Explanation:
Okay, so I need to solve the equation 2/3 equals 4 divided by (x minus 5). Let me write that down first to make sure I have it right: 2/3 = 4/(x - 5). Alright, so my goal is to find the value of x that makes this equation true.

Let me think about how to approach this. It looks like a rational equation, meaning there's a fraction on one side and a variable in the denominator. To solve for x, I probably need to get rid of the fractions or cross-multiply to make it easier. Let me recall: if I have two fractions equal to each other, like a/b = c/d, then cross-multiplying gives me a*d = b*c. So maybe I can apply that here.

So applying cross-multiplication to 2/3 = 4/(x - 5), that would mean multiplying the numerator of one side by the denominator of the other side. Let me do that step by step.

Left side numerator is 2, right side denominator is (x - 5). So 2 times (x - 5) should equal the right side numerator, which is 4, times the left side denominator, which is 3. Let me write that out:

2*(x - 5) = 4*3

Let me compute both sides. On the left, 2 times (x - 5) is 2x - 10. On the right, 4 times 3 is 12. So now the equation is:

2x - 10 = 12

Alright, now I need to solve for x. Let me isolate x. First, I can add 10 to both sides to get rid of the -10 on the left.

Adding 10 to both sides:

2x - 10 + 10 = 12 + 10

Simplifying:

2x = 22

Now, to get x by itself, I divide both sides by 2.

2x / 2 = 22 / 2

Which simplifies to:

x = 11

Wait, so is x equal to 11? Let me check my steps again to make sure I didn't make any mistakes.

Starting with cross-multiplication: 2*(x - 5) = 4*3. That seems right. Then expanding the left side: 2x - 10 = 12. Then adding 10 to both sides gives 2x = 22, so x = 11.

Hmm. Let me verify the solution by plugging x = 11 back into the original equation to see if both sides are equal.

Original equation: 2/3 = 4/(x - 5). Substitute x with 11:

Left side is 2/3. Right side is 4/(11 - 5) = 4/6. Simplify 4/6 by dividing numerator and denominator by 2: 2/3.

Yes, so both sides are equal. That checks out. Therefore, x = 11 is indeed the correct solution.

But wait, let me think again if there's another way to approach this problem, just to confirm. Maybe by solving the equation step by step without cross-multiplying? Let me see.

Original equation: 2/3 = 4/(x - 5).

If I want to solve for x, maybe I can start by taking reciprocals of both sides? Wait, but the equation is 2/3 equals 4 over something. So if I take reciprocals, it would be 3/2 = (x - 5)/4. Then cross-multiplying again would give me 3*4 = 2*(x - 5), which is 12 = 2x - 10, leading to the same equation as before. So that method also leads to the same result.

Alternatively, maybe I can first write the equation as 2/3 = 4/(x - 5) and then solve for x - 5. Let's try that.

Starting with 2/3 = 4/(x - 5). Let me solve for (x - 5). Multiply both sides by (x - 5) to get rid of the denominator.

That would give me (2/3)*(x - 5) = 4.

Then, to solve for x - 5, I can multiply both sides by 3/2 to get rid of the fraction.

So: (x - 5) = 4*(3/2).

Calculating 4*(3/2) is 6. Therefore, x - 5 = 6.

Adding 5 to both sides: x = 6 + 5 = 11.

Same answer again. So that's good. All methods lead to x = 11.

Let me check if there are any restrictions on x. The original equation has a denominator of (x - 5). Therefore, x cannot be 5, because that would make the denominator zero, which is undefined. However, our solution is x = 11, which is not 5, so it's valid.

Therefore, x = 11 is the only solution, and it's valid.

Is there any other possible solution? Let me see. Since this is a linear equation after cross-multiplying, there should be only one solution. So I don't think there are other solutions.

Alternatively, if we think about the original equation, 2/3 is equal to 4 divided by something. So that something must be equal to 4 divided by 2/3, which is 4*(3/2) = 6. Therefore, x - 5 = 6, so x = 11. Same result.

So no matter how I approach it, I get x = 11.

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

**Final Answer**
The solution is \boxed{11}.

To solve the given equation:

$$
\frac{2}{3} = \frac{4}{x - 5}
$$

We can proceed by first analyzing the structure of the equation. It is a **rational equation** where a constant is equal to a fraction involving a variable in the denominator.

---

### Step 1: Cross-Multiply to Eliminate the Denominator

To eliminate the denominator on the right-hand side, we use **cross-multiplication**. That is:

$$
2(x - 5) = 3 \cdot 4
$$

Simplify both sides:

- Left: $ 2(x - 5) = 2x - 10 $
- Right: $ 3 \cdot 4 = 12 $

So the equation becomes:

$$
2x - 10 = 12
$$

---

### Step 2: Solve for $ x $

Add 10 to both sides:

$$
2x = 22
$$

Now divide both sides by 2:

$$
x = 11
$$

---

### Step 3: Verify the Solution

We should **check** the solution by substituting $ x = 11 $ back into the original equation:

- Left-hand side: $ \frac{2}{3} $
- Right-hand side: $ \frac{4}{x - 5} = \frac{4}{11 - 5} = \frac{4}{6} = \frac{2}{3} $

Both sides are equal, confirming the solution is correct.

---

### Step 4: Domain Consideration

The denominator in the original equation is $ x - 5 $, so we must ensure:

$$
x - 5 \ne 0 \Rightarrow x \ne 5
$$

Since our solution is $ x = 11 $, it is valid and not excluded from the domain.

---

### Final Answer

$$
\boxed{11}
$$

[Assistant]
Explanation:
Okay, so I need to solve the equation 2/3 equals 4 divided by (x minus 5). Let me write that down first to make sure I have it right: 2/3 = 4/(x - 5). Alright, so my goal is to find the value of x that makes this equation true.

Let me think about how to approach this. It looks like a rational equation, meaning there's a fraction on one side and a variable in the denominator. To solve for x, I probably need to get rid of the fractions or cross-multiply to make it easier. Let me recall: if I have two fractions equal to each other, like a/b = c/d, then cross-multiplying gives me a*d = b*c. So maybe I can apply that here.

So applying cross-multiplication to 2/3 = 4/(x - 5), that would mean multiplying the numerator of one side by the denominator of the other side. Let me do that step by step.

Left side numerator is 2, right side denominator is (x - 5). So 2 times (x - 5) should equal the right side numerator, which is 4, times the left side denominator, which is 3. Let me write that out:

2*(x - 5) = 4*3

Let me compute both sides. On the left, 2 times (x - 5) is 2x - 10. On the right, 4 times 3 is 12. So now the equation is:

2x - 10 = 12

Alright, now I need to solve for x. Let me isolate x. First, I can add 10 to both sides to get rid of the -10 on the left.

Adding 10 to both sides:

2x - 10 + 10 = 12 + 10

Simplifying:

2x = 22

Now, to get x by itself, I divide both sides by 2.

2x / 2 = 22 / 2

Which simplifies to:

x = 11

Wait, so is x equal to 11? Let me check my steps again to make sure I didn't make any mistakes.

Starting with cross-multiplication: 2*(x - 5) = 4*3. That seems right. Then expanding the left side: 2x - 10 = 12. Then adding 10 to both sides gives 2x = 22, so x = 11.

Hmm. Let me verify the solution by plugging x = 11 back into the original equation to see if both sides are equal.

Original equation: 2/3 = 4/(x - 5). Substitute x with 11:

Left side is 2/3. Right side is 4/(11 - 5) = 4/6. Simplify 4/6 by dividing numerator and denominator by 2: 2/3.

Yes, so both sides are equal. That checks out. Therefore, x = 11 is indeed the correct solution.

But wait, let me think again if there's another way to approach this problem, just to confirm. Maybe by solving the equation step by step without cross-multiplying? Let me see.

Original equation: 2/3 = 4/(x - 5).

If I want to solve for x, maybe I can start by taking reciprocals of both sides? Wait, but the equation is 2/3 equals 4 over something. So if I take reciprocals, it would be 3/2 = (x - 5)/4. Then cross-multiplying again would give me 3*4 = 2*(x - 5), which is 12 = 2x - 10, leading to the same equation as before. So that method also leads to the same result.

Alternatively, maybe I can first write the equation as 2/3 = 4/(x - 5) and then solve for x - 5. Let's try that.

Starting with 2/3 = 4/(x - 5). Let me solve for (x - 5). Multiply both sides by (x - 5) to get rid of the denominator.

That would give me (2/3)*(x - 5) = 4.

Then, to solve for x - 5, I can multiply both sides by 3/2 to get rid of the fraction.

So: (x - 5) = 4*(3/2).

Calculating 4*(3/2) is 6. Therefore, x - 5 = 6.

Adding 5 to both sides: x = 6 + 5 = 11.

Same answer again. So that's good. All methods lead to x = 11.

Let me check if there are any restrictions on x. The original equation has a denominator of (x - 5). Therefore, x cannot be 5, because that would make the denominator zero, which is undefined. However, our solution is x = 11, which is not 5, so it's valid.

Therefore, x = 11 is the only solution, and it's valid.

Is there any other possible solution? Let me see. Since this is a linear equation after cross-multiplying, there should be only one solution. So I don't think there are other solutions.

Alternatively, if we think about the original equation, 2/3 is equal to 4 divided by something. So that something must be equal to 4 divided by 2/3, which is 4*(3/2) = 6. Therefore, x - 5 = 6, so x = 11. Same result.

So no matter how I approach it, I get x = 11.

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

**Final Answer**
The solution is \boxed{11}.

To solve the given equation:

$$
\frac{2}{3} = \frac{4}{x - 5}
$$

We can proceed by first analyzing the structure of the equation. It is a **rational equation** where a constant is equal to a fraction involving a variable in the denominator.

---

### Step 1: Cross-Multiply to Eliminate the Denominator

To eliminate the denominator on the right-hand side, we use **cross-multiplication**. That is:

$$
2(x - 5) = 3 \cdot 4
$$

Simplify both sides:

- Left: $ 2(x - 5) = 2x - 10 $
- Right: $ 3 \cdot 4 = 12 $

So the equation becomes:

$$
2x - 10 = 12
$$

---

### Step 2: Solve for $ x $

Add 10 to both sides:

$$
2x = 22
$$

Now divide both sides by 2:

$$
x = 11
$$

---

### Step 3: Verify the Solution

We should **check** the solution by substituting $ x = 11 $ back into the original equation:

- Left-hand side: $ \frac{2}{3} $
- Right-hand side: $ \frac{4}{x - 5} = \frac{4}{11 - 5} = \frac{4}{6} = \frac{2}{3} $

Both sides are equal, confirming the solution is correct.

---

### Step 4: Domain Consideration

The denominator in the original equation is $ x - 5 $, so we must ensure:

$$
x - 5 \ne 0 \Rightarrow x \ne 5
$$

Since our solution is $ x = 11 $, it is valid and not excluded from the domain.

---

### Final Answer

$$
\boxed{11}
$$

Solution: Multiplying both sides by $x-5$ and by 3 gives $2(x-5) = 4(3)$.  Expanding the left side gives $2x-10 = 12$.  Adding 10 to both sides gives $2x = 22$ and dividing by 2 gives $x = \boxed{11}$.


Explanation:
Okay, let's see. I need to find the volume of a cone. The formula given is V equals one-third times B times h. Alright, so B is the area of the base, and h is the height. They told me that the area of the base is 30 square units, and the height is 6.5 units. So I just need to plug those numbers into the formula, right?

Wait, let me make sure I remember the formula correctly. Volume of a cone... yeah, it's one-third times the area of the base times the height. That's different from the volume of a cylinder, which is just B times h. So cones are a third of that. Okay, that makes sense.

