Explanation:
Okay, so Mikka wants to order a pizza with two different toppings, and he has 8 different toppings to choose from. The question is asking how many different pizzas he could order. Hmm, let me think about how to approach this.

First, I remember that when dealing with combinations, the order doesn't matter. Like, if you choose topping A and topping B, that's the same as choosing topping B and topping A. So, this seems like a combination problem rather than a permutation problem.

But wait, let me make sure. If the pizza is just two toppings, regardless of the order they're placed on the pizza, then yes, the order doesn't matter. So combinations are the way to go. But maybe I should also think about whether there are any restrictions or other factors. The problem says two different toppings, so we can't have the same topping twice. So each pizza must have exactly two distinct toppings.

So, the problem reduces to: How many ways can we choose 2 toppings out of 8, where the order doesn't matter?

The formula for combinations is C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items we want to choose. In this case, n is 8, and k is 2. Let me plug those numbers into the formula.

So, C(8, 2) = 8! / (2! * (8 - 2)!) = 8! / (2! * 6!).

Hmm, simplifying that. I know that 8! is 8 × 7 × 6!, so we can write that as:

C(8, 2) = (8 × 7 × 6!) / (2! × 6!)

Then, the 6! in the numerator and denominator cancels out, leaving:

C(8, 2) = (8 × 7) / 2!

Since 2! is 2 × 1 = 2, this becomes:

(8 × 7) / 2 = 56 / 2 = 28

So, that would be 28 different ways.

Wait, let me check if there's another way to think about it. For example, if I consider permutations first. If order did matter, then the number of ways to choose 2 toppings out of 8 would be P(n, k) = n! / (n - k)! = 8! / 6! = 8 × 7 = 56. But since order doesn't matter, each pair is counted twice in that permutation count. For example, choosing A then B is the same as choosing B then A. So, to get the combination count, we divide by 2!, which is 2. So 56 / 2 = 28. That matches the previous result.

So, that seems consistent.

Alternatively, maybe I can list them out for a smaller number and see if the formula works. Let's say there are 3 toppings: A, B, C. How many ways to choose 2? It should be 3: AB, AC, BC. Using the formula C(3,2) = 3! / (2! * 1!) = 6 / 2 = 3. Correct. If I use the permutation approach: P(3,2) = 3*2=6, then divide by 2! to get 3. Yep.

So, applying that to 8 toppings, it's 28.

But let me think if there's any chance that the problem is considering the order of toppings? For example, if the pizza has a first topping and a second topping, maybe? But in reality, pizza toppings are not ordered. So, if you have pepperoni and mushrooms, it's the same as mushrooms and pepperoni. So, combinations are the right approach here.

Another way: think of each topping as a choice. For the first topping, he has 8 options. Then, for the second topping, he has 7 options left. So, 8*7=56. But again, this counts each pizza twice, once for each order. So, divide by 2 to get 28.

Yes, that's the same as before.

So, all methods lead me to 28.

Wait, but maybe the problem is a trick question? Like, is there any possibility that some toppings are the same? Wait, the problem says "two different toppings," and he has 8 different toppings. So, each topping is unique. So, no duplicates. So, the answer should be 28.

Alternatively, if the question had said "two toppings, which can be the same or different," then it would be different. But since it's specified that they are different, combinations are the way to go.

Therefore, I think the answer is 28.

**Final Answer**
The number of different pizzas Mikka could order is \boxed{28}.

To determine how many different pizzas Mikka could order, we need to analyze the problem logically and apply the appropriate mathematical concept.

---

### Problem Breakdown

Mikka wants to order a pizza with **two different toppings**, and he has **8 different toppings** to choose from. The key point here is that:

- **The order** in which the toppings are selected does **not matter** (a pizza with toppings A and B is the same as one with B and A).
- **Each topping must be unique**, and **no repetition** is allowed (he can't choose the same topping twice).

