjugate 5 - √3 would rationalize the denominator. But in this case, we aren't dealing with a fraction, just the product of the number and its conjugate.

But regardless, the product is a rational number, which is 22. That makes sense because when you multiply a number by its conjugate, you eliminate the radical. So in this case, the radical √3 is eliminated, resulting in a rational number.

Let me check with actual decimal approximations to see if the product is indeed 22. Let me compute 5 + √3 numerically. The square root of 3 is approximately 1.732. So 5 + 1.732 is approximately 6.732. Then the radical conjugate is 5 - √3, which is approximately 5 - 1.732 = 3.268. Multiplying these two approximate numbers: 6.732 * 3.268. Let me compute that.

First, approximate 6.732 * 3.268. Let me do 6.732 * 3 = 20.196, and 6.732 * 0.268. Let me compute 6.732 * 0.2 = 1.3464, 6.732 * 0.06 = 0.40392, and 6.732 * 0.008 = 0.053856. Adding those together: 1.3464 + 0.40392 = 1.75032 + 0.053856 ≈ 1.804176. Then total product is approximately 20.196 + 1.804176 ≈ 22.000176. Wow, that's really close to 22. So that numerical approximation confirms that the exact value is indeed 22.

Therefore, I can be confident that the product of 5 + √3 and its radical conjugate 5 - √3 is 22.

Just to recap, the key steps were:

1. Identify the radical conjugate of 5 + √3 as 5 - √3.
2. Recognize that multiplying these two binomials is a difference of squares.
3. Apply the formula (a + b)(a - b) = a² - b².
4. Calculate 5² - (√3)² = 25 - 3 = 22.
5. Verify by expanding the multiplication step-by-step, confirming that the radical terms cancel out.
6. Cross-check with decimal approximations to ensure the result is accurate.

All steps lead to the same conclusion, so I think that's solid. I don't see any mistakes in my reasoning. It's always good to check with multiple methods, especially when dealing with radicals, to make sure that the operations are performed correctly.

Another thing to note is that this concept is useful in various areas of algebra, such as simplifying expressions, rationalizing denominators, and solving quadratic equations. For instance, if you have a quadratic equation with irrational roots, the product of the roots can be found using the formula related to the coefficients, and in some cases, that product is exactly the product of the number and its conjugate.

In this problem, even though it's straightforward, it's a good exercise to practice working with radicals and their conjugates. It reinforces the understanding of how to manipulate expressions involving square roots and how to simplify them effectively.

So, in summary, the product is 22, and all methods of calculation confirm this result. I think that's thorough enough. No steps seem to be missing, and all checks align. Therefore, I can confidently say the answer is 22.

**Final Answer**
The product of the number $5+\sqrt{3}$ and its radical conjugate is \boxed{22}.

To compute the product of the number $ 5 + \sqrt{3} $ and its radical conjugate, we begin by identifying the radical conjugate of the expression.

---

### Step 1: Identify the Radical Conjugate

The **radical conjugate** of an expression of the form $ a + b\sqrt{c} $ is $ a - b\sqrt{c} $. In this case, the given number is:

$$
5 + \sqrt{3}
$$

So, its radical conjugate is:

$$
5 - \sqrt{3}
$$

---

### Step 2: Multiply the Expression by Its Conjugate

We now compute the product:

$$
(5 + \sqrt{3})(5 - \sqrt{3})
$$

This is a classic **difference of squares** expression, which follows the identity:

$$
(a + b)(a - b) = a^2 - b^2
$$

Here, $ a = 5 $ and $ b = \sqrt{3} $, so:

$$
(5 + \sqrt{3})(5 - \sqrt{3}) = 5^2 - (\sqrt{3})^2
$$

---

### Step 3: Simplify the Expression

Now compute each term:

- $ 5^2 = 25 $
- $ (\sqrt{3})^2 = 3 $

So the product becomes:

$$
25 - 3 = 22
$$

---

### Step 4: Verification via Expansion

To double-check, we can expand the product using the distributive property (FOIL method):

$$
(5 + \sqrt{3})(5 - \sqrt{3}) = 5 \cdot 5 + 5 \cdot (-\sqrt{3}) + \sqrt{3} \cdot 5 + \sqrt{3} \cdot (-\sqrt{3})
$$

$$
= 25 - 5\sqrt{3} + 5\sqrt{3} - 3
$$

The middle terms $ -5\sqrt{3} $ and $ +5\sqrt{3} $ cancel each other, leaving:

$$
25 - 3 = 22
$$

---

### Final Answer

The product of the number $ 5 + \sqrt{3} $ and its radical conjugate is:

$$
\boxed{22}
$$

Solution: The radical conjugate of this number is $5-\sqrt{3},$ so the product of the two numbers is \[(5+\sqrt3)(5-\sqrt3) = 5^2 - (\sqrt3)^2 = 25 - 3 = \boxed{22}.\]


Explanation:
Okay, so I need to solve the equation $10^x - 10 = 9990$. Let me think about how to approach this. Hmm, first, maybe I should isolate the term with the exponent. That usually helps with exponential equations.

So, starting with the original equation:

$10^x - 10 = 9990$

I can add 10 to both sides to get rid of the -10 on the left. Let me do that:

$10^x - 10 + 10 = 9990 + 10$

Simplifying both sides:

$10^x = 10000$

Alright, so now I have $10^x = 10000$. I need to solve for x. Since the base is 10, maybe I can express 10000 as a power of 10. Let me recall that $10^1 = 10$, $10^2 = 100$, $10^3 = 1000$, $10^4 = 10000$. Oh, right! So 10000 is $10^4$.

Therefore, substituting back into the equation:

$10^x = 10^4$

Since the bases are the same, the exponents must be equal. That means:

$x = 4$

Wait, let me check if that's correct. Let me substitute x = 4 back into the original equation to verify.

Original equation: $10^x - 10 = 9990$

Substituting x = 4:

Left side: $10^4 - 10 = 10000 - 10 = 9990$

Right side: 9990

Yes, both sides are equal. So x = 4 is indeed the solution.

But let me think again if there's another way to approach this problem, just to make sure I didn't miss anything. Maybe using logarithms? Let me try that method as well.