So, given that B is 30, and h is 6.5. So substituting into the formula, V equals one-third times 30 times 6.5. Let me write that down step by step.

First, calculate one-third of 30. Let me do that part first. One-third of 30... Well, 30 divided by 3 is 10. So that part is straightforward. Then I have to multiply that by 6.5. So 10 times 6.5. Hmm, 10 times 6 is 60, and 10 times 0.5 is 5, so adding those together, 60 + 5 equals 65. So is the volume 65 cubic units?

Wait, let me check my steps again to make sure I didn't make any mistakes. First, area of the base is 30, height is 6.5. Formula is 1/3 * B * h. So 1/3 * 30 is indeed 10. Then 10 * 6.5. Let me verify that multiplication again. 6.5 can be thought of as 6 + 0.5. So 10*6 is 60, 10*0.5 is 5, so total is 65. Yeah, that seems right.

Alternatively, maybe I can do it another way. Let me convert 6.5 into a fraction. 6.5 is the same as 13/2. So then, 1/3 * 30 * 13/2. Let's compute that. 1/3 times 30 is 10, as before. Then 10 times 13/2. Which is (10/1) * (13/2) = (10*13)/2 = 130/2 = 65. Yep, same answer. Okay, so that checks out.

Alternatively, if I do all the multiplication at once. Let's see, 1/3 * 30 * 6.5. Let me compute 30 * 6.5 first. 30 times 6 is 180, and 30 times 0.5 is 15, so adding those gives 180 + 15 = 195. Then 195 divided by 3 is... 195 divided by 3. Let me compute that. 3 goes into 19 five times (15), remainder 4. 3 goes into 4 once (3), remainder 1. Then 3 goes into 1 zero times with a decimal. So 5.0... Wait, no, maybe I should do it differently. 3 times 65 is 195. Because 3*60 is 180, 3*5 is 15. So 180+15=195. Therefore, 195 divided by 3 is 65. Yep, same result.

So all three methods give me 65. That seems pretty solid. So the volume should be 65 cubic units.

Wait, hold on, but let me just make sure that I didn't mix up any numbers. The base area is 30, height is 6.5. So no, I didn't confuse radius or diameter or anything? Wait, the formula is given as V = 1/3 * B * h, where B is the area of the base. So they already gave me the area of the base, so I don't need to calculate that from radius or anything. So that's straightforward. So if B is 30, then I just use that. So I think that's correct.

Alternatively, if someone thought that B was the radius, but no, the problem says "area of the base," so that's definitely 30. So no problem there. The height is given as 6.5 units, which is a decimal, but that's okay. So the multiplication still works out. So I think that 65 is the correct answer here.

Just to recap, the formula is volume equals one-third times base area times height. Plug in the numbers, do the arithmetic, and get 65. All checks confirm that. So I think that's the answer.

**Final Answer**
The volume of the cone is \boxed{65} cubic units.

To determine the volume of a cone, we use the formula:

$$
V = \frac{1}{3}Bh
$$

Where:
- $ V $ is the volume (in cubic units),
- $ B $ is the area of the base (in square units),
- $ h $ is the height (in units).

---

### Step 1: Identify the given values
From the problem, we are told:
- The area of the base, $ B = 30 $ square units,
- The height, $ h = 6.5 $ units.

---

### Step 2: Substitute the known values into the formula

$$
V = \frac{1}{3} \times 30 \times 6.5
$$

---

### Step 3: Simplify the expression step by step

First, multiply $ \frac{1}{3} \times 30 $:

$$
\frac{1}{3} \times 30 = 10
$$

Next, multiply the result by 6.5:

$$
10 \times 6.5 = 65
$$

---

### Step 4: Final Answer

$$
V = \boxed{65}
$$

So, the volume of the cone is **65 cubic units**.

[Assistant]
Explanation:
Okay, let's see. I need to find the volume of a cone. The formula given is V equals one-third times B times h. Alright, so B is the area of the base, and h is the height. They told me that the area of the base is 30 square units, and the height is 6.5 units. So I just need to plug those numbers into the formula, right?

Wait, let me make sure I remember the formula correctly. Volume of a cone... yeah, it's one-third times the area of the base times the height. That's different from the volume of a cylinder, which is just B times h. So cones are a third of that. Okay, that makes sense.

So, given that B is 30, and h is 6.5. So substituting into the formula, V equals one-third times 30 times 6.5. Let me write that down step by step.

First, calculate one-third of 30. Let me do that part first. One-third of 30... Well, 30 divided by 3 is 10. So that part is straightforward. Then I have to multiply that by 6.5. So 10 times 6.5. Hmm, 10 times 6 is 60, and 10 times 0.5 is 5, so adding those together, 60 + 5 equals 65. So is the volume 65 cubic units?

Wait, let me check my steps again to make sure I didn't make any mistakes. First, area of the base is 30, height is 6.5. Formula is 1/3 * B * h. So 1/3 * 30 is indeed 10. Then 10 * 6.5. Let me verify that multiplication again. 6.5 can be thought of as 6 + 0.5. So 10*6 is 60, 10*0.5 is 5, so total is 65. Yeah, that seems right.

Alternatively, maybe I can do it another way. Let me convert 6.5 into a fraction. 6.5 is the same as 13/2. So then, 1/3 * 30 * 13/2. Let's compute that. 1/3 times 30 is 10, as before. Then 10 times 13/2. Which is (10/1) * (13/2) = (10*13)/2 = 130/2 = 65. Yep, same answer. Okay, so that checks out.

Alternatively, if I do all the multiplication at once. Let's see, 1/3 * 30 * 6.5. Let me compute 30 * 6.5 first. 30 times 6 is 180, and 30 times 0.5 is 15, so adding those gives 180 + 15 = 195. Then 195 divided by 3 is... 195 divided by 3. Let me compute that. 3 goes into 19 five times (15), remainder 4. 3 goes into 4 once (3), remainder 1. Then 3 goes into 1 zero times with a decimal. So 5.0... Wait, no, maybe I should do it differently. 3 times 65 is 195. Because 3*60 is 180, 3*5 is 15. So 180+15=195. Therefore, 195 divided by 3 is 65. Yep, same result.

So all three methods give me 65. That seems pretty solid. So the volume should be 65 cubic units.

Wait, hold on, but let me just make sure that I didn't mix up any numbers. The base area is 30, height is 6.5. So no, I didn't confuse radius or diameter or anything? Wait, the formula is given as V = 1/3 * B * h, where B is the area of the base. So they already gave me the area of the base, so I don't need to calculate that from radius or anything. So that's straightforward. So if B is 30, then I just use that. So I think that's correct.

Alternatively, if someone thought that B was the radius, but no, the problem says "area of the base," so that's definitely 30. So no problem there. The height is given as 6.5 units, which is a decimal, but that's okay. So the multiplication still works out. So I think that 65 is the correct answer here.

Just to recap, the formula is volume equals one-third times base area times height. Plug in the numbers, do the arithmetic, and get 65. All checks confirm that. So I think that's the answer.

**Final Answer**
The volume of the cone is \boxed{65} cubic units.

To determine the volume of a cone, we use the formula:

$$
V = \frac{1}{3}Bh
$$

Where:
- $ V $ is the volume (in cubic units),
- $ B $ is the area of the base (in square units),
- $ h $ is the height (in units).

---

### Step 1: Identify the given values
From the problem, we are told:
- The area of the base, $ B = 30 $ square units,
- The height, $ h = 6.5 $ units.

---

### Step 2: Substitute the known values into the formula

$$
V = \frac{1}{3} \times 30 \times 6.5
$$

---

### Step 3: Simplify the expression step by step

First, multiply $ \frac{1}{3} \times 30 $:

$$
\frac{1}{3} \times 30 = 10
$$

Next, multiply the result by 6.5:

$$
10 \times 6.5 = 65
$$

---

### Step 4: Final Answer

$$
V = \boxed{65}
$$

So, the volume of the cone is **65 cubic units**.

Solution: We are given that $B = 30$ and $h = 6.5$ and asked to find $\frac{1}{3}Bh$.  We find that \[\frac{1}{3}Bh = \frac{1}{3}(30)(6.5) = (10)(6.5) = \boxed{65}.\]


Explanation:
Okay, so Louis and Jack are sharing a pie, and I need to figure out how much of the pie they ate altogether. Let me think... Louis ate 1/5 of the pie, and Jack ate 2/3 of the pie. Hmm, so I guess I need to add those two fractions together to find the total amount they ate.

But wait, adding fractions requires a common denominator, right? Let me recall... To add fractions, you have to have the same denominator. So, the denominators here are 5 and 3. What's the least common denominator for 5 and 3? Well, 5 times 3 is 15, and since they don't have any common factors other than 1, 15 must be the least common denominator.

Okay, so let me convert both fractions to have 15 as the denominator. Starting with Louis's part: 1/5. To convert 1/5 to a denominator of 15, I need to multiply both the numerator and the denominator by 3, because 5 times 3 is 15. So, 1 times 3 is 3, and 5 times 3 is 15. That makes Louis's portion 3/15 of the pie.

Now for Jack's portion: 2/3. To convert that to a denominator of 15, I need to multiply numerator and denominator by 5, since 3 times 5 is 15. So, 2 times 5 is 10, and 3 times 5 is 15. Therefore, Jack's portion is 10/15 of the pie.

Now I can add them together: 3/15 + 10/15. Since the denominators are the same, I just add the numerators. 3 + 10 is 13, so that would be 13/15.

Wait a second, let me check if that makes sense. If Louis ate 1/5, which is 3/15, and Jack ate 2/3, which is 10/15, adding them gives 13/15. So together they ate 13/15 of the pie.

But let me think again—does that make sense? If the whole pie is 1, then 1/5 is 0.2 and 2/3 is approximately 0.666... Adding those together gives roughly 0.866..., and 13/15 is approximately 0.866..., so the decimal check works out.

Alternatively, maybe I can convert the fractions to decimals and add them. Let me try that.

1 divided by 5 is 0.2.

2 divided by 3 is approximately 0.666666...

Adding 0.2 and 0.666... gives 0.866..., which is the same as 13/15 because 13 divided by 15 is equal to 0.866... So that's consistent.

Therefore, I think the answer is 13/15.

But just to make sure I didn't make any mistakes in my fraction addition. Let me go through it again.

Original fractions: 1/5 and 2/3.

Find the common denominator: 15.

Convert 1/5 to 3/15 (multiply numerator and denominator by 3).

Convert 2/3 to 10/15 (multiply numerator and denominator by 5).

Add numerators: 3 + 10 = 13.

Result: 13/15.

Yes, that seems correct.

Alternatively, if I didn't convert to a common denominator, could I do something else? Maybe cross-multiplying? Wait, no, cross-multiplying is for multiplying fractions, not adding.

So, adding fractions must have common denominators.

Another way to think about it: the pie is divided into parts. Louis eats 1/5, so if I imagine the pie divided into 15 equal parts, Louis eats 3 parts (since 1/5 is 3/15) and Jack eats 10 parts (since 2/3 is 10/15). Together, they eat 3 + 10 = 13 parts out of 15.

So 13/15 is the total.

Is there any chance that they ate more than the whole pie? Let me check. If they ate 13/15, that's less than 1, which is the whole pie. So that makes sense.

Alternatively, if they had eaten more than the whole pie, that would be impossible, but here 13/15 is less than 1.

So, yes, that seems to be the correct answer.

Therefore, the amount of the pie that Louis and Jack eat altogether is 13/15.