So, the problem is essentially asking:

> *How many unique pairs of toppings can be formed from 8 distinct toppings, where the order does not matter?*

This is a classic **combination** problem.

---

### Applying the Combination Formula

The formula for combinations is:

$$
C(n, k) = \frac{n!}{k!(n - k)!}
$$

Where:
- $ n $ is the total number of items (here, 8 toppings),
- $ k $ is the number of items to choose (here, 2 toppings).

Plugging in the values:
$$
C(8, 2) = \frac{8!}{2!(8 - 2)!} = \frac{8!}{2! \cdot 6!}
$$

We can simplify this:
$$
\frac{8 \times 7 \times 6!}{2 \times 1 \times 6!} = \frac{56}{2} = 28
$$

---

### Verification with Alternative Approach

Another way to think about this is:

1. **First topping choice**: 8 options
2. **Second topping choice**: 7 remaining options

So, there are $ 8 \times 7 = 56 $ ordered selections, but since the order does **not matter**, each pair is counted **twice** (once as AB, once as BA). Hence, we divide by 2:

$$
\frac{8 \times 7}{2} = \frac{56}{2} = 28
$$

This confirms our result.

---

### Final Answer

The number of different pizzas Mikka could order is:

$$
\boxed{28}
$$

[Assistant]
Explanation:
Okay, so Mikka wants to order a pizza with two different toppings, and he has 8 different toppings to choose from. The question is asking how many different pizzas he could order. Hmm, let me think about how to approach this.

First, I remember that when dealing with combinations, the order doesn't matter. Like, if you choose topping A and topping B, that's the same as choosing topping B and topping A. So, this seems like a combination problem rather than a permutation problem.

But wait, let me make sure. If the pizza is just two toppings, regardless of the order they're placed on the pizza, then yes, the order doesn't matter. So combinations are the way to go. But maybe I should also think about whether there are any restrictions or other factors. The problem says two different toppings, so we can't have the same topping twice. So each pizza must have exactly two distinct toppings.

So, the problem reduces to: How many ways can we choose 2 toppings out of 8, where the order doesn't matter?

The formula for combinations is C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items we want to choose. In this case, n is 8, and k is 2. Let me plug those numbers into the formula.

So, C(8, 2) = 8! / (2! * (8 - 2)!) = 8! / (2! * 6!).

Hmm, simplifying that. I know that 8! is 8 × 7 × 6!, so we can write that as:

C(8, 2) = (8 × 7 × 6!) / (2! × 6!)

Then, the 6! in the numerator and denominator cancels out, leaving:

C(8, 2) = (8 × 7) / 2!

Since 2! is 2 × 1 = 2, this becomes:

(8 × 7) / 2 = 56 / 2 = 28

So, that would be 28 different ways.

Wait, let me check if there's another way to think about it. For example, if I consider permutations first. If order did matter, then the number of ways to choose 2 toppings out of 8 would be P(n, k) = n! / (n - k)! = 8! / 6! = 8 × 7 = 56. But since order doesn't matter, each pair is counted twice in that permutation count. For example, choosing A then B is the same as choosing B then A. So, to get the combination count, we divide by 2!, which is 2. So 56 / 2 = 28. That matches the previous result.

So, that seems consistent.

Alternatively, maybe I can list them out for a smaller number and see if the formula works. Let's say there are 3 toppings: A, B, C. How many ways to choose 2? It should be 3: AB, AC, BC. Using the formula C(3,2) = 3! / (2! * 1!) = 6 / 2 = 3. Correct. If I use the permutation approach: P(3,2) = 3*2=6, then divide by 2! to get 3. Yep.

So, applying that to 8 toppings, it's 28.

But let me think if there's any chance that the problem is considering the order of toppings? For example, if the pizza has a first topping and a second topping, maybe? But in reality, pizza toppings are not ordered. So, if you have pepperoni and mushrooms, it's the same as mushrooms and pepperoni. So, combinations are the right approach here.