Starting again from $10^x = 10000$. Taking the logarithm base 10 of both sides:

$\log_{10}(10^x) = \log_{10}(10000)$

Simplifying the left side using logarithm properties: $\log_{10}(10^x) = x$, because $\log_b(b^k) = k$.

On the right side, $\log_{10}(10000)$. Since 10000 is $10^4$, this becomes $\log_{10}(10^4) = 4$.

Therefore, x = 4. Same result. So that confirms it again.

Alternatively, if I didn't recognize that 10000 is $10^4$, I could use natural logarithm or common logarithm. Let me try using natural logarithm for practice.

Starting from $10^x = 10000$. Taking natural logarithm on both sides:

$\ln(10^x) = \ln(10000)$

Using the power rule for logarithms: $x \ln(10) = \ln(10000)$

Therefore, solving for x:

$x = \frac{\ln(10000)}{\ln(10)}$

Now, calculating $\ln(10000)$. Since 10000 is $10^4$, $\ln(10^4) = 4 \ln(10)$. Therefore:

$x = \frac{4 \ln(10)}{\ln(10)} = 4$.

Same answer again. So regardless of the method, I get x = 4.

Is there any chance that there could be another solution? For example, sometimes with logarithms, you have to consider different branches or something? But in this case, since the original equation is $10^x = 10000$, and 10^x is a one-to-one function, there's only one solution. So x must be 4.

Let me also think about the graph of the function $f(x) = 10^x - 10$. It's an exponential growth function shifted down by 10 units. The equation $10^x - 10 = 9990$ is equivalent to finding the x-intercept of $f(x) - 9990 = 0$, which would be a single point since the function is strictly increasing. Therefore, only one solution exists, which is x = 4.

Another way to check is to consider the value of x. If x is 4, then 10^4 is 10000, which minus 10 is 9990. Perfect. If I tried x = 3, then 10^3 is 1000, minus 10 is 990, which is way less than 9990. If x = 5, 10^5 is 100000, minus 10 is 99990, which is way more than 9990. So x must be between 4 and 5? Wait, but we already found x = 4 gives exactly 9990. Wait, no, hold on. Wait, 10^4 is 10000, so 10^4 - 10 is 9990. So x = 4 is exactly the solution.

Wait, but if I thought x was between 4 and 5, that would be a mistake. Wait, no. Let me check again. If x is 4, then 10^4 is 10000. If x is 4.0, then 10^4.0 is 10000. So that's correct. So there's no need to think about decimal values here.

Alternatively, if the equation was something like 10^x = 9990 + 10 = 10000, so x is 4.

So, all methods lead to x = 4. Therefore, I think that's the correct answer.

But just to make sure, let me try plugging in x = 4 into the original equation again:

Left side: 10^4 - 10 = 10000 - 10 = 9990. Right side is 9990. Yep, matches.

Therefore, I can be confident that x = 4 is the correct solution.

I don't think I made any mistakes here. The problem seems straightforward once you isolate the exponential term and recognize the power of 10. It's important to check the solution by substituting back in, which I did, and it works.

Another thing to consider: sometimes when dealing with exponents, especially with different bases, you might need to use logarithms, but in this case, since the base is 10 and the right-hand side is a power of 10, it's straightforward.

So, in conclusion, after solving step by step and verifying through multiple methods, the value of x is 4.

**Final Answer**
The value of $ x $ is \boxed{4}.

To solve the equation:

$$
10^x - 10 = 9990
$$

we begin by isolating the exponential term. This is a standard approach for equations involving exponents.

---

### Step 1: Isolate the Exponential Term

Add 10 to both sides of the equation:

$$
10^x - 10 + 10 = 9990 + 10
$$

$$
10^x = 10000
$$

---

### Step 2: Express the Right-Hand Side as a Power of 10

We recognize that:

$$
10000 = 10^4
$$

So the equation becomes:

$$
10^x = 10^4
$$

---

### Step 3: Equate the Exponents

Since the bases are the same and the equation is in the form $ a^b = a^c $, we can equate the exponents:

$$
x = 4
$$

---

### Step 4: Verify the Solution

Substitute $ x = 4 $ back into the original equation:

$$
10^4 - 10 = 10000 - 10 = 9990
$$

This matches the right-hand side of the original equation, confirming that the solution is correct.

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to solve the equation $10^x - 10 = 9990$. Let me think about how to approach this. Hmm, first, maybe I should isolate the term with the exponent. That usually helps with exponential equations.

So, starting with the original equation:

$10^x - 10 = 9990$

I can add 10 to both sides to get rid of the -10 on the left. Let me do that:

$10^x - 10 + 10 = 9990 + 10$

Simplifying both sides:

$10^x = 10000$

Alright, so now I have $10^x = 10000$. I need to solve for x. Since the base is 10, maybe I can express 10000 as a power of 10. Let me recall that $10^1 = 10$, $10^2 = 100$, $10^3 = 1000$, $10^4 = 10000$. Oh, right! So 10000 is $10^4$.

Therefore, substituting back into the equation:

$10^x = 10^4$

Since the bases are the same, the exponents must be equal. That means:

$x = 4$

Wait, let me check if that's correct. Let me substitute x = 4 back into the original equation to verify.

Original equation: $10^x - 10 = 9990$

Substituting x = 4:

Left side: $10^4 - 10 = 10000 - 10 = 9990$

Right side: 9990

Yes, both sides are equal. So x = 4 is indeed the solution.

But let me think again if there's another way to approach this problem, just to make sure I didn't miss anything. Maybe using logarithms? Let me try that method as well.

Starting again from $10^x = 10000$. Taking the logarithm base 10 of both sides:

$\log_{10}(10^x) = \log_{10}(10000)$

Simplifying the left side using logarithm properties: $\log_{10}(10^x) = x$, because $\log_b(b^k) = k$.

On the right side, $\log_{10}(10000)$. Since 10000 is $10^4$, this becomes $\log_{10}(10^4) = 4$.

Therefore, x = 4. Same result. So that confirms it again.

Alternatively, if I didn't recognize that 10000 is $10^4$, I could use natural logarithm or common logarithm. Let me try using natural logarithm for practice.