**Final Answer**
The total amount of the pie that Louis and Jack eat altogether is \boxed{\dfrac{13}{15}}.

To determine how much of the pie Louis and Jack ate altogether, we begin by adding their individual portions:
- Louis eats $ \frac{1}{5} $ of the pie, and
- Jack eats $ \frac{2}{3} $ of the pie.

---

### Step 1: Identify the Common Denominator

The denominators of the two fractions are 5 and 3. To add these fractions, we need a common denominator. The least common denominator (LCD) of 5 and 3 is **15**.

---

### Step 2: Convert Fractions to Equivalent Denominators

We express each fraction with a denominator of 15:

- $ \frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15} $
- $ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} $

---

### Step 3: Add the Fractions

Now that both fractions have the same denominator, we add their numerators:

$$
\frac{3}{15} + \frac{10}{15} = \frac{3 + 10}{15} = \frac{13}{15}
$$

---

### Step 4: Interpret the Result

The result, $ \frac{13}{15} $, means that Louis and Jack together ate **13 out of 15 equal parts** of the pie. This is less than the whole pie (which is 1, or $ \frac{15}{15} $), and makes sense in the context of the problem.

---

### Final Answer

$$
\boxed{\dfrac{13}{15}}
$$

[Assistant]
Explanation:
Okay, so Louis and Jack are sharing a pie, and I need to figure out how much of the pie they ate altogether. Let me think... Louis ate 1/5 of the pie, and Jack ate 2/3 of the pie. Hmm, so I guess I need to add those two fractions together to find the total amount they ate.

But wait, adding fractions requires a common denominator, right? Let me recall... To add fractions, you have to have the same denominator. So, the denominators here are 5 and 3. What's the least common denominator for 5 and 3? Well, 5 times 3 is 15, and since they don't have any common factors other than 1, 15 must be the least common denominator.

Okay, so let me convert both fractions to have 15 as the denominator. Starting with Louis's part: 1/5. To convert 1/5 to a denominator of 15, I need to multiply both the numerator and the denominator by 3, because 5 times 3 is 15. So, 1 times 3 is 3, and 5 times 3 is 15. That makes Louis's portion 3/15 of the pie.

Now for Jack's portion: 2/3. To convert that to a denominator of 15, I need to multiply numerator and denominator by 5, since 3 times 5 is 15. So, 2 times 5 is 10, and 3 times 5 is 15. Therefore, Jack's portion is 10/15 of the pie.

Now I can add them together: 3/15 + 10/15. Since the denominators are the same, I just add the numerators. 3 + 10 is 13, so that would be 13/15.

Wait a second, let me check if that makes sense. If Louis ate 1/5, which is 3/15, and Jack ate 2/3, which is 10/15, adding them gives 13/15. So together they ate 13/15 of the pie.

But let me think again—does that make sense? If the whole pie is 1, then 1/5 is 0.2 and 2/3 is approximately 0.666... Adding those together gives roughly 0.866..., and 13/15 is approximately 0.866..., so the decimal check works out.

Alternatively, maybe I can convert the fractions to decimals and add them. Let me try that.

1 divided by 5 is 0.2.

2 divided by 3 is approximately 0.666666...

Adding 0.2 and 0.666... gives 0.866..., which is the same as 13/15 because 13 divided by 15 is equal to 0.866... So that's consistent.

Therefore, I think the answer is 13/15.

But just to make sure I didn't make any mistakes in my fraction addition. Let me go through it again.

Original fractions: 1/5 and 2/3.

Find the common denominator: 15.

Convert 1/5 to 3/15 (multiply numerator and denominator by 3).

Convert 2/3 to 10/15 (multiply numerator and denominator by 5).

Add numerators: 3 + 10 = 13.

Result: 13/15.

Yes, that seems correct.

Alternatively, if I didn't convert to a common denominator, could I do something else? Maybe cross-multiplying? Wait, no, cross-multiplying is for multiplying fractions, not adding.

So, adding fractions must have common denominators.

Another way to think about it: the pie is divided into parts. Louis eats 1/5, so if I imagine the pie divided into 15 equal parts, Louis eats 3 parts (since 1/5 is 3/15) and Jack eats 10 parts (since 2/3 is 10/15). Together, they eat 3 + 10 = 13 parts out of 15.

So 13/15 is the total.

Is there any chance that they ate more than the whole pie? Let me check. If they ate 13/15, that's less than 1, which is the whole pie. So that makes sense.

Alternatively, if they had eaten more than the whole pie, that would be impossible, but here 13/15 is less than 1.

So, yes, that seems to be the correct answer.

Therefore, the amount of the pie that Louis and Jack eat altogether is 13/15.

**Final Answer**
The total amount of the pie that Louis and Jack eat altogether is \boxed{\dfrac{13}{15}}.

To determine how much of the pie Louis and Jack ate altogether, we begin by adding their individual portions:
- Louis eats $ \frac{1}{5} $ of the pie, and
- Jack eats $ \frac{2}{3} $ of the pie.

---

### Step 1: Identify the Common Denominator

The denominators of the two fractions are 5 and 3. To add these fractions, we need a common denominator. The least common denominator (LCD) of 5 and 3 is **15**.

---

### Step 2: Convert Fractions to Equivalent Denominators

We express each fraction with a denominator of 15:

- $ \frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15} $
- $ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} $

---

### Step 3: Add the Fractions

Now that both fractions have the same denominator, we add their numerators:

$$
\frac{3}{15} + \frac{10}{15} = \frac{3 + 10}{15} = \frac{13}{15}
$$

---

### Step 4: Interpret the Result

The result, $ \frac{13}{15} $, means that Louis and Jack together ate **13 out of 15 equal parts** of the pie. This is less than the whole pie (which is 1, or $ \frac{15}{15} $), and makes sense in the context of the problem.

---

### Final Answer

$$
\boxed{\dfrac{13}{15}}
$$

Solution: The denominators $5$ and $3$ have a common multiple of $15$.  We use this to write $\frac{1}{5}\cdot \frac{3}{3}=\frac{3}{15}$ and $\frac{2}{3}\cdot \frac{5}{5}=\frac{10}{15}$.  Then, we can add the fractions by adding the numerators and keeping the denominator.  We have $$\frac{1}{5}+\frac{2}{3}=\frac{3}{15}+\frac{10}{15}=\frac{3+10}{15}=\boxed{\frac{13}{15}}.$$


Explanation:
Okay, so I need to find the sixth term of an arithmetic sequence where the fourth term is 200 and the eighth term is 500. Let me remember what an arithmetic sequence is. It's a sequence where each term increases by a constant difference. So, if I can figure out what that common difference is, I can find any term in the sequence.

First, let me recall the formula for the nth term of an arithmetic sequence. I think it's something like a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. Yeah, that sounds right. So, if I can write equations for the fourth and eighth terms, I can solve for the unknowns, which are a_1 and d. Then, once I have those, I can find the sixth term.

Let me write down what I know. The fourth term, which is a_4, is 200. Using the formula, that would be a_1 + (4 - 1)d = a_1 + 3d = 200. Similarly, the eighth term, a_8, is 500. So, that would be a_1 + (8 - 1)d = a_1 + 7d = 500.

So now I have two equations:

1. a_1 + 3d = 200
2. a_1 + 7d = 500

Hmm, I can solve this system of equations to find a_1 and d. Let me subtract the first equation from the second equation to eliminate a_1.

So, (a_1 + 7d) - (a_1 + 3d) = 500 - 200

Simplifying the left side: a_1 - a_1 + 7d - 3d = 4d

And the right side is 300. So, 4d = 300. Therefore, d = 300 / 4 = 75.

Oh, so the common difference is 75. Let me check that. If d is 75, then let me plug back into one of the equations to find a_1. Let's take the first equation: a_1 + 3*75 = 200. 3*75 is 225. So, a_1 = 200 - 225 = -25. Wait, so the first term is -25? Let me check with the second equation to make sure.

Second equation: a_1 + 7d = -25 + 7*75. Let me calculate 7*75. 7*70 is 490, and 7*5 is 35, so total is 525. Then -25 + 525 is 500. Yes, that matches. So that checks out.

Okay, so now I know that the first term is -25 and the common difference is 75. Now I need to find the sixth term. Let's use the formula again.

The sixth term, a_6, is a_1 + (6 - 1)d = a_1 + 5d. Plugging in the known values: -25 + 5*75. Let me calculate 5*75. 5*70 is 350, 5*5 is 25, so total is 375. Then, -25 + 375 is 350.

Wait, so the sixth term is 350? Let me verify that another way. Let me list out the terms from the fourth term onward.

We know the fourth term is 200. Since each term increases by 75, then the fifth term would be 200 + 75 = 275. The sixth term would be 275 + 75 = 350. Then the seventh term is 350 + 75 = 425, and the eighth term is 425 + 75 = 500. Yep, that matches the given eighth term. So that seems right.

Alternatively, maybe I can think in terms of the difference between the fourth and eighth term. The eighth term is four terms after the fourth term. So, the difference between term 8 and term 4 is 4d, which we already found to be 300. Which is 500 - 200 = 300, so that's consistent with 4d=300, so d=75.

So, since the sixth term is two terms after the fourth term, then the sixth term would be the fourth term plus 2d. So, 200 + 2*75 = 200 + 150 = 350. That also gives me 350.

So, all methods lead me to the sixth term being 350. Therefore, I think that's the correct answer.

But just to make sure I didn't make any calculation errors. Let me go through the equations again.

Original equations:

a_1 + 3d = 200

a_1 + 7d = 500

Subtract first from second: 4d = 300 => d=75. Then a_1 = 200 - 3*75 = 200 - 225 = -25. Correct.

Then sixth term: a_6 = a_1 + 5d = -25 + 5*75. Let me compute 5*75 again. 75*5: 70*5=350, 5*5=25, so 350 +25=375. Then -25 + 375 = 350. Correct.

Alternatively, since term 4 is 200, term 5 is 200 +75=275, term 6 is 275 +75=350. Yep.

Alternatively, term 8 is term 4 + 4d = 200 + 300 = 500. Correct.

So, seems consistent. So unless I made a mistake in the arithmetic, which I don't think I did, the sixth term is 350.

I think that's solid. So the answer should be 350.

**Final Answer**
The sixth term is \boxed{350}.

To find the sixth term of an arithmetic sequence, we begin with the key properties of such a sequence. In an arithmetic progression, each term increases by a fixed difference $ d $, and the general formula for the $ n $-th term is:

$$
a_n = a_1 + (n - 1)d
$$

We are given:

- The fourth term: $ a_4 = 200 $, so:
  $$
  a_1 + 3d = 200 \quad \text{(1)}
  $$

- The eighth term: $ a_8 = 500 $, so:
  $$
  a_1 + 7d = 500 \quad \text{(2)}
  $$

---

### Step 1: Eliminate $ a_1 $ to solve for $ d $

Subtract equation (1) from equation (2):

$$
(a_1 + 7d) - (a_1 + 3d) = 500 - 200
$$
$$
4d = 300 \Rightarrow d = \frac{300}{4} = 75
$$

---

### Step 2: Solve for $ a_1 $

Plug $ d = 75 $ back into equation (1):

$$
a_1 + 3(75) = 200 \Rightarrow a_1 + 225 = 200 \Rightarrow a_1 = 200 - 225 = -25
$$

---

### Step 3: Find the Sixth Term

Use the general formula with $ n = 6 $:

$$
a_6 = a_1 + 5d = -25 + 5(75) = -25 + 375 = 350
$$

---

### Final Answer

$$
\boxed{350}
$$

[Assistant]
Explanation:
Okay, so I need to find the sixth term of an arithmetic sequence where the fourth term is 200 and the eighth term is 500. Let me remember what an arithmetic sequence is. It's a sequence where each term increases by a constant difference. So, if I can figure out what that common difference is, I can find any term in the sequence.