Another way: think of each topping as a choice. For the first topping, he has 8 options. Then, for the second topping, he has 7 options left. So, 8*7=56. But again, this counts each pizza twice, once for each order. So, divide by 2 to get 28.

Yes, that's the same as before.

So, all methods lead me to 28.

Wait, but maybe the problem is a trick question? Like, is there any possibility that some toppings are the same? Wait, the problem says "two different toppings," and he has 8 different toppings. So, each topping is unique. So, no duplicates. So, the answer should be 28.

Alternatively, if the question had said "two toppings, which can be the same or different," then it would be different. But since it's specified that they are different, combinations are the way to go.

Therefore, I think the answer is 28.

**Final Answer**
The number of different pizzas Mikka could order is \boxed{28}.

To determine how many different pizzas Mikka could order, we need to analyze the problem logically and apply the appropriate mathematical concept.

---

### Problem Breakdown

Mikka wants to order a pizza with **two different toppings**, and he has **8 different toppings** to choose from. The key point here is that:

- **The order** in which the toppings are selected does **not matter** (a pizza with toppings A and B is the same as one with B and A).
- **Each topping must be unique**, and **no repetition** is allowed (he can't choose the same topping twice).

So, the problem is essentially asking:

> *How many unique pairs of toppings can be formed from 8 distinct toppings, where the order does not matter?*

This is a classic **combination** problem.

---

### Applying the Combination Formula

The formula for combinations is:

$$
C(n, k) = \frac{n!}{k!(n - k)!}
$$

Where:
- $ n $ is the total number of items (here, 8 toppings),
- $ k $ is the number of items to choose (here, 2 toppings).

Plugging in the values:
$$
C(8, 2) = \frac{8!}{2!(8 - 2)!} = \frac{8!}{2! \cdot 6!}
$$

We can simplify this:
$$
\frac{8 \times 7 \times 6!}{2 \times 1 \times 6!} = \frac{56}{2} = 28
$$

---

### Verification with Alternative Approach

Another way to think about this is:

1. **First topping choice**: 8 options
2. **Second topping choice**: 7 remaining options

So, there are $ 8 \times 7 = 56 $ ordered selections, but since the order does **not matter**, each pair is counted **twice** (once as AB, once as BA). Hence, we divide by 2:

$$
\frac{8 \times 7}{2} = \frac{56}{2} = 28
$$

This confirms our result.

---

### Final Answer

The number of different pizzas Mikka could order is:

$$
\boxed{28}
$$

Solution: There are 8 choices of toppings, and we need to choose 2 distinct toppings. This is represented by the number of 2-element subsets of an 8 -element set. We use the binomial coefficient ${8 \choose 2} = \boxed{28}$ to compute this.


Explanation:
Okay, let's see. The problem says that when the two-digit integer "AB" is cubed, the value is 912,673. I need to find A + B, where AB is the two-digit number. Alright, let me break this down step by step.

First, I need to figure out what the two-digit number AB is. Since AB is a two-digit number, that means it's between 10 and 99, inclusive. When we cube it, we get 912,673. So essentially, I need to find the cube root of 912,673 and check if it's a two-digit integer. If it is, then A and B are its tens and units digits, respectively, and then I can add them together.

Let me start by recalling that the cube of a number n is n³. So, if AB³ = 912,673, then AB = ∛912,673. Let me try to approximate the cube root of 912,673.

First, I know that 10³ = 1,000 and 100³ = 1,000,000. So 912,673 is between 10³ and 100³, which makes sense because AB is a two-digit number. So the cube root of 912,673 should be between 10 and 100. However, since we know it's a two-digit number, we can narrow it down further.

Let me think about numbers around 97 or so? Wait, maybe I should find a better way to approximate this cube root.

Alternatively, perhaps I can figure out the cube root by estimating. Let me try some numbers.

First, let's note that 90³ = 729,000. 95³ is... 95*95 = 9025, then 9025*95. Let me compute that. 9025*95: 9025*100 is 902,500 minus 9025*5 which is 45,125, so 902,500 - 45,125 = 857,375. So 95³ = 857,375. That's still less than 912,673.