Starting from $10^x = 10000$. Taking natural logarithm on both sides:

$\ln(10^x) = \ln(10000)$

Using the power rule for logarithms: $x \ln(10) = \ln(10000)$

Therefore, solving for x:

$x = \frac{\ln(10000)}{\ln(10)}$

Now, calculating $\ln(10000)$. Since 10000 is $10^4$, $\ln(10^4) = 4 \ln(10)$. Therefore:

$x = \frac{4 \ln(10)}{\ln(10)} = 4$.

Same answer again. So regardless of the method, I get x = 4.

Is there any chance that there could be another solution? For example, sometimes with logarithms, you have to consider different branches or something? But in this case, since the original equation is $10^x = 10000$, and 10^x is a one-to-one function, there's only one solution. So x must be 4.

Let me also think about the graph of the function $f(x) = 10^x - 10$. It's an exponential growth function shifted down by 10 units. The equation $10^x - 10 = 9990$ is equivalent to finding the x-intercept of $f(x) - 9990 = 0$, which would be a single point since the function is strictly increasing. Therefore, only one solution exists, which is x = 4.

Another way to check is to consider the value of x. If x is 4, then 10^4 is 10000, which minus 10 is 9990. Perfect. If I tried x = 3, then 10^3 is 1000, minus 10 is 990, which is way less than 9990. If x = 5, 10^5 is 100000, minus 10 is 99990, which is way more than 9990. So x must be between 4 and 5? Wait, but we already found x = 4 gives exactly 9990. Wait, no, hold on. Wait, 10^4 is 10000, so 10^4 - 10 is 9990. So x = 4 is exactly the solution.

Wait, but if I thought x was between 4 and 5, that would be a mistake. Wait, no. Let me check again. If x is 4, then 10^4 is 10000. If x is 4.0, then 10^4.0 is 10000. So that's correct. So there's no need to think about decimal values here.

Alternatively, if the equation was something like 10^x = 9990 + 10 = 10000, so x is 4.

So, all methods lead to x = 4. Therefore, I think that's the correct answer.

But just to make sure, let me try plugging in x = 4 into the original equation again:

Left side: 10^4 - 10 = 10000 - 10 = 9990. Right side is 9990. Yep, matches.

Therefore, I can be confident that x = 4 is the correct solution.

I don't think I made any mistakes here. The problem seems straightforward once you isolate the exponential term and recognize the power of 10. It's important to check the solution by substituting back in, which I did, and it works.

Another thing to consider: sometimes when dealing with exponents, especially with different bases, you might need to use logarithms, but in this case, since the base is 10 and the right-hand side is a power of 10, it's straightforward.

So, in conclusion, after solving step by step and verifying through multiple methods, the value of x is 4.

**Final Answer**
The value of $ x $ is \boxed{4}.

To solve the equation:

$$
10^x - 10 = 9990
$$

we begin by isolating the exponential term. This is a standard approach for equations involving exponents.

---

### Step 1: Isolate the Exponential Term

Add 10 to both sides of the equation:

$$
10^x - 10 + 10 = 9990 + 10
$$

$$
10^x = 10000
$$

---

### Step 2: Express the Right-Hand Side as a Power of 10

We recognize that:

$$
10000 = 10^4
$$

So the equation becomes:

$$
10^x = 10^4
$$

---

### Step 3: Equate the Exponents

Since the bases are the same and the equation is in the form $ a^b = a^c $, we can equate the exponents:

$$
x = 4
$$

---

### Step 4: Verify the Solution

Substitute $ x = 4 $ back into the original equation:

$$
10^4 - 10 = 10000 - 10 = 9990
$$

This matches the right-hand side of the original equation, confirming that the solution is correct.

---

### Final Answer

$$
\boxed{4}
$$

Solution: Since $10^x - 10 = 9990,$ we have $$10^x = 9990+10=10000.$$If $10^x = 10000,$ then $x=\boxed{4},$ since $10000$ ends in four zeroes.


Explanation:
Okay, so I need to find the sum of the measures of angle J and angle H in this circle J diagram. Let me try to visualize the Asymptote figure they provided. There's a circle with center J, and points H, O, and N. HO and HN are tangent to the circle at O and N respectively. The diagram shows J connected to N and O, and then H connected to N and O, forming a quadrilateral JNHO? Wait, the Asymptote code draws J--N--H--O--cycle, so that's a quadrilateral with vertices J, N, H, O. So, the figure is a quadrilateral where two sides are tangents from H to the circle, and the other two sides are radii from J to N and O?

Wait, but tangents from a common external point are equal in length. So HO and HN are both tangents from H to the circle, so their lengths should be equal. Also, since HO and HN are tangent to the circle at O and N, then JO and JN are radii perpendicular to the tangents at the points of tangency. So, JO is perpendicular to HO, and JN is perpendicular to HN. That means angles JOH and JNH are right angles (90 degrees each).

So, quadrilateral JNHO has two right angles at O and N. Let me try to sketch this mentally. The circle is centered at J, with radius JO and JN. Points O and N are points of tangency from H. So, H is outside the circle, and HO and HN are tangent segments. Then, connecting J to N and J to O, forming two radii. Then connecting H to N and H to O. So quadrilateral JNHO has sides JO, ON, NH, and HJ? Wait, no. Wait, the Asymptote code draws J--N--H--O--cycle. So the quadrilateral is J connected to N, N connected to H, H connected to O, and O connected back to J. So the sides are JN, NH, HO, and OJ.

So, in this quadrilateral, we have two right angles at O and N. Because JO is perpendicular to HO, and JN is perpendicular to HN. Therefore, angles at O and N are 90 degrees. So, quadrilateral JNHO has two right angles. Now, the question is asking for the sum of the measures of angle J and angle H. So angle at J (which is angle OJN?) and angle at H (which is angle NHO?).

Wait, let me clarify. In quadrilateral JNHO, the vertices are J, N, H, O. So angle J is the angle at vertex J, which is between sides JN and JO. Similarly, angle H is the angle at vertex H, which is between sides HN and HO. So, we need to find m∠J + m∠H.

Since we know two angles of the quadrilateral (at O and N are 90 degrees each), maybe we can use the fact that the sum of the interior angles of a quadrilateral is 360 degrees. So, if we can find the measures of angles at J and H, then their sum would be 360 - 90 - 90 = 180 degrees? Wait, that would mean that m∠J + m∠H = 180 degrees? But that seems too straightforward. Is that correct?