First, let me recall the formula for the nth term of an arithmetic sequence. I think it's something like a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. Yeah, that sounds right. So, if I can write equations for the fourth and eighth terms, I can solve for the unknowns, which are a_1 and d. Then, once I have those, I can find the sixth term.

Let me write down what I know. The fourth term, which is a_4, is 200. Using the formula, that would be a_1 + (4 - 1)d = a_1 + 3d = 200. Similarly, the eighth term, a_8, is 500. So, that would be a_1 + (8 - 1)d = a_1 + 7d = 500.

So now I have two equations:

1. a_1 + 3d = 200
2. a_1 + 7d = 500

Hmm, I can solve this system of equations to find a_1 and d. Let me subtract the first equation from the second equation to eliminate a_1.

So, (a_1 + 7d) - (a_1 + 3d) = 500 - 200

Simplifying the left side: a_1 - a_1 + 7d - 3d = 4d

And the right side is 300. So, 4d = 300. Therefore, d = 300 / 4 = 75.

Oh, so the common difference is 75. Let me check that. If d is 75, then let me plug back into one of the equations to find a_1. Let's take the first equation: a_1 + 3*75 = 200. 3*75 is 225. So, a_1 = 200 - 225 = -25. Wait, so the first term is -25? Let me check with the second equation to make sure.

Second equation: a_1 + 7d = -25 + 7*75. Let me calculate 7*75. 7*70 is 490, and 7*5 is 35, so total is 525. Then -25 + 525 is 500. Yes, that matches. So that checks out.

Okay, so now I know that the first term is -25 and the common difference is 75. Now I need to find the sixth term. Let's use the formula again.

The sixth term, a_6, is a_1 + (6 - 1)d = a_1 + 5d. Plugging in the known values: -25 + 5*75. Let me calculate 5*75. 5*70 is 350, 5*5 is 25, so total is 375. Then, -25 + 375 is 350.

Wait, so the sixth term is 350? Let me verify that another way. Let me list out the terms from the fourth term onward.

We know the fourth term is 200. Since each term increases by 75, then the fifth term would be 200 + 75 = 275. The sixth term would be 275 + 75 = 350. Then the seventh term is 350 + 75 = 425, and the eighth term is 425 + 75 = 500. Yep, that matches the given eighth term. So that seems right.

Alternatively, maybe I can think in terms of the difference between the fourth and eighth term. The eighth term is four terms after the fourth term. So, the difference between term 8 and term 4 is 4d, which we already found to be 300. Which is 500 - 200 = 300, so that's consistent with 4d=300, so d=75.

So, since the sixth term is two terms after the fourth term, then the sixth term would be the fourth term plus 2d. So, 200 + 2*75 = 200 + 150 = 350. That also gives me 350.

So, all methods lead me to the sixth term being 350. Therefore, I think that's the correct answer.

But just to make sure I didn't make any calculation errors. Let me go through the equations again.

Original equations:

a_1 + 3d = 200

a_1 + 7d = 500

Subtract first from second: 4d = 300 => d=75. Then a_1 = 200 - 3*75 = 200 - 225 = -25. Correct.

Then sixth term: a_6 = a_1 + 5d = -25 + 5*75. Let me compute 5*75 again. 75*5: 70*5=350, 5*5=25, so 350 +25=375. Then -25 + 375 = 350. Correct.

Alternatively, since term 4 is 200, term 5 is 200 +75=275, term 6 is 275 +75=350. Yep.

Alternatively, term 8 is term 4 + 4d = 200 + 300 = 500. Correct.

So, seems consistent. So unless I made a mistake in the arithmetic, which I don't think I did, the sixth term is 350.

I think that's solid. So the answer should be 350.

**Final Answer**
The sixth term is \boxed{350}.

To find the sixth term of an arithmetic sequence, we begin with the key properties of such a sequence. In an arithmetic progression, each term increases by a fixed difference $ d $, and the general formula for the $ n $-th term is:

$$
a_n = a_1 + (n - 1)d
$$

We are given:

- The fourth term: $ a_4 = 200 $, so:
  $$
  a_1 + 3d = 200 \quad \text{(1)}
  $$

- The eighth term: $ a_8 = 500 $, so:
  $$
  a_1 + 7d = 500 \quad \text{(2)}
  $$

---

### Step 1: Eliminate $ a_1 $ to solve for $ d $

Subtract equation (1) from equation (2):

$$
(a_1 + 7d) - (a_1 + 3d) = 500 - 200
$$
$$
4d = 300 \Rightarrow d = \frac{300}{4} = 75
$$

---

### Step 2: Solve for $ a_1 $

Plug $ d = 75 $ back into equation (1):

$$
a_1 + 3(75) = 200 \Rightarrow a_1 + 225 = 200 \Rightarrow a_1 = 200 - 225 = -25
$$

---

### Step 3: Find the Sixth Term

Use the general formula with $ n = 6 $:

$$
a_6 = a_1 + 5d = -25 + 5(75) = -25 + 375 = 350
$$

---

### Final Answer

$$
\boxed{350}
$$

Solution: The sixth term is exactly halfway between the fourth and the eighth in the arithmetic sequence, so it is the average of the two terms. Therefore, the sixth term is $(200 + 500)/2 = \boxed{350}$.  We also could have found the common difference by noting that there are four steps between the fourth term and the eighth term.  So, if $d$ is the common difference, we have $4d = 500-200 = 300$. Therefore, we find $d=75$.  The sixth term is two steps after the fourth, or $200 + 2d = \boxed{350}$.


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

First, I remember that when you raise a power to another power, you multiply the exponents. So, (a^m)^n is the same as a^(m*n). Let me verify that. Yeah, that seems right. For example, (2^3)^2 would be 2^(3*2) = 2^6, which is 64. If I calculate it directly, (2^3)^2 is 8 squared, which is also 64. Okay, so that rule holds.

Applying that to the problem here: (3^k)^6. According to the rule, this should be 3^(k*6). So, the left side simplifies to 3^(6k). The equation given is 3^(6k) = 3^6.

Now, if the bases are the same and the expressions are equal, then their exponents must be equal as well. That is, if a^b = a^c and a ≠ 0 or 1, then b must equal c. Since 3 is not 0 or 1, this rule applies here. Therefore, 6k must equal 6.

So, solving for k, I can divide both sides of the equation 6k = 6 by 6. That gives k = 6 / 6, which simplifies to k = 1.

Wait, let me double-check. If k is 1, then (3^1)^6 is 3^6, which is exactly the right-hand side of the equation. So that works.

Is there any other possible value of k that could satisfy this equation? Let me think. Suppose k was something else, say 0. Let's test k=0. Then (3^0)^6 is (1)^6 = 1, but 3^6 is 729. So that's not equal. What if k was 2? Then (3^2)^6 is 9^6, which is way bigger than 3^6. So that doesn't work. If k was negative, like -1, then (3^-1)^6 = (1/3)^6 = 1/729, which is not equal to 3^6. So no, it seems like only k=1 works.

Another way to look at it: Let's take the logarithm of both sides. Taking natural log on both sides: ln(3^(6k)) = ln(3^6). Then, using the logarithm power rule, that becomes 6k * ln(3) = 6 * ln(3). Dividing both sides by ln(3) (which is not zero), we get 6k = 6, so again k=1.

Alternatively, maybe using exponents properties. Since both sides have base 3, we can set the exponents equal. Therefore, 6k = 6. Solving for k gives k=1.

So all methods lead to k=1. Therefore, I think the answer is 1.

But let me check if there are any special cases or if I made an incorrect assumption. The original equation is (3^k)^6 = 3^6. If k was a fraction or something? For example, if k was 1/6. Let's test that. Then (3^(1/6))^6 = 3^( (1/6)*6 ) = 3^1 = 3, but the right side is 3^6. So 3 vs 3^6. Not equal. So no.

Alternatively, if k was 6, then (3^6)^6 = 3^(36), which is way bigger than 3^6. So no.

What if k was something else? Let me think. Suppose k is 1, then as we saw, it works. If k is any other number, the exponent 6k will not equal 6 unless k is 1.

Wait, unless... Wait, if 6k is congruent to 6 modulo something? But exponents with base 3 aren't periodic in the real numbers, unless you consider complex numbers or something. But the problem seems to be in real numbers. So unless there's some other trick here.

Wait, the equation is 3^(6k) = 3^6. So unless the exponents are equal, which they only are if 6k = 6, since 3 is a positive real number not equal to 1. Therefore, the only solution is k=1.

Alternatively, if we thought about the equation 3^(6k) - 3^6 = 0. Factor out 3^6: 3^6(3^(6k - 6) - 1) = 0. Since 3^6 is never zero, we have 3^(6k -6) -1 = 0. So 3^(6k -6) =1. Then 3^(anything) is 1 only when the exponent is 0. So 6k -6=0. Therefore, 6k=6, k=1.

So again, same result.

Therefore, all possible ways of looking at the problem lead to k=1.

So I think that's the answer. Let me just confirm once more with substitution. Original equation: (3^k)^6 = 3^6. If k=1, left side is (3^1)^6 = 3^6. Which equals the right side. Perfect.

No other solutions. Therefore, k is 1.

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

To solve the equation:

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

we begin by applying the **power of a power rule** of exponents, which states:

$$
(a^m)^n = a^{mn}
$$

---

### Step 1: Apply the Rule

Apply this to the left-hand side:

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

Now set this equal to the right-hand side:

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

---

### Step 2: Use the One-to-One Property of Exponential Functions

When the bases are the same and the expressions are equal, their **exponents must be equal**:

$$
6k = 6
$$

---

### Step 3: Solve for $ k $

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

---

### Step 4: Verify the Solution

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

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

$$
3^6 = 3^6
$$

This confirms that 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 exponents work here.

First, I remember that when you raise a power to another power, you multiply the exponents. So, (a^m)^n is the same as a^(m*n). Let me verify that. Yeah, that seems right. For example, (2^3)^2 would be 2^(3*2) = 2^6, which is 64. If I calculate it directly, (2^3)^2 is 8 squared, which is also 64. Okay, so that rule holds.

Applying that to the problem here: (3^k)^6. According to the rule, this should be 3^(k*6). So, the left side simplifies to 3^(6k). The equation given is 3^(6k) = 3^6.

Now, if the bases are the same and the expressions are equal, then their exponents must be equal as well. That is, if a^b = a^c and a ≠ 0 or 1, then b must equal c. Since 3 is not 0 or 1, this rule applies here. Therefore, 6k must equal 6.

So, solving for k, I can divide both sides of the equation 6k = 6 by 6. That gives k = 6 / 6, which simplifies to k = 1.

Wait, let me double-check. If k is 1, then (3^1)^6 is 3^6, which is exactly the right-hand side of the equation. So that works.

Is there any other possible value of k that could satisfy this equation? Let me think. Suppose k was something else, say 0. Let's test k=0. Then (3^0)^6 is (1)^6 = 1, but 3^6 is 729. So that's not equal. What if k was 2? Then (3^2)^6 is 9^6, which is way bigger than 3^6. So that doesn't work. If k was negative, like -1, then (3^-1)^6 = (1/3)^6 = 1/729, which is not equal to 3^6. So no, it seems like only k=1 works.

Another way to look at it: Let's take the logarithm of both sides. Taking natural log on both sides: ln(3^(6k)) = ln(3^6). Then, using the logarithm power rule, that becomes 6k * ln(3) = 6 * ln(3). Dividing both sides by ln(3) (which is not zero), we get 6k = 6, so again k=1.