How about 96³? Let me compute 96³. Let's do 96*96 first. 96² is 9216. Then multiply by 96: 9216*96. Let me compute that. 9216*100 is 921,600 minus 9216*4 which is 36,864. So 921,600 - 36,864 = 884,736. So 96³ = 884,736. Still less than 912,673.

Next, 97³. Let me calculate 97² first. 97² = 9409. Then multiply by 97: 9409*97. Let me do 9409*(100 - 3) = 940900 - 3*9409. 3*9409 = 28,227. So 940,900 - 28,227 = 912,673. Wait, that's exactly the number we have here!

So 97³ = 912,673. Therefore, AB is 97. Let me check again to make sure I didn't make a calculation error.

So 97 squared: 97*97. Let me compute that again. 100*97 = 9700, minus 3*97 = 291, so 9700 - 291 = 9409. Yes, that's correct. Then 9409*97. Let me check that again. 9409*97. Alternatively, maybe I can use another method. Let's see:

9409*97 = 9409*(90 + 7) = 9409*90 + 9409*7.

Calculate 9409*90: 9409*9*10. 9409*9: Let me compute 9000*9 = 81,000, 400*9=3,600, 9*9=81. So 81,000 + 3,600 = 84,600 + 81 = 84,681. Then multiply by 10: 846,810.

Now 9409*7: 9000*7 = 63,000; 400*7 = 2,800; 9*7 = 63. So 63,000 + 2,800 = 65,800 + 63 = 65,863.

Adding those two results: 846,810 + 65,863. Let's do 846,810 + 60,000 = 906,810; then +5,863 = 912,673. Yes, that's exactly the number given. So 97³ is indeed 912,673. Therefore, the two-digit number AB is 97. Therefore, A is 9 and B is 7. Therefore, A + B is 9 + 7 = 16.

Wait, but let me check if there's another two-digit number that could cube to this. Wait, since 97 is between 96 and 98, and we saw that 96³ is 884,736 and 98³ would be... Let me check 98³ just to confirm.

98³: 98*98 = 9604; then 9604*98. Let me calculate 9604*100 - 9604*2 = 960,400 - 19,208 = 941,192. So 98³ is 941,192, which is bigger than 912,673. So between 96 and 98, only 97 gives us the exact value. So AB is definitely 97. Therefore, A is 9 and B is 7, so A + B is 16. Let me just make sure that there isn't a different two-digit number that could cube to that. But since cube function is strictly increasing, there's only one number in that range that would give that value. So I think that's solid.

Therefore, the answer is 16.

**Final Answer**
The value of $A + B$ is \boxed{16}.

To solve the problem, we are given that the cube of the two-digit integer "AB" equals 912,673. Our goal is to find the sum $ A + B $, where "AB" represents the two-digit number formed by the digits A and B.

---

### Step 1: Understanding the Relationship

We interpret the two-digit number "AB" as a variable representing a number between 10 and 99. Let's denote this number as $ n = 10A + B $. We are told that:

$$
n^3 = 912,673
$$

This implies:

$$
n = \sqrt[3]{912,673}
$$

---

### Step 2: Estimating the Cube Root

We estimate the cube root of 912,673 by testing some values.

- $ 95^3 = 857,375 $
- $ 96^3 = 884,736 $
- $ 97^3 = 912,673 $ ✅ (matches the given cube)

Thus:

$$
n = 97
$$

This means the two-digit number "AB" is 97, where:
- $ A = 9 $
- $ B = 7 $

---

### Step 3: Calculating $ A + B $

$$
A + B = 9 + 7 = 16
$$

---

### Final Answer

$$
\boxed{16}
$$

[Assistant]
Explanation:
Okay, let's see. The problem says that when the two-digit integer "AB" is cubed, the value is 912,673. I need to find A + B, where AB is the two-digit number. Alright, let me break this down step by step.