Wait, let me check again. The sum of the interior angles of any quadrilateral is 360 degrees. If two of the angles are 90 degrees each, then the sum of the other two angles is 360 - 90 - 90 = 180 degrees. Therefore, m∠J + m∠H = 180 degrees. So is the answer 180? But maybe I need to verify this with another method to be sure.

Alternatively, maybe there's some properties of tangents and circles that can be used here. Let me think. Since HO and HN are tangents from H to the circle, then HO = HN. Therefore, triangle HON is an isosceles triangle with HO = HN. Also, JO and JN are radii, so JO = JN. Wait, but JO and JN are both radii of the circle, so they are equal. Therefore, triangle JON is also an isosceles triangle with JO = JN.

But how does that help with angles at J and H?

Alternatively, maybe we can consider triangles JHO and JHN? Wait, but H is outside the circle, and JO and JN are radii. Let me think about the angles. Since JO is perpendicular to HO, angle JOH is 90 degrees. Similarly, angle JNH is 90 degrees. So, if I consider triangle JHO, which has a right angle at O, and triangle JHN, which has a right angle at N.

But angle at J in triangle JHO is angle OJH, and angle at J in triangle JHN is angle NJH. Wait, but angle J in the quadrilateral is the angle between JN and JO, which is angle OJN. Wait, but if we consider triangle JHO and triangle JHN, then angle OJH and angle NJH are parts of angle J in the quadrilateral. Wait, maybe I need to think about the entire angle at J.

Alternatively, maybe using coordinates. The Asymptote code defines J at (0,0), H at (6,0), and then constructs the circle with center J and radius 3. Then, it finds the midpoint M of JH, which is at (3,0), and draws a circle with center M and radius 3. The intersection points of the two circles are O and N. Then draws J--N--H--O--cycle.

So, maybe I can compute coordinates for O and N? Let me try that. Let me try to find coordinates of O and N.

First, circle J is centered at (0,0) with radius 3. The other circle is centered at M, which is midpoint of JH. Since J is (0,0) and H is (6,0), midpoint M is (3,0). The radius of the second circle is 3 as well. So, the two circles are:

Circle J: x² + y² = 9

Circle M: (x - 3)² + y² = 9

To find their intersection points, solve these equations. Subtract the first equation from the second:

(x - 3)² + y² - (x² + y²) = 9 - 9 => (x² -6x +9 + y²) - x² - y² = 0 => -6x +9 = 0 => x = 9/6 = 3/2 = 1.5

So, the points of intersection lie on x = 1.5. Substitute back into circle J's equation:

(1.5)² + y² = 9 => 2.25 + y² = 9 => y² = 6.75 => y = ±√(27/4) = ±(3√3)/2 ≈ ±2.598

Therefore, the points O and N are (1.5, (3√3)/2) and (1.5, -(3√3)/2). The Asymptote code labels O as tangentPoints[0] and N as tangentPoints[1], so depending on the order, but since the diagram labels O at NE (northeast) and N at S (south), so O is (1.5, (3√3)/2) and N is (1.5, -(3√3)/2).

So, coordinates:

J = (0,0)

H = (6,0)

O = (1.5, (3√3)/2)

N = (1.5, -(3√3)/2)

Now, we can compute angles at J and H.

First, angle at J: angle OJN. Wait, but in the quadrilateral JNHO, angle at J is between JN and JO. So, vectors from J to N and from J to O. Let me compute vectors JN and JO.

Vector JN is from J(0,0) to N(1.5, -(3√3)/2): (1.5, -(3√3)/2)

Vector JO is from J(0,0) to O(1.5, (3√3)/2): (1.5, (3√3)/2)

So, angle between vectors JN and JO. Since these vectors are symmetric with respect to the x-axis, the angle between them can be calculated. Let me compute the angle between them.

The formula for the angle between two vectors u and v is:

cosθ = (u • v)/(|u| |v|)

First, compute the dot product:

u • v = (1.5)(1.5) + [-(3√3)/2][(3√3)/2] = 2.25 - [ (9*3)/4 ] = 2.25 - (27/4) = 2.25 - 6.75 = -4.5

The magnitude of each vector:

|u| = |v| = sqrt( (1.5)^2 + [ (3√3)/2 ]^2 )

Compute (1.5)^2 = 2.25

[ (3√3)/2 ]^2 = (9*3)/4 = 27/4 = 6.75

So |u| = sqrt(2.25 + 6.75) = sqrt(9) = 3. Which makes sense because JN and JO are radii of the circle, so their lengths are 3.

Therefore, cosθ = (-4.5)/(3*3) = -4.5/9 = -0.5

Therefore, θ = arccos(-0.5) = 120 degrees. So angle at J is 120 degrees.

Now, angle at H: angle NHO. Let me find that. So angle at H between points N, H, O. So vectors from H to N and from H to O.

Coordinates:

H is at (6,0)

N is at (1.5, -(3√3)/2)

O is at (1.5, (3√3)/2)

So vectors:

HN = N - H = (1.5 - 6, -(3√3)/2 - 0) = (-4.5, -(3√3)/2)

HO = O - H = (1.5 - 6, (3√3)/2 - 0) = (-4.5, (3√3)/2)

So angle between vectors HN and HO. Let me compute this angle.

Again, using the dot product formula.

First, compute the dot product:

HN • HO = (-4.5)(-4.5) + [ -(3√3)/2 * (3√3)/2 ] = 20.25 - [ (9*3)/4 ] = 20.25 - 6.75 = 13.5

The magnitude of HN and HO:

Since HN and HO are both tangent segments from H to the circle, their lengths should be equal. Let me confirm:

Length of HN: sqrt( (-4.5)^2 + [ -(3√3)/2 ]^2 ) = sqrt(20.25 + 6.75) = sqrt(27) = 3√3

Similarly for HO. So |HN| = |HO| = 3√3

Therefore, cosφ = (13.5)/( (3√3)(3√3) ) = 13.5 / (9*3) = 13.5 / 27 = 0.5

Therefore, φ = arccos(0.5) = 60 degrees. Therefore, angle at H is 60 degrees.

Therefore, the sum of angle J (120 degrees) and angle H (60 degrees) is 180 degrees. So the answer is 180 degrees.