Alternatively, maybe using exponents properties. Since both sides have base 3, we can set the exponents equal. Therefore, 6k = 6. Solving for k gives k=1.

So all methods lead to k=1. Therefore, I think the answer is 1.

But let me check if there are any special cases or if I made an incorrect assumption. The original equation is (3^k)^6 = 3^6. If k was a fraction or something? For example, if k was 1/6. Let's test that. Then (3^(1/6))^6 = 3^( (1/6)*6 ) = 3^1 = 3, but the right side is 3^6. So 3 vs 3^6. Not equal. So no.

Alternatively, if k was 6, then (3^6)^6 = 3^(36), which is way bigger than 3^6. So no.

What if k was something else? Let me think. Suppose k is 1, then as we saw, it works. If k is any other number, the exponent 6k will not equal 6 unless k is 1.

Wait, unless... Wait, if 6k is congruent to 6 modulo something? But exponents with base 3 aren't periodic in the real numbers, unless you consider complex numbers or something. But the problem seems to be in real numbers. So unless there's some other trick here.

Wait, the equation is 3^(6k) = 3^6. So unless the exponents are equal, which they only are if 6k = 6, since 3 is a positive real number not equal to 1. Therefore, the only solution is k=1.

Alternatively, if we thought about the equation 3^(6k) - 3^6 = 0. Factor out 3^6: 3^6(3^(6k - 6) - 1) = 0. Since 3^6 is never zero, we have 3^(6k -6) -1 = 0. So 3^(6k -6) =1. Then 3^(anything) is 1 only when the exponent is 0. So 6k -6=0. Therefore, 6k=6, k=1.

So again, same result.

Therefore, all possible ways of looking at the problem lead to k=1.

So I think that's the answer. Let me just confirm once more with substitution. Original equation: (3^k)^6 = 3^6. If k=1, left side is (3^1)^6 = 3^6. Which equals the right side. Perfect.

No other solutions. Therefore, k is 1.

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

To solve the equation:

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

we begin by applying the **power of a power rule** of exponents, which states:

$$
(a^m)^n = a^{mn}
$$

---

### Step 1: Apply the Rule

Apply this to the left-hand side:

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

Now set this equal to the right-hand side:

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

---

### Step 2: Use the One-to-One Property of Exponential Functions

When the bases are the same and the expressions are equal, their **exponents must be equal**:

$$
6k = 6
$$

---

### Step 3: Solve for $ k $

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

---

### Step 4: Verify the Solution

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

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

$$
3^6 = 3^6
$$

This confirms that 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, let's see. Hillary has eleven coins, all dimes and nickels, and they add up to 75 cents. I need to figure out how many nickels she has. Hmm, okay, so I guess I need to set up some equations here. Let me think.

First, let me recall the values of the coins. A nickel is 5 cents, and a dime is 10 cents. So, if I let the number of nickels be some variable, maybe n, and the number of dimes be d, then I can write two equations based on the information given.

The first equation should be about the total number of coins. She has eleven coins in total. So, the number of nickels plus the number of dimes equals 11. That would be:

n + d = 11

Okay, that seems straightforward. Now, the second equation is about the total value of the coins. Each nickel is 5 cents, so the total value from nickels would be 5n cents. Similarly, each dime is 10 cents, so the total value from dimes would be 10d cents. The total of these should be 75 cents. So:

5n + 10d = 75

Alright, so now I have a system of two equations:

1. n + d = 11
2. 5n + 10d = 75

Now, I need to solve this system. Let me think about how to do that. Maybe substitution? Since the first equation is simple, I can solve for one variable in terms of the other and substitute into the second equation. Let me try solving the first equation for n.

From equation 1: n = 11 - d

Then, substitute this into equation 2:

5(11 - d) + 10d = 75

Okay, let me compute that. First, distribute the 5:

5*11 - 5d + 10d = 75

Which is 55 - 5d + 10d = 75

Combine like terms: 55 + 5d = 75

Then, subtract 55 from both sides:

5d = 75 - 55

Which is 5d = 20

Divide both sides by 5:

d = 4

So, there are 4 dimes. Then, since the total number of coins is 11, the number of nickels would be 11 - 4 = 7.

Wait, let me check if that's correct. Let me verify the total value.

If there are 7 nickels, that's 7*5 = 35 cents. And 4 dimes is 4*10 = 40 cents. Adding them together, 35 + 40 = 75 cents. Yes, that matches. And the total number of coins is 7 + 4 = 11. Perfect, that checks out.

But let me try solving it another way just to make sure I didn't make any mistakes. Maybe using elimination?

Starting again with the equations:

n + d = 11

5n + 10d = 75

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

5n + 5d = 55

Then subtract that from the second equation:

(5n + 10d) - (5n + 5d) = 75 - 55

Which simplifies to:

0n + 5d = 20 => 5d = 20 => d = 4

Same result as before. So, again, n = 11 - 4 = 7. So that seems consistent.

Alternatively, maybe I can think about it in terms of cents. Since there are 11 coins, and they add up to 75 cents. Let me think about how much each type contributes.

If all coins were nickels, the total 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 5 cents more than a nickel). So, how many replacements do we need to get 20 cents more?

20 divided by 5 is 4. So, replacing 4 nickels with dimes would add 20 cents. Therefore, there are 4 dimes and 11 - 4 = 7 nickels. That's another way to think about it, and it also gives the same answer.

So, all methods lead to 7 nickels. Therefore, I think the answer is 7.

Wait, just to ensure, let me try another approach. Let me let n be the number of nickels, so the number of dimes is 11 - n. Then total value is 5n + 10(11 - n) = 75.

So, let's compute:

5n + 110 - 10n = 75

Combine like terms: -5n + 110 = 75

Subtract 110: -5n = 75 - 110 => -5n = -35

Divide both sides by -5: n = (-35)/(-5) = 7. Yep, same answer. So that's consistent.

Therefore, regardless of the method, the number of nickels is 7. So, I think that's solid.

But just to make sure, let me check once more with actual numbers. If there are 7 nickels and 4 dimes:

Nickels: 7 * 5 = 35 cents

Dimes: 4 * 10 = 40 cents

Total: 35 + 40 = 75 cents. Perfect. And total coins: 7 + 4 = 11. Perfect. So, it's all correct.

Hmm, seems like there's no mistake here. Therefore, the answer should be 7 nickels.

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

To determine how many nickels Hillary has, we are given the following information:

- She has **11 coins total**, which are all either **nickels (5 cents)** or **dimes (10 cents)**.
- The **total value of the coins is 75 cents**.

---

### Step 1: Define Variables

Let:

- $ n $ = number of **nickels**
- $ d $ = number of **dimes**

From the problem, we can set up the following system of equations:

1. **Total number of coins:**
$$
n + d = 11
$$

2. **Total value of coins:**
$$
5n + 10d = 75
$$

---

### Step 2: Solve the System of Equations

We can solve this by substitution. From the first equation, solve for $ n $:

$$
n = 11 - d
$$

Substitute into the second equation:

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

Simplify:

$$
55 - 5d + 10d = 75
$$
$$
55 + 5d = 75
$$
$$
5d = 20
$$
$$
d = 4
$$

Now substitute back to find $ n $:

$$
n = 11 - d = 11 - 4 = 7
$$

---

### Step 3: Verify the Solution

- Nickels: $ 7 \times 5 = 35 $ cents
- Dimes: $ 4 \times 10 = 40 $ cents
- Total: $ 35 + 40 = 75 $ cents
- Total coins: $ 7 + 4 = 11 $

All conditions are satisfied.

---

### Final Answer

$$
\boxed{7}
$$

[Assistant]
Explanation:
Okay, let's see. Hillary has eleven coins, all dimes and nickels, and they add up to 75 cents. I need to figure out how many nickels she has. Hmm, okay, so I guess I need to set up some equations here. Let me think.

First, let me recall the values of the coins. A nickel is 5 cents, and a dime is 10 cents. So, if I let the number of nickels be some variable, maybe n, and the number of dimes be d, then I can write two equations based on the information given.

The first equation should be about the total number of coins. She has eleven coins in total. So, the number of nickels plus the number of dimes equals 11. That would be:

n + d = 11

Okay, that seems straightforward. Now, the second equation is about the total value of the coins. Each nickel is 5 cents, so the total value from nickels would be 5n cents. Similarly, each dime is 10 cents, so the total value from dimes would be 10d cents. The total of these should be 75 cents. So:

5n + 10d = 75

Alright, so now I have a system of two equations:

1. n + d = 11
2. 5n + 10d = 75

Now, I need to solve this system. Let me think about how to do that. Maybe substitution? Since the first equation is simple, I can solve for one variable in terms of the other and substitute into the second equation. Let me try solving the first equation for n.

From equation 1: n = 11 - d

Then, substitute this into equation 2:

5(11 - d) + 10d = 75

Okay, let me compute that. First, distribute the 5:

5*11 - 5d + 10d = 75

Which is 55 - 5d + 10d = 75

Combine like terms: 55 + 5d = 75

Then, subtract 55 from both sides:

5d = 75 - 55

Which is 5d = 20

Divide both sides by 5:

d = 4

So, there are 4 dimes. Then, since the total number of coins is 11, the number of nickels would be 11 - 4 = 7.

Wait, let me check if that's correct. Let me verify the total value.

If there are 7 nickels, that's 7*5 = 35 cents. And 4 dimes is 4*10 = 40 cents. Adding them together, 35 + 40 = 75 cents. Yes, that matches. And the total number of coins is 7 + 4 = 11. Perfect, that checks out.

But let me try solving it another way just to make sure I didn't make any mistakes. Maybe using elimination?

Starting again with the equations:

n + d = 11

5n + 10d = 75

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

5n + 5d = 55

Then subtract that from the second equation:

(5n + 10d) - (5n + 5d) = 75 - 55

Which simplifies to:

0n + 5d = 20 => 5d = 20 => d = 4

Same result as before. So, again, n = 11 - 4 = 7. So that seems consistent.

Alternatively, maybe I can think about it in terms of cents. Since there are 11 coins, and they add up to 75 cents. Let me think about how much each type contributes.

If all coins were nickels, the total 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 5 cents more than a nickel). So, how many replacements do we need to get 20 cents more?

20 divided by 5 is 4. So, replacing 4 nickels with dimes would add 20 cents. Therefore, there are 4 dimes and 11 - 4 = 7 nickels. That's another way to think about it, and it also gives the same answer.

So, all methods lead to 7 nickels. Therefore, I think the answer is 7.

Wait, just to ensure, let me try another approach. Let me let n be the number of nickels, so the number of dimes is 11 - n. Then total value is 5n + 10(11 - n) = 75.

So, let's compute:

5n + 110 - 10n = 75

Combine like terms: -5n + 110 = 75

Subtract 110: -5n = 75 - 110 => -5n = -35

Divide both sides by -5: n = (-35)/(-5) = 7. Yep, same answer. So that's consistent.

Therefore, regardless of the method, the number of nickels is 7. So, I think that's solid.

But just to make sure, let me check once more with actual numbers. If there are 7 nickels and 4 dimes:

Nickels: 7 * 5 = 35 cents

Dimes: 4 * 10 = 40 cents

Total: 35 + 40 = 75 cents. Perfect. And total coins: 7 + 4 = 11. Perfect. So, it's all correct.

Hmm, seems like there's no mistake here. Therefore, the answer should be 7 nickels.

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

To determine how many nickels Hillary has, we are given the following information:

- She has **11 coins total**, which are all either **nickels (5 cents)** or **dimes (10 cents)**.
- The **total value of the coins is 75 cents**.

---

### Step 1: Define Variables

Let:

- $ n $ = number of **nickels**
- $ d $ = number of **dimes**

From the problem, we can set up the following system of equations:

1. **Total number of coins:**
$$
n + d = 11
$$

2. **Total value of coins:**
$$
5n + 10d = 75
$$

---

### Step 2: Solve the System of Equations

We can solve this by substitution. From the first equation, solve for $ n $:

$$
n = 11 - d
$$

Substitute into the second equation:

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

Simplify:

$$
55 - 5d + 10d = 75
$$
$$
55 + 5d = 75
$$
$$
5d = 20
$$
$$
d = 4
$$

Now substitute back to find $ n $:

$$
n = 11 - d = 11 - 4 = 7
$$

---

### Step 3: Verify the Solution

- Nickels: $ 7 \times 5 = 35 $ cents
- Dimes: $ 4 \times 10 = 40 $ cents
- Total: $ 35 + 40 = 75 $ cents
- Total coins: $ 7 + 4 = 11 $

All conditions are satisfied.

---

### 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, let me try to solve this problem. The equation given is 441 + 2(21)(19) + 361 = x. I need to find the value of x. Let me take it step by step.

First, let me write down the equation again to make sure I have it right:

441 + 2*(21)*(19) + 361 = x.

Alright, so there are three terms here: 441, then 2 times 21 times 19, and then 361. I need to calculate each part and then add them together.

Starting with the first term, 441. That's straightforward. I know that 21 squared is 441 because 21*21 is 441. Let me check that: 20*20 is 400, plus 20 + 20 + 1 is 441. Yep, that's right. So 441 is 21².

Then the second term is 2*(21)*(19). Hmm, let me think about how to compute that. Maybe I can simplify this part first. Let me recall that 2*21*19 can be thought of as 2*21*19. Alternatively, since 21 and 19 are both numbers close to 20, maybe there's a way to simplify this using some algebraic identity?

Wait, I remember that (a + b)² = a² + 2ab + b². Let me check if this problem resembles that formula. Let me see. If I consider 21 and 19 as numbers around 20, perhaps?

Wait, 21 is 20 + 1 and 19 is 20 - 1. So, if I let a = 20, then 21 = a + 1 and 19 = a - 1. Then, if I compute (a + 1)² + 2*(a + 1)*(a - 1) + (a - 1)², does that relate to something?

Wait, maybe not exactly. Let me think again. The original problem is 441 + 2*(21)*(19) + 361. Let me note that 441 is 21², and 361 is 19², because 19*19 is 361. Let me check that: 20*20 is 400, minus 20 + 20 - 1 is 400 - 39 = 361? Wait, no. Actually, 19*19 is 361. Let me calculate 20*20 - 2*20 + 1 = 400 - 40 + 1 = 361. Yes, that's correct. So 361 is 19². Therefore, the entire expression is 21² + 2*(21)*(19) + 19².

Oh! Wait a second, that looks exactly like the expansion of (a + b)² where a is 21 and b is 19. Let me recall that (a + b)² = a² + 2ab + b². So in this case, if we let a = 21 and b = 19, then the entire expression is (21 + 19)². Let me compute that.

21 + 19 is 40. Therefore, 40 squared is 1600. Therefore, x = 1600. Wait, so is that correct? Let me verify by computing each term separately to make sure I didn't make a mistake.

First, compute 441 + 2*(21)*(19) + 361.

Compute 2*(21)*(19). Let me compute 21*19 first. 21*19. Hmm, 20*19 is 380, and 1*19 is 19, so 380 + 19 = 399. Then multiply by 2: 399*2 = 798.

So the second term is 798. Then add 441 and 361. Let me add them up step by step:

First, 441 + 798. Let me compute that. 400 + 700 = 1100. 41 + 98 = 139. Therefore, total is 1100 + 139 = 1239.

Then add 361 to 1239. 1239 + 361. Let me break that down. 1239 + 300 = 1539, then +61 = 1539 + 61 = 1600. Okay, that matches. So x = 1600.

But wait, let me check again using the formula. If it's (21 + 19)^2, which is 40^2, that's 1600. So both methods give me the same result. So that seems solid. So perhaps the answer is 1600.

But let me check once more just to be thorough. Maybe I made a mistake when calculating 21*19. Let me compute 21*19 again. 21*19 can also be calculated as (20 + 1)*(20 - 1) = 20² - 1² = 400 - 1 = 399. Oh, that's another way to compute it. So that's 399. Multiply by 2: 798. Then 441 + 798 + 361.

Alternatively, maybe adding 441 + 361 first? Let me try that. 441 + 361. 400 + 300 = 700, 41 + 61 = 102, so total is 700 + 102 = 802. Then add 798: 802 + 798. 800 + 798 = 1598, plus 2 is 1600. Yep, same result.

Therefore, regardless of the method, the answer is 1600. So x = 1600.

Alternatively, if I didn't notice the perfect square, I could have just calculated each term step by step. But recognizing the pattern of the expansion of (a + b)^2 makes it easier.

But just to make sure there is no trick here. The original problem is 441 + 2*(21)*(19) + 361. Let me check if there's any possibility of misreading the problem. For example, maybe it was supposed to be 441 + 2*21*19 + 361. Which is exactly what it is. So no, there's no ambiguity there. The parentheses are there to indicate that the multiplication happens before addition. So yes, the calculation is correct.

Alternatively, if someone didn't recognize the perfect square, they could have done the arithmetic step by step, as I did above, and still arrived at the same answer.

Therefore, I'm confident that x equals 1600.

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

To solve the equation:

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

we can approach it in two ways: either by directly evaluating each term or by recognizing a pattern that simplifies the computation.

---

### **Step-by-Step Evaluation**

Let's first compute each component of the expression:

1. **First term**:
   $ 441 = 21^2 $

2. **Second term**:
   $ 2 \cdot 21 \cdot 19 = 2 \cdot (21 \cdot 19) $

   Compute $ 21 \cdot 19 $:
   $$
   21 \cdot 19 = (20 + 1)(20 - 1) = 20^2 - 1 = 400 - 1 = 399
   $$
   Then,
   $$
   2 \cdot 399 = 798
   $$

3. **Third term**:
   $ 361 = 19^2 $

---

### **Adding the Components**

Now, sum all the terms:

$$
441 + 798 + 361
$$

Break this down:

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

---

### **Recognizing the Pattern (Alternate Approach)**

Alternatively, we can observe that the expression:

$$
21^2 + 2 \cdot 21 \cdot 19 + 19^2
$$

is in the form $ a^2 + 2ab + b^2 $, which is the expansion of:

$$
(a + b)^2
$$

Let $ a = 21 $ and $ b = 19 $. Then:

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

This confirms the result without computing each term individually.

---

### **Conclusion**

Both methods lead to the same result. Therefore, the value of $ x $ is:

$$
\boxed{1600}
$$

[Assistant]
Explanation:
Okay, let me try to solve this problem. The equation given is 441 + 2(21)(19) + 361 = x. I need to find the value of x. Let me take it step by step.

First, let me write down the equation again to make sure I have it right:

441 + 2*(21)*(19) + 361 = x.

Alright, so there are three terms here: 441, then 2 times 21 times 19, and then 361. I need to calculate each part and then add them together.

Starting with the first term, 441. That's straightforward. I know that 21 squared is 441 because 21*21 is 441. Let me check that: 20*20 is 400, plus 20 + 20 + 1 is 441. Yep, that's right. So 441 is 21².

Then the second term is 2*(21)*(19). Hmm, let me think about how to compute that. Maybe I can simplify this part first. Let me recall that 2*21*19 can be thought of as 2*21*19. Alternatively, since 21 and 19 are both numbers close to 20, maybe there's a way to simplify this using some algebraic identity?

Wait, I remember that (a + b)² = a² + 2ab + b². Let me check if this problem resembles that formula. Let me see. If I consider 21 and 19 as numbers around 20, perhaps?

Wait, 21 is 20 + 1 and 19 is 20 - 1. So, if I let a = 20, then 21 = a + 1 and 19 = a - 1. Then, if I compute (a + 1)² + 2*(a + 1)*(a - 1) + (a - 1)², does that relate to something?

Wait, maybe not exactly. Let me think again. The original problem is 441 + 2*(21)*(19) + 361. Let me note that 441 is 21², and 361 is 19², because 19*19 is 361. Let me check that: 20*20 is 400, minus 20 + 20 - 1 is 400 - 39 = 361? Wait, no. Actually, 19*19 is 361. Let me calculate 20*20 - 2*20 + 1 = 400 - 40 + 1 = 361. Yes, that's correct. So 361 is 19². Therefore, the entire expression is 21² + 2*(21)*(19) + 19².

Oh! Wait a second, that looks exactly like the expansion of (a + b)² where a is 21 and b is 19. Let me recall that (a + b)² = a² + 2ab + b². So in this case, if we let a = 21 and b = 19, then the entire expression is (21 + 19)². Let me compute that.

21 + 19 is 40. Therefore, 40 squared is 1600. Therefore, x = 1600. Wait, so is that correct? Let me verify by computing each term separately to make sure I didn't make a mistake.

First, compute 441 + 2*(21)*(19) + 361.

Compute 2*(21)*(19). Let me compute 21*19 first. 21*19. Hmm, 20*19 is 380, and 1*19 is 19, so 380 + 19 = 399. Then multiply by 2: 399*2 = 798.

So the second term is 798. Then add 441 and 361. Let me add them up step by step:

First, 441 + 798. Let me compute that. 400 + 700 = 1100. 41 + 98 = 139. Therefore, total is 1100 + 139 = 1239.

Then add 361 to 1239. 1239 + 361. Let me break that down. 1239 + 300 = 1539, then +61 = 1539 + 61 = 1600. Okay, that matches. So x = 1600.

But wait, let me check again using the formula. If it's (21 + 19)^2, which is 40^2, that's 1600. So both methods give me the same result. So that seems solid. So perhaps the answer is 1600.

But let me check once more just to be thorough. Maybe I made a mistake when calculating 21*19. Let me compute 21*19 again. 21*19 can also be calculated as (20 + 1)*(20 - 1) = 20² - 1² = 400 - 1 = 399. Oh, that's another way to compute it. So that's 399. Multiply by 2: 798. Then 441 + 798 + 361.

Alternatively, maybe adding 441 + 361 first? Let me try that. 441 + 361. 400 + 300 = 700, 41 + 61 = 102, so total is 700 + 102 = 802. Then add 798: 802 + 798. 800 + 798 = 1598, plus 2 is 1600. Yep, same result.

Therefore, regardless of the method, the answer is 1600. So x = 1600.

Alternatively, if I didn't notice the perfect square, I could have just calculated each term step by step. But recognizing the pattern of the expansion of (a + b)^2 makes it easier.

But just to make sure there is no trick here. The original problem is 441 + 2*(21)*(19) + 361. Let me check if there's any possibility of misreading the problem. For example, maybe it was supposed to be 441 + 2*21*19 + 361. Which is exactly what it is. So no, there's no ambiguity there. The parentheses are there to indicate that the multiplication happens before addition. So yes, the calculation is correct.

Alternatively, if someone didn't recognize the perfect square, they could have done the arithmetic step by step, as I did above, and still arrived at the same answer.

Therefore, I'm confident that x equals 1600.

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

To solve the equation:

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

we can approach it in two ways: either by directly evaluating each term or by recognizing a pattern that simplifies the computation.