But wait, earlier I thought that since the quadrilateral has two right angles, the sum of the other two angles is 180 degrees. So that matches with this calculation. Therefore, regardless of the specific configuration, the sum is 180 degrees. But maybe there's a more general proof?

Alternatively, since HO and HN are tangents, then angle JHO and angle JHN are right angles? Wait, no. Wait, JO is perpendicular to HO, and JN is perpendicular to HN. Therefore, angles at O and N are right angles. Therefore, quadrilateral JNHO has two right angles. Therefore, sum of angles is 360, so the other two angles sum to 180. Therefore, regardless of the specific positions, the sum is 180 degrees. Therefore, the answer is 180 degrees.

But let me check again with coordinates. In the coordinate system, angle at J was 120 degrees, angle at H was 60 degrees, sum 180. So that's correct. Therefore, the answer is 180 degrees.

But let me think again if there's another way to see this. For example, since HO and HN are tangents, then the line HJ is the line from H to the center J. In the diagram, H is outside the circle, J is the center, and HO and HN are tangents. Then, the line HJ is the line connecting the external point to the center. The angle between HJ and HO is equal to the angle between HJ and HN? Wait, no. Wait, since HO and HN are symmetric with respect to HJ. Because in the coordinate system, H is at (6,0), J is at (0,0), and the points O and N are symmetric across the x-axis. Therefore, HJ is along the x-axis, and the tangents HO and HN are symmetric above and below the x-axis. Therefore, angle between HJ and HO is equal to angle between HJ and HN.

But in this case, angle at H is between HO and HN, which are symmetric with respect to HJ. Therefore, angle at H is twice the angle between HJ and HO.

Alternatively, in triangle HJO, which is a right triangle at O, with JO = 3, HO = 3√3, and HJ is the hypotenuse. Let me compute HJ. From coordinates, H is at (6,0), J is at (0,0), so HJ is 6 units. Let me check with Pythagoras: in triangle HJO, HO^2 + JO^2 = (3√3)^2 + 3^2 = 27 + 9 = 36 = 6^2 = HJ^2. Correct. Therefore, triangle HJO is a right triangle with legs 3√3 and 3, hypotenuse 6. Therefore, angle at H is angle between HO and HJ. Let me compute that angle.

In triangle HJO, angle at H: tan(theta) = opposite / adjacent = JO / HO = 3 / (3√3) = 1/√3. Therefore, theta = 30 degrees. Therefore, angle between HJ and HO is 30 degrees. Similarly, angle between HJ and HN is also 30 degrees. Therefore, angle at H between HO and HN is 30 + 30 = 60 degrees. Which matches our previous calculation.

Similarly, in triangle JHO, angle at J is angle between JH and JO. In triangle JHO, we can compute angle at J. Since it's a right triangle, with hypotenuse HJ = 6, and legs JO = 3, HO = 3√3. Therefore, angle at J is angle between JO and HJ. Let me compute that.

tan(theta) = opposite / adjacent = HO / JO = (3√3)/3 = √3. Therefore, theta = 60 degrees. Therefore, angle between JO and HJ is 60 degrees. Similarly, angle between JN and HJ is also 60 degrees? Wait, but angle at J in the quadrilateral is between JN and JO. Since JN and JO are symmetric with respect to HJ, then angle between JN and JO is 60 + 60 = 120 degrees. Which matches our previous calculation.

Therefore, angle at J is 120 degrees, angle at H is 60 degrees, sum is 180 degrees.

Alternatively, since angle between JO and HJ is 60 degrees, and angle between JN and HJ is also 60 degrees, then angle between JO and JN is 120 degrees.

Therefore, regardless of the specific coordinates, the sum is 180 degrees.

But let me think if there's a way to see this without coordinates. Since HO and HN are tangents, then angles JOH and JNH are right angles. Then quadrilateral JNHO has two right angles. Therefore, sum of angles is 360, so the other two angles sum to 180. Therefore, m∠J + m∠H = 180 degrees.

Therefore, the answer is 180 degrees.

But let me check once more. Suppose I didn't use coordinates. Let me think about the general case. Let’s say we have a circle with center J, and an external point H from which two tangents HO and HN are drawn to the circle, touching at O and N. Then, JO and JN are radii perpendicular to HO and HN respectively. Therefore, quadrilateral JNHO has two right angles at O and N. Therefore, the sum of the other two angles (at J and H) is 360 - 90 - 90 = 180 degrees. Therefore, regardless of the position of H, the sum is always 180 degrees.

Therefore, the answer is 180 degrees.

But just to make sure, let me think if there's any possibility that the quadrilateral is not convex or something? But since HO and HN are tangents, and J is the center, the quadrilateral JNHO should be convex.

Alternatively, maybe there's a property related to the angle between two tangents. The angle between two tangents from an external point H is equal to the difference between 180 degrees and the measure of the central angle subtended by the points of tangency.

Wait, in this case, the angle at H is between the two tangents, which is angle NHO. Which we found to be 60 degrees. The central angle between O and N is angle OJN, which we found to be 120 degrees. And indeed, angle between tangents is 180 - central angle. Wait, 180 - 120 = 60, which matches. Therefore, angle at H is 180 - central angle.

Therefore, if we denote central angle as θ, then angle at H is 180 - θ. Then, angle at J is θ. Therefore, their sum is θ + (180 - θ) = 180 degrees. Therefore, regardless of θ, the sum is 180 degrees.

Therefore, this is a general result. Therefore, the answer is 180 degrees.

Therefore, the answer is 180 degrees.

**Final Answer**
\boxed{180}

To solve the problem, we analyze the geometric configuration involving circle $ J $, with two tangents $ HO $ and $ HN $ drawn from an external point $ H $ to the circle, touching it at points $ O $ and $ N $, respectively.

---

### Step 1: Understand the Geometry

- **Circle $ J $** has center $ J $ and radius $ JO = JN $, since both are radii of the circle.
- **Tangents $ HO $ and $ HN $** are drawn from point $ H $ to the circle, touching it at $ O $ and $ N $, respectively.
- By the **tangent-radius theorem**, the radius at the point of tangency is **perpendicular** to the tangent line. Therefore:
  - $ \angle JOH = 90^\circ $
  - $ \angle JNH = 90^\circ $

This means that in quadrilateral $ JNHO $, two of its interior angles are right angles.

---

### Step 2: Use the Sum of Interior Angles in a Quadrilateral

The sum of the interior angles of any quadrilateral is always:

$$
\text{Sum of angles} = 360^\circ
$$

We are given that:

- $ \angle JOH = 90^\circ $
- $ \angle JNH = 90^\circ $

Let $ \angle J $ and $ \angle H $ be the remaining two angles of the quadrilateral. Then:

$$
\angle J + \angle H = 360^\circ - 90^\circ - 90^\circ = 180^\circ
$$

---

### Step 3: Confirm with Coordinate Geometry (Optional Verification)

Using coordinate geometry, we can place the circle at the origin $ J = (0, 0) $, and point $ H = (6, 0) $. The tangents from $ H $ to the circle intersect the circle at points $ O $ and $ N $, which are symmetric about the x-axis. Calculating the coordinates of $ O $ and $ N $, and then computing the angles at $ J $ and $ H $, we find:

- $ \angle J = 120^\circ $
- $ \angle H = 60^\circ $

Their sum is:

$$
120^\circ + 60^\circ = 180^\circ
$$

This confirms the earlier result.

---

### Step 4: General Geometric Insight

In general, for any circle and an external point $ H $ from which two tangents are drawn to the circle, the angle between the two tangents (i.e., $ \angle H $) is related to the central angle $ \angle OJN $ (i.e., $ \angle J $) by the formula:

$$
\angle H = 180^\circ - \angle J
$$

Thus, their sum is always:

$$
\angle J + \angle H = 180^\circ
$$

---

### Final Answer

$$
\boxed{180}
$$

[Assistant]
Explanation:
Okay, so I need to find the sum of the measures of angle J and angle H in this circle J diagram. Let me try to visualize the Asymptote figure they provided. There's a circle with center J, and points H, O, and N. HO and HN are tangent to the circle at O and N respectively. The diagram shows J connected to N and O, and then H connected to N and O, forming a quadrilateral JNHO? Wait, the Asymptote code draws J--N--H--O--cycle, so that's a quadrilateral with vertices J, N, H, O. So, the figure is a quadrilateral where two sides are tangents from H to the circle, and the other two sides are radii from J to N and O?

Wait, but tangents from a common external point are equal in length. So HO and HN are both tangents from H to the circle, so their lengths should be equal. Also, since HO and HN are tangent to the circle at O and N, then JO and JN are radii perpendicular to the tangents at the points of tangency. So, JO is perpendicular to HO, and JN is perpendicular to HN. That means angles JOH and JNH are right angles (90 degrees each).

So, quadrilateral JNHO has two right angles at O and N. Let me try to sketch this mentally. The circle is centered at J, with radius JO and JN. Points O and N are points of tangency from H. So, H is outside the circle, and HO and HN are tangent segments. Then, connecting J to N and J to O, forming two radii. Then connecting H to N and H to O. So quadrilateral JNHO has sides JO, ON, NH, and HJ? Wait, no. Wait, the Asymptote code draws J--N--H--O--cycle. So the quadrilateral is J connected to N, N connected to H, H connected to O, and O connected back to J. So the sides are JN, NH, HO, and OJ.

So, in this quadrilateral, we have two right angles at O and N. Because JO is perpendicular to HO, and JN is perpendicular to HN. Therefore, angles at O and N are 90 degrees. So, quadrilateral JNHO has two right angles. Now, the question is asking for the sum of the measures of angle J and angle H. So angle at J (which is angle OJN?) and angle at H (which is angle NHO?).

Wait, let me clarify. In quadrilateral JNHO, the vertices are J, N, H, O. So angle J is the angle at vertex J, which is between sides JN and JO. Similarly, angle H is the angle at vertex H, which is between sides HN and HO. So, we need to find m∠J + m∠H.

Since we know two angles of the quadrilateral (at O and N are 90 degrees each), maybe we can use the fact that the sum of the interior angles of a quadrilateral is 360 degrees. So, if we can find the measures of angles at J and H, then their sum would be 360 - 90 - 90 = 180 degrees? Wait, that would mean that m∠J + m∠H = 180 degrees? But that seems too straightforward. Is that correct?

Wait, let me check again. The sum of the interior angles of any quadrilateral is 360 degrees. If two of the angles are 90 degrees each, then the sum of the other two angles is 360 - 90 - 90 = 180 degrees. Therefore, m∠J + m∠H = 180 degrees. So is the answer 180? But maybe I need to verify this with another method to be sure.

Alternatively, maybe there's some properties of tangents and circles that can be used here. Let me think. Since HO and HN are tangents from H to the circle, then HO = HN. Therefore, triangle HON is an isosceles triangle with HO = HN. Also, JO and JN are radii, so JO = JN. Wait, but JO and JN are both radii of the circle, so they are equal. Therefore, triangle JON is also an isosceles triangle with JO = JN.

But how does that help with angles at J and H?

Alternatively, maybe we can consider triangles JHO and JHN? Wait, but H is outside the circle, and JO and JN are radii. Let me think about the angles. Since JO is perpendicular to HO, angle JOH is 90 degrees. Similarly, angle JNH is 90 degrees. So, if I consider triangle JHO, which has a right angle at O, and triangle JHN, which has a right angle at N.

But angle at J in triangle JHO is angle OJH, and angle at J in triangle JHN is angle NJH. Wait, but angle J in the quadrilateral is the angle between JN and JO, which is angle OJN. Wait, but if we consider triangle JHO and triangle JHN, then angle OJH and angle NJH are parts of angle J in the quadrilateral. Wait, maybe I need to think about the entire angle at J.

Alternatively, maybe using coordinates. The Asymptote code defines J at (0,0), H at (6,0), and then constructs the circle with center J and radius 3. Then, it finds the midpoint M of JH, which is at (3,0), and draws a circle with center M and radius 3. The intersection points of the two circles are O and N. Then draws J--N--H--O--cycle.

So, maybe I can compute coordinates for O and N? Let me try that. Let me try to find coordinates of O and N.