---

### **Step-by-Step Evaluation**

Let's first compute each component of the expression:

1. **First term**:
   $ 441 = 21^2 $

2. **Second term**:
   $ 2 \cdot 21 \cdot 19 = 2 \cdot (21 \cdot 19) $

   Compute $ 21 \cdot 19 $:
   $$
   21 \cdot 19 = (20 + 1)(20 - 1) = 20^2 - 1 = 400 - 1 = 399
   $$
   Then,
   $$
   2 \cdot 399 = 798
   $$

3. **Third term**:
   $ 361 = 19^2 $

---

### **Adding the Components**

Now, sum all the terms:

$$
441 + 798 + 361
$$

Break this down:

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

---

### **Recognizing the Pattern (Alternate Approach)**

Alternatively, we can observe that the expression:

$$
21^2 + 2 \cdot 21 \cdot 19 + 19^2
$$

is in the form $ a^2 + 2ab + b^2 $, which is the expansion of:

$$
(a + b)^2
$$

Let $ a = 21 $ and $ b = 19 $. Then:

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

This confirms the result without computing each term individually.

---

### **Conclusion**

Both methods lead to the same result. Therefore, the value of $ x $ is:

$$
\boxed{1600}
$$

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. Let me think about how to approach this.

First, I know that there are three equations here with three variables, so maybe I can solve for each variable step by step. Let me write down the equations again to have them clear:

1. xy = 24
2. xz = 48
3. yz = 72

Hmm, so each equation involves the product of two variables. Maybe I can find the values of x, y, z by dividing equations or multiplying them? Let me think.

If I multiply all three equations together, that might give me something useful. Let me try that:

(xy)(xz)(yz) = 24 * 48 * 72

Calculating the left side: (xy)(xz)(yz) = x^2 y^2 z^2. Because each variable is multiplied twice. So that's (xyz)^2.

On the right side, let me compute 24 * 48 * 72. Hmm, that might be a bit tedious. Let me break it down:

First, multiply 24 and 48. 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. Let me do this step by step. Maybe break 72 into 70 + 2. So:

1152 * 70 = 1152 * 7 * 10. Let me calculate 1152 * 7 first. 1000*7=7000, 152*7=1064, so total is 7000 + 1064 = 8064. Then multiply by 10: 80640.

Then 1152 * 2 = 2304. So total is 80640 + 2304 = 82944.

So (xyz)^2 = 82944. Therefore, xyz = sqrt(82944). Let me compute that square root.

Hmm, sqrt(82944). Let me think. Let me try dividing by some squares. Let me note that 288 squared is... Let me check. 288^2 is (200 + 88)^2, but maybe that's too complicated. Alternatively, let me divide 82944 by 144, since 144 is a square. 82944 / 144. Let me compute that.

Divide by 144: 82944 / 144. Let me divide numerator and denominator by 12: 82944 / 12 = 6912, 144 / 12 = 12. So now 6912 / 12 = 576. So that's 576. Therefore sqrt(82944) is sqrt(144 * 576) which is sqrt(144)*sqrt(576) = 12 * 24 = 288.

So xyz = 288.

Okay, so now that I know xyz = 288, maybe I can find x, y, z individually by dividing by the products given.

For example, if I have xyz = 288, and I know that xy = 24, then z would be (xyz)/(xy) = 288 / 24 = 12.

Similarly, if I want to find x, since yz = 72, then x = (xyz)/(yz) = 288 / 72 = 4.

Likewise, to find y, since xz = 48, then y = (xyz)/(xz) = 288 / 48 = 6.

So let me check:

x = 4, y = 6, z = 12.

Check if these satisfy the original equations:

xy = 4 * 6 = 24 ✔

xz = 4 * 12 = 48 ✔

yz = 6 * 12 = 72 ✔

Perfect! All equations are satisfied. Therefore, x + y + z = 4 + 6 + 12 = 22.

Wait, let me just make sure I didn't make any calculation errors. Let me verify the multiplication steps again.

Starting with multiplying the three equations: (xy)(xz)(yz) = 24*48*72.

Yes, that gives (xyz)^2 = 24*48*72. Then I calculated that as 82944, and sqrt(82944) is 288. Then xyz = 288.

Then dividing by each product:

z = 288 / 24 = 12.

x = 288 / 72 = 4.

y = 288 / 48 = 6.

Yes, that seems correct. Then adding them up gives 4 + 6 + 12 = 22.

Alternatively, I could solve the equations step by step without multiplying all three. Let me try that approach as another method to confirm.

Starting with 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.

So z is twice y. Then, since yz = 72, substitute z with 2y: y*(2y) = 72 => 2y² = 72 => y² = 36 => y = 6 (since y is positive).

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

So again, x = 4, y = 6, z = 12.

Same result. So x + y + z = 22.

Alternatively, maybe another approach. Let me try solving for variables step by step.

From xy = 24, so x = 24 / y.

From xz = 48, substituting x gives (24 / y) * z = 48 => 24z / y = 48 => z / y = 2 => z = 2y.

Same as before. Then substitute z = 2y into yz = 72: y*(2y) = 72 => 2y² = 72 => y² = 36 => y = 6.

Same steps again. So x = 24 / 6 = 4, z = 12.

Therefore, all methods lead to the same answer.

Wait, but just to make sure, let me check if there's another way someone might approach this problem and maybe get confused.

Another thought: If I take the equations:

xy = 24,

xz = 48,

yz = 72.

Let me solve for each variable in terms of another. For example, from the first equation, x = 24/y. From the second equation, x = 48/z. So 24/y = 48/z => cross multiplying, 24z = 48y => z = 2y.

Which is the same as before. Then from the third equation, yz = 72 => y*(2y) = 72 => same as before.

Alternatively, from the first equation, y = 24/x. From the third equation, y = 72/z. So 24/x = 72/z => 24z = 72x => z = 3x.

Wait, hold on, this seems conflicting with previous result where z = 2y. Wait, but if z = 3x, and earlier we had z = 2y, but since x = 24/y, then z = 3*(24/y) = 72/y. But from z = 2y, we have z = 2y. Therefore, 72/y = 2y => 72 = 2y² => y² = 36 => y = 6. Which is same as before. So that also leads to y = 6, z = 12, x = 4.

So here, if I get z = 3x from equating two expressions for y.

But then, if z = 3x, and from the second equation xz = 48, substitute z: x*(3x) = 3x² = 48 => x² = 16 => x = 4. Which is again same result.

So that's another way. So regardless of the approach, we get the same answer.

So, seems like the answer is 22.

But just to be thorough, maybe trying to solve using substitution. Let me try that.

Let me take the first equation, xy = 24. So x = 24/y. Then plug into the second equation, which is xz = 48. So (24/y)*z = 48 => z = 48y / 24 = 2y. So z = 2y. Then plug into the third equation yz = 72. So y*(2y) = 2y² = 72 => y² = 36 => y = 6. Then z = 2*6 = 12. Then x = 24 / 6 = 4.

Same result again.

Alternatively, suppose I wanted to use ratios. Let me think.

Let me denote the equations again:

xy = 24,

xz = 48,

yz = 72.

Let me take the ratio of the second equation to the first: (xz)/(xy) = 48/24 => z/y = 2 => z = 2y.

Similarly, take the ratio of the third equation to the second: (yz)/(xz) = 72/48 => y/x = 3/2 => y = (3/2)x.

So now, we have z = 2y and y = (3/2)x.

So substitute y into z: z = 2*(3/2)x = 3x.

So now, we have z = 3x.

Now, let's substitute into one of the original equations. For example, take the second equation xz = 48. Substitute z = 3x: x*(3x) = 3x² = 48 => x² = 16 => x = 4.

Then y = (3/2)x = (3/2)*4 = 6.

Then z = 3x = 12.

Same result.

Alternatively, maybe using substitution in another equation. Let me check with the first equation. If x = 4, then y = 24/x = 6. Then z = 48/x = 48/4 = 12.

Yes.

Therefore, regardless of the method, all roads lead to x = 4, y = 6, z = 12, so x + y + z = 22.

But just to be 100% sure, maybe plug in the values into all equations again:

xy = 4*6 = 24 ✔

xz = 4*12 = 48 ✔

yz = 6*12 = 72 ✔

Perfect.

So there's no mistake here.

Alternatively, maybe I can think of this as a system of equations and solve it step by step. Let me try that.

We have three equations:

1. xy = 24

2. xz = 48

3. yz = 72

Let me solve for x from equation 1: x = 24/y

Substitute x into equation 2: (24/y) * z = 48 => 24z = 48y => z = 2y (same as before)

Substitute z = 2y into equation 3: y*(2y) = 72 => 2y² = 72 => y² = 36 => y = 6 (since positive)

Then, since z = 2y, z = 12, and x = 24/y = 24/6 = 4.

So same answer again.

Therefore, I think this is solid. The answer is 22.

Wait, but let me think if there's another way to get confused here. For example, if someone thought that multiplying all three equations would give something else, but no, that method was correct.

Alternatively, if someone tried to solve for variables individually without multiplying, but that seems more complicated.

Alternatively, someone might try to take square roots or something else, but I don't think so.

Alternatively, maybe trying to find ratios between variables. For example, since xy = 24, xz = 48, so dividing gives z/y = 2, so z = 2y. Similarly, from xz = 48 and yz = 72, dividing gives x/y = 48/72 = 2/3 => x = (2/3)y. Then if x = (2/3)y, then from xy = 24, substitute: (2/3)y * y = 24 => (2/3)y² = 24 => y² = 24*(3/2) = 36 => y = 6.

Same result.

Therefore, even through different methods, same answer.

Another thought: maybe using logarithms? Probably overcomplicating, but just for fun.

Take the equations:

ln(xy) = ln24 => ln x + ln y = ln24

ln(xz) = ln48 => ln x + ln z = ln48

ln(yz) = ln72 => ln y + ln z = ln72

Let me denote a = ln x, b = ln y, c = ln z. So:

a + b = ln24

a + c = ln48

b + c = ln72

So we have three equations:

1. a + b = ln24

2. a + c = ln48

3. b + c = ln72

We can solve for a, b, c. Let me do that.

Subtract equation 1 from equation 2: (a + c) - (a + b) = ln48 - ln24 => c - b = ln(48/24) = ln2.

So c - b = ln2.

From equation 3: b + c = ln72.

So we have:

c - b = ln2

and

c + b = ln72

Adding both equations: 2c = ln2 + ln72 = ln(2*72) = ln144. Therefore, c = (ln144)/2 = ln(144^(1/2)) = ln(12). Therefore, c = ln12. Therefore, z = e^c = 12.

Similarly, from equation 3: b + c = ln72 => b = ln72 - c = ln72 - ln12 = ln(72/12) = ln6. Therefore, b = ln6 => y = e^b = 6.

Then from equation 1: a + b = ln24 => a = ln24 - ln6 = ln(24/6) = ln4 => a = ln4 => x = e^a = 4.

Same result! So even using logarithms, we get x = 4, y = 6, z = 12.

Therefore, x + y + z = 22.

Wow, that was thorough. I think I've verified it multiple ways.

Therefore, the answer should be 22.

**Final Answer**
\boxed{22}

To solve the system of equations:

$$
\begin{align*}
(1)\quad xy &= 24 \\
(2)\quad xz &= 48 \\
(3)\quad yz &= 72
\end{align*}
$$

we are given that $ x $, $ y $, and $ z $ are all positive real numbers. The goal is to find the sum $ x + y + z $.

---

### Step 1: Use the Product of All Three Equations

Multiply all three equations together:

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

The left-hand side becomes:

$$
x^2 y^2 z^2 = (xyz)^2
$$

So:

$$
(xyz)^2 = 24 \cdot 48 \cdot 72 = 82944
$$

Taking the square root of both sides:

$$
xyz = \sqrt{82944} = 288
$$

---

### Step 2: Solve for Each Variable

We now know the product $ xyz = 288 $ and can use this to find the individual values of $ x $, $ y $, and $ z $.

Using the known values:

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

---

### Step 3: Verify the Solution

Check if $ x = 4 $, $ y = 6 $, and $ z = 12 $ satisfy all original equations:

- $ xy = 4 \cdot 6 = 24 $ ✔
- $ xz = 4 \cdot 12 = 48 $ ✔
- $ yz = 6 \cdot 12 = 72 $ ✔

All conditions are satisfied.

---

### Step 4: Find 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. Let me think about how to approach this.

First, I know that there are three equations here with three variables, so maybe I can solve for each variable step by step. Let me write down the equations again to have them clear:

1. xy = 24
2. xz = 48
3. yz = 72

Hmm, so each equation involves the product of two variables. Maybe I can find the values of x, y, z by dividing equations or multiplying them? Let me think.

If I multiply all three equations together, that might give me something useful. Let me try that:

(xy)(xz)(yz) = 24 * 48 * 72

Calculating the left side: (xy)(xz)(yz) = x^2 y^2 z^2. Because each variable is multiplied twice. So that's (xyz)^2.

On the right side, let me compute 24 * 48 * 72. Hmm, that might be a bit tedious. Let me break it down:

First, multiply 24 and 48. 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. Let me do this step by step. Maybe break 72 into 70 + 2. So:

1152 * 70 = 1152 * 7 * 10. Let me calculate 1152 * 7 first. 1000*7=7000, 152*7=1064, so total is 7000 + 1064 = 8064. Then multiply by 10: 80640.

Then 1152 * 2 = 2304. So total is 80640 + 2304 = 82944.

So (xyz)^2 = 82944. Therefore, xyz = sqrt(82944). Let me compute that square root.

Hmm, sqrt(82944). Let me think. Let me try dividing by some squares. Let me note that 288 squared is... Let me check. 288^2 is (200 + 88)^2, but maybe that's too complicated. Alternatively, let me divide 82944 by 144, since 144 is a square. 82944 / 144. Let me compute that.

Divide by 144: 82944 / 144. Let me divide numerator and denominator by 12: 82944 / 12 = 6912, 144 / 12 = 12. So now 6912 / 12 = 576. So that's 576. Therefore sqrt(82944) is sqrt(144 * 576) which is sqrt(144)*sqrt(576) = 12 * 24 = 288.

So xyz = 288.

Okay, so now that I know xyz = 288, maybe I can find x, y, z individually by dividing by the products given.

For example, if I have xyz = 288, and I know that xy = 24, then z would be (xyz)/(xy) = 288 / 24 = 12.

Similarly, if I want to find x, since yz = 72, then x = (xyz)/(yz) = 288 / 72 = 4.

Likewise, to find y, since xz = 48, then y = (xyz)/(xz) = 288 / 48 = 6.

So let me check:

x = 4, y = 6, z = 12.

Check if these satisfy the original equations:

xy = 4 * 6 = 24 ✔

xz = 4 * 12 = 48 ✔

yz = 6 * 12 = 72 ✔

Perfect! All equations are satisfied. Therefore, x + y + z = 4 + 6 + 12 = 22.

Wait, let me just make sure I didn't make any calculation errors. Let me verify the multiplication steps again.

Starting with multiplying the three equations: (xy)(xz)(yz) = 24*48*72.

Yes, that gives (xyz)^2 = 24*48*72. Then I calculated that as 82944, and sqrt(82944) is 288. Then xyz = 288.

Then dividing by each product:

z = 288 / 24 = 12.

x = 288 / 72 = 4.

y = 288 / 48 = 6.

Yes, that seems correct. Then adding them up gives 4 + 6 + 12 = 22.

Alternatively, I could solve the equations step by step without multiplying all three. Let me try that approach as another method to confirm.

Starting with 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.

So z is twice y. Then, since yz = 72, substitute z with 2y: y*(2y) = 72 => 2y² = 72 => y² = 36 => y = 6 (since y is positive).

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

So again, x = 4, y = 6, z = 12.

Same result. So x + y + z = 22.

Alternatively, maybe another approach. Let me try solving for variables step by step.

From xy = 24, so x = 24 / y.

From xz = 48, substituting x gives (24 / y) * z = 48 => 24z / y = 48 => z / y = 2 => z = 2y.

Same as before. Then substitute z = 2y into yz = 72: y*(2y) = 72 => 2y² = 72 => y² = 36 => y = 6.

Same steps again. So x = 24 / 6 = 4, z = 12.

Therefore, all methods lead to the same answer.

Wait, but just to make sure, let me check if there's another way someone might approach this problem and maybe get confused.

Another thought: If I take the equations:

xy = 24,

xz = 48,

yz = 72.

Let me solve for each variable in terms of another. For example, from the first equation, x = 24/y. From the second equation, x = 48/z. So 24/y = 48/z => cross multiplying, 24z = 48y => z = 2y.

Which is the same as before. Then from the third equation, yz = 72 => y*(2y) = 72 => same as before.

Alternatively, from the first equation, y = 24/x. From the third equation, y = 72/z. So 24/x = 72/z => 24z = 72x => z = 3x.

Wait, hold on, this seems conflicting with previous result where z = 2y. Wait, but if z = 3x, and earlier we had z = 2y, but since x = 24/y, then z = 3*(24/y) = 72/y. But from z = 2y, we have z = 2y. Therefore, 72/y = 2y => 72 = 2y² => y² = 36 => y = 6. Which is same as before. So that also leads to y = 6, z = 12, x = 4.

So here, if I get z = 3x from equating two expressions for y.

But then, if z = 3x, and from the second equation xz = 48, substitute z: x*(3x) = 3x² = 48 => x² = 16 => x = 4. Which is again same result.

So that's another way. So regardless of the approach, we get the same answer.

So, seems like the answer is 22.

But just to be thorough, maybe trying to solve using substitution. Let me try that.

Let me take the first equation, xy = 24. So x = 24/y. Then plug into the second equation, which is xz = 48. So (24/y)*z = 48 => z = 48y / 24 = 2y. So z = 2y. Then plug into the third equation yz = 72. So y*(2y) = 2y² = 72 => y² = 36 => y = 6. Then z = 2*6 = 12. Then x = 24 / 6 = 4.

Same result again.

Alternatively, suppose I wanted to use ratios. Let me think.

Let me denote the equations again:

xy = 24,

xz = 48,

yz = 72.

Let me take the ratio of the second equation to the first: (xz)/(xy) = 48/24 => z/y = 2 => z = 2y.

Similarly, take the ratio of the third equation to the second: (yz)/(xz) = 72/48 => y/x = 3/2 => y = (3/2)x.

So now, we have z = 2y and y = (3/2)x.

So substitute y into z: z = 2*(3/2)x = 3x.

So now, we have z = 3x.

Now, let's substitute into one of the original equations. For example, take the second equation xz = 48. Substitute z = 3x: x*(3x) = 3x² = 48 => x² = 16 => x = 4.

Then y = (3/2)x = (3/2)*4 = 6.

Then z = 3x = 12.

Same result.

Alternatively, maybe using substitution in another equation. Let me check with the first equation. If x = 4, then y = 24/x = 6. Then z = 48/x = 48/4 = 12.

Yes.

Therefore, regardless of the method, all roads lead to x = 4, y = 6, z = 12, so x + y + z = 22.

But just to be 100% sure, maybe plug in the values into all equations again:

xy = 4*6 = 24 ✔

xz = 4*12 = 48 ✔

yz = 6*12 = 72 ✔

Perfect.

So there's no mistake here.

Alternatively, maybe I can think of this as a system of equations and solve it step by step. Let me try that.

We have three equations:

1. xy = 24

2. xz = 48

3. yz = 72

Let me solve for x from equation 1: x = 24/y

Substitute x into equation 2: (24/y) * z = 48 => 24z = 48y => z = 2y (same as before)

Substitute z = 2y into equation 3: y*(2y) = 72 => 2y² = 72 => y² = 36 => y = 6 (since positive)

Then, since z = 2y, z = 12, and x = 24/y = 24/6 = 4.

So same answer again.

Therefore, I think this is solid. The answer is 22.

Wait, but let me think if there's another way to get confused here. For example, if someone thought that multiplying all three equations would give something else, but no, that method was correct.

Alternatively, if someone tried to solve for variables individually without multiplying, but that seems more complicated.

Alternatively, someone might try to take square roots or something else, but I don't think so.

Alternatively, maybe trying to find ratios between variables. For example, since xy = 24, xz = 48, so dividing gives z/y = 2, so z = 2y. Similarly, from xz = 48 and yz = 72, dividing gives x/y = 48/72 = 2/3 => x = (2/3)y. Then if x = (2/3)y, then from xy = 24, substitute: (2/3)y * y = 24 => (2/3)y² = 24 => y² = 24*(3/2) = 36 => y = 6.

Same result.

Therefore, even through different methods, same answer.

Another thought: maybe using logarithms? Probably overcomplicating, but just for fun.

Take the equations:

ln(xy) = ln24 => ln x + ln y = ln24

ln(xz) = ln48 => ln x + ln z = ln48

ln(yz) = ln72 => ln y + ln z = ln72

Let me denote a = ln x, b = ln y, c = ln z. So:

a + b = ln24

a + c = ln48

b + c = ln72

So we have three equations:

1. a + b = ln24

2. a + c = ln48

3. b + c = ln72

We can solve for a, b, c. Let me do that.

Subtract equation 1 from equation 2: (a + c) - (a + b) = ln48 - ln24 => c - b = ln(48/24) = ln2.

So c - b = ln2.

From equation 3: b + c = ln72.

So we have:

c - b = ln2

and

c + b = ln72

Adding both equations: 2c = ln2 + ln72 = ln(2*72) = ln144. Therefore, c = (ln144)/2 = ln(144^(1/2)) = ln(12). Therefore, c = ln12. Therefore, z = e^c = 12.

Similarly, from equation 3: b + c = ln72 => b = ln72 - c = ln72 - ln12 = ln(72/12) = ln6. Therefore, b = ln6 => y = e^b = 6.

Then from equation 1: a + b = ln24 => a = ln24 - ln6 = ln(24/6) = ln4 => a = ln4 => x = e^a = 4.

Same result! So even using logarithms, we get x = 4, y = 6, z = 12.

Therefore, x + y + z = 22.

Wow, that was thorough. I think I've verified it multiple ways.

Therefore, the answer should be 22.

**Final Answer**
\boxed{22}

To solve the system of equations:

$$
\begin{align*}
(1)\quad xy &= 24 \\
(2)\quad xz &= 48 \\
(3)\quad yz &= 72
\end{align*}
$$

we are given that $ x $, $ y $, and $ z $ are all positive real numbers. The goal is to find the sum $ x + y + z $.

---

### Step 1: Use the Product of All Three Equations

Multiply all three equations together:

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

The left-hand side becomes:

$$
x^2 y^2 z^2 = (xyz)^2
$$

So:

$$
(xyz)^2 = 24 \cdot 48 \cdot 72 = 82944
$$

Taking the square root of both sides:

$$
xyz = \sqrt{82944} = 288
$$

---

### Step 2: Solve for Each Variable

We now know the product $ xyz = 288 $ and can use this to find the individual values of $ x $, $ y $, and $ z $.

Using the known values:

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

---

### Step 3: Verify the Solution

Check if $ x = 4 $, $ y = 6 $, and $ z = 12 $ satisfy all original equations:

- $ xy = 4 \cdot 6 = 24 $ ✔
- $ xz = 4 \cdot 12 = 48 $ ✔
- $ yz = 6 \cdot 12 = 72 $ ✔

All conditions are satisfied.

---

### Step 4: Find 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, $x + y + z = \boxed{22}.$