First, circle J is centered at (0,0) with radius 3. The other circle is centered at M, which is midpoint of JH. Since J is (0,0) and H is (6,0), midpoint M is (3,0). The radius of the second circle is 3 as well. So, the two circles are:

Circle J: x² + y² = 9

Circle M: (x - 3)² + y² = 9

To find their intersection points, solve these equations. Subtract the first equation from the second:

(x - 3)² + y² - (x² + y²) = 9 - 9 => (x² -6x +9 + y²) - x² - y² = 0 => -6x +9 = 0 => x = 9/6 = 3/2 = 1.5

So, the points of intersection lie on x = 1.5. Substitute back into circle J's equation:

(1.5)² + y² = 9 => 2.25 + y² = 9 => y² = 6.75 => y = ±√(27/4) = ±(3√3)/2 ≈ ±2.598

Therefore, the points O and N are (1.5, (3√3)/2) and (1.5, -(3√3)/2). The Asymptote code labels O as tangentPoints[0] and N as tangentPoints[1], so depending on the order, but since the diagram labels O at NE (northeast) and N at S (south), so O is (1.5, (3√3)/2) and N is (1.5, -(3√3)/2).

So, coordinates:

J = (0,0)

H = (6,0)

O = (1.5, (3√3)/2)

N = (1.5, -(3√3)/2)

Now, we can compute angles at J and H.

First, angle at J: angle OJN. Wait, but in the quadrilateral JNHO, angle at J is between JN and JO. So, vectors from J to N and from J to O. Let me compute vectors JN and JO.

Vector JN is from J(0,0) to N(1.5, -(3√3)/2): (1.5, -(3√3)/2)

Vector JO is from J(0,0) to O(1.5, (3√3)/2): (1.5, (3√3)/2)

So, angle between vectors JN and JO. Since these vectors are symmetric with respect to the x-axis, the angle between them can be calculated. Let me compute the angle between them.

The formula for the angle between two vectors u and v is:

cosθ = (u • v)/(|u| |v|)

First, compute the dot product:

u • v = (1.5)(1.5) + [-(3√3)/2][(3√3)/2] = 2.25 - [ (9*3)/4 ] = 2.25 - (27/4) = 2.25 - 6.75 = -4.5

The magnitude of each vector:

|u| = |v| = sqrt( (1.5)^2 + [ (3√3)/2 ]^2 )

Compute (1.5)^2 = 2.25

[ (3√3)/2 ]^2 = (9*3)/4 = 27/4 = 6.75

So |u| = sqrt(2.25 + 6.75) = sqrt(9) = 3. Which makes sense because JN and JO are radii of the circle, so their lengths are 3.

Therefore, cosθ = (-4.5)/(3*3) = -4.5/9 = -0.5

Therefore, θ = arccos(-0.5) = 120 degrees. So angle at J is 120 degrees.

Now, angle at H: angle NHO. Let me find that. So angle at H between points N, H, O. So vectors from H to N and from H to O.

Coordinates:

H is at (6,0)

N is at (1.5, -(3√3)/2)

O is at (1.5, (3√3)/2)

So vectors:

HN = N - H = (1.5 - 6, -(3√3)/2 - 0) = (-4.5, -(3√3)/2)

HO = O - H = (1.5 - 6, (3√3)/2 - 0) = (-4.5, (3√3)/2)

So angle between vectors HN and HO. Let me compute this angle.

Again, using the dot product formula.

First, compute the dot product:

HN • HO = (-4.5)(-4.5) + [ -(3√3)/2 * (3√3)/2 ] = 20.25 - [ (9*3)/4 ] = 20.25 - 6.75 = 13.5

The magnitude of HN and HO:

Since HN and HO are both tangent segments from H to the circle, their lengths should be equal. Let me confirm:

Length of HN: sqrt( (-4.5)^2 + [ -(3√3)/2 ]^2 ) = sqrt(20.25 + 6.75) = sqrt(27) = 3√3

Similarly for HO. So |HN| = |HO| = 3√3

Therefore, cosφ = (13.5)/( (3√3)(3√3) ) = 13.5 / (9*3) = 13.5 / 27 = 0.5

Therefore, φ = arccos(0.5) = 60 degrees. Therefore, angle at H is 60 degrees.

Therefore, the sum of angle J (120 degrees) and angle H (60 degrees) is 180 degrees. So the answer is 180 degrees.

But wait, earlier I thought that since the quadrilateral has two right angles, the sum of the other two angles is 180 degrees. So that matches with this calculation. Therefore, regardless of the specific configuration, the sum is 180 degrees. But maybe there's a more general proof?

Alternatively, since HO and HN are tangents, then angle JHO and angle JHN are right angles? Wait, no. Wait, JO is perpendicular to HO, and JN is perpendicular to HN. Therefore, angles at O and N are right angles. Therefore, quadrilateral JNHO has two right angles. Therefore, sum of angles is 360, so the other two angles sum to 180. Therefore, regardless of the specific positions, the sum is 180 degrees. Therefore, the answer is 180 degrees.

But let me check again with coordinates. In the coordinate system, angle at J was 120 degrees, angle at H was 60 degrees, sum 180. So that's correct. Therefore, the answer is 180 degrees.

But let me think again if there's another way to see this. For example, since HO and HN are tangents, then the line HJ is the line from H to the center J. In the diagram, H is outside the circle, J is the center, and HO and HN are tangents. Then, the line HJ is the line connecting the external point to the center. The angle between HJ and HO is equal to the angle between HJ and HN? Wait, no. Wait, since HO and HN are symmetric with respect to HJ. Because in the coordinate system, H is at (6,0), J is at (0,0), and the points O and N are symmetric across the x-axis. Therefore, HJ is along the x-axis, and the tangents HO and HN are symmetric above and below the x-axis. Therefore, angle between HJ and HO is equal to angle between HJ and HN.

But in this case, angle at H is between HO and HN, which are symmetric with respect to HJ. Therefore, angle at H is twice the angle between HJ and HO.

Alternatively, in triangle HJO, which is a right triangle at O, with JO = 3, HO = 3√3, and HJ is the hypotenuse. Let me compute HJ. From coordinates, H is at (6,0), J is at (0,0), so HJ is 6 units. Let me check with Pythagoras: in triangle HJO, HO^2 + JO^2 = (3√3)^2 + 3^2 = 27 + 9 = 36 = 6^2 = HJ^2. Correct. Therefore, triangle HJO is a right triangle with legs 3√3 and 3, hypotenuse 6. Therefore, angle at H is angle between HO and HJ. Let me compute that angle.

In triangle HJO, angle at H: tan(theta) = opposite / adjacent = JO / HO = 3 / (3√3) = 1/√3. Therefore, theta = 30 degrees. Therefore, angle between HJ and HO is 30 degrees. Similarly, angle between HJ and HN is also 30 degrees. Therefore, angle at H between HO and HN is 30 + 30 = 60 degrees. Which matches our previous calculation.

Similarly, in triangle JHO, angle at J is angle between JH and JO. In triangle JHO, we can compute angle at J. Since it's a right triangle, with hypotenuse HJ = 6, and legs JO = 3, HO = 3√3. Therefore, angle at J is angle between JO and HJ. Let me compute that.

tan(theta) = opposite / adjacent = HO / JO = (3√3)/3 = √3. Therefore, theta = 60 degrees. Therefore, angle between JO and HJ is 60 degrees. Similarly, angle between JN and HJ is also 60 degrees? Wait, but angle at J in the quadrilateral is between JN and JO. Since JN and JO are symmetric with respect to HJ, then angle between JN and JO is 60 + 60 = 120 degrees. Which matches our previous calculation.

Therefore, angle at J is 120 degrees, angle at H is 60 degrees, sum is 180 degrees.

Alternatively, since angle between JO and HJ is 60 degrees, and angle between JN and HJ is also 60 degrees, then angle between JO and JN is 120 degrees.

Therefore, regardless of the specific coordinates, the sum is 180 degrees.

But let me think if there's a way to see this without coordinates. Since HO and HN are tangents, then angles JOH and JNH are right angles. Then quadrilateral JNHO has two right angles. Therefore, sum of angles is 360, so the other two angles sum to 180. Therefore, m∠J + m∠H = 180 degrees.

Therefore, the answer is 180 degrees.

But let me check once more. Suppose I didn't use coordinates. Let me think about the general case. Let’s say we have a circle with center J, and an external point H from which two tangents HO and HN are drawn to the circle, touching at O and N. Then, JO and JN are radii perpendicular to HO and HN respectively. Therefore, quadrilateral JNHO has two right angles at O and N. Therefore, the sum of the other two angles (at J and H) is 360 - 90 - 90 = 180 degrees. Therefore, regardless of the position of H, the sum is always 180 degrees.

Therefore, the answer is 180 degrees.

But just to make sure, let me think if there's any possibility that the quadrilateral is not convex or something? But since HO and HN are tangents, and J is the center, the quadrilateral JNHO should be convex.

Alternatively, maybe there's a property related to the angle between two tangents. The angle between two tangents from an external point H is equal to the difference between 180 degrees and the measure of the central angle subtended by the points of tangency.

Wait, in this case, the angle at H is between the two tangents, which is angle NHO. Which we found to be 60 degrees. The central angle between O and N is angle OJN, which we found to be 120 degrees. And indeed, angle between tangents is 180 - central angle. Wait, 180 - 120 = 60, which matches. Therefore, angle at H is 180 - central angle.

Therefore, if we denote central angle as θ, then angle at H is 180 - θ. Then, angle at J is θ. Therefore, their sum is θ + (180 - θ) = 180 degrees. Therefore, regardless of θ, the sum is 180 degrees.

Therefore, this is a general result. Therefore, the answer is 180 degrees.

Therefore, the answer is 180 degrees.

**Final Answer**
\boxed{180}

To solve the problem, we analyze the geometric configuration involving circle $ J $, with two tangents $ HO $ and $ HN $ drawn from an external point $ H $ to the circle, touching it at points $ O $ and $ N $, respectively.

---

### Step 1: Understand the Geometry

- **Circle $ J $** has center $ J $ and radius $ JO = JN $, since both are radii of the circle.
- **Tangents $ HO $ and $ HN $** are drawn from point $ H $ to the circle, touching it at $ O $ and $ N $, respectively.
- By the **tangent-radius theorem**, the radius at the point of tangency is **perpendicular** to the tangent line. Therefore:
  - $ \angle JOH = 90^\circ $
  - $ \angle JNH = 90^\circ $

This means that in quadrilateral $ JNHO $, two of its interior angles are right angles.

---

### Step 2: Use the Sum of Interior Angles in a Quadrilateral

The sum of the interior angles of any quadrilateral is always:

$$
\text{Sum of angles} = 360^\circ
$$

We are given that:

- $ \angle JOH = 90^\circ $
- $ \angle JNH = 90^\circ $

Let $ \angle J $ and $ \angle H $ be the remaining two angles of the quadrilateral. Then:

$$
\angle J + \angle H = 360^\circ - 90^\circ - 90^\circ = 180^\circ
$$

---

### Step 3: Confirm with Coordinate Geometry (Optional Verification)

Using coordinate geometry, we can place the circle at the origin $ J = (0, 0) $, and point $ H = (6, 0) $. The tangents from $ H $ to the circle intersect the circle at points $ O $ and $ N $, which are symmetric about the x-axis. Calculating the coordinates of $ O $ and $ N $, and then computing the angles at $ J $ and $ H $, we find:

- $ \angle J = 120^\circ $
- $ \angle H = 60^\circ $

Their sum is:

$$
120^\circ + 60^\circ = 180^\circ
$$

This confirms the earlier result.

---

### Step 4: General Geometric Insight

In general, for any circle and an external point $ H $ from which two tangents are drawn to the circle, the angle between the two tangents (i.e., $ \angle H $) is related to the central angle $ \angle OJN $ (i.e., $ \angle J $) by the formula:

$$
\angle H = 180^\circ - \angle J
$$

Thus, their sum is always:

$$
\angle J + \angle H = 180^\circ
$$

---

### Final Answer

$$
\boxed{180}
$$

Solution: Since $\overline{OH}$ and $\overline{NH}$ are tangent to radii of the circle at $O$ and $N$, we have $\angle O =\angle N = 90^\circ$.  The sum of the measures of the interior angles of quadrilateral $JOHN$ is $360^\circ$, so $\angle J + \angle H = 360^\circ - \angle O - \angle N = \boxed{180^\circ}$.