First, I need to figure out what the two-digit number AB is. Since AB is a two-digit number, that means it's between 10 and 99, inclusive. When we cube it, we get 912,673. So essentially, I need to find the cube root of 912,673 and check if it's a two-digit integer. If it is, then A and B are its tens and units digits, respectively, and then I can add them together.

Let me start by recalling that the cube of a number n is n³. So, if AB³ = 912,673, then AB = ∛912,673. Let me try to approximate the cube root of 912,673.

First, I know that 10³ = 1,000 and 100³ = 1,000,000. So 912,673 is between 10³ and 100³, which makes sense because AB is a two-digit number. So the cube root of 912,673 should be between 10 and 100. However, since we know it's a two-digit number, we can narrow it down further.

Let me think about numbers around 97 or so? Wait, maybe I should find a better way to approximate this cube root.

Alternatively, perhaps I can figure out the cube root by estimating. Let me try some numbers.

First, let's note that 90³ = 729,000. 95³ is... 95*95 = 9025, then 9025*95. Let me compute that. 9025*95: 9025*100 is 902,500 minus 9025*5 which is 45,125, so 902,500 - 45,125 = 857,375. So 95³ = 857,375. That's still less than 912,673.

How about 96³? Let me compute 96³. Let's do 96*96 first. 96² is 9216. Then multiply by 96: 9216*96. Let me compute that. 9216*100 is 921,600 minus 9216*4 which is 36,864. So 921,600 - 36,864 = 884,736. So 96³ = 884,736. Still less than 912,673.

Next, 97³. Let me calculate 97² first. 97² = 9409. Then multiply by 97: 9409*97. Let me do 9409*(100 - 3) = 940900 - 3*9409. 3*9409 = 28,227. So 940,900 - 28,227 = 912,673. Wait, that's exactly the number we have here!

So 97³ = 912,673. Therefore, AB is 97. Let me check again to make sure I didn't make a calculation error.

So 97 squared: 97*97. Let me compute that again. 100*97 = 9700, minus 3*97 = 291, so 9700 - 291 = 9409. Yes, that's correct. Then 9409*97. Let me check that again. 9409*97. Alternatively, maybe I can use another method. Let's see:

9409*97 = 9409*(90 + 7) = 9409*90 + 9409*7.

Calculate 9409*90: 9409*9*10. 9409*9: Let me compute 9000*9 = 81,000, 400*9=3,600, 9*9=81. So 81,000 + 3,600 = 84,600 + 81 = 84,681. Then multiply by 10: 846,810.

Now 9409*7: 9000*7 = 63,000; 400*7 = 2,800; 9*7 = 63. So 63,000 + 2,800 = 65,800 + 63 = 65,863.

Adding those two results: 846,810 + 65,863. Let's do 846,810 + 60,000 = 906,810; then +5,863 = 912,673. Yes, that's exactly the number given. So 97³ is indeed 912,673. Therefore, the two-digit number AB is 97. Therefore, A is 9 and B is 7. Therefore, A + B is 9 + 7 = 16.

Wait, but let me check if there's another two-digit number that could cube to this. Wait, since 97 is between 96 and 98, and we saw that 96³ is 884,736 and 98³ would be... Let me check 98³ just to confirm.

98³: 98*98 = 9604; then 9604*98. Let me calculate 9604*100 - 9604*2 = 960,400 - 19,208 = 941,192. So 98³ is 941,192, which is bigger than 912,673. So between 96 and 98, only 97 gives us the exact value. So AB is definitely 97. Therefore, A is 9 and B is 7, so A + B is 16. Let me just make sure that there isn't a different two-digit number that could cube to that. But since cube function is strictly increasing, there's only one number in that range that would give that value. So I think that's solid.

Therefore, the answer is 16.

**Final Answer**
The value of $A + B$ is \boxed{16}.

To solve the problem, we are given that the cube of the two-digit integer "AB" equals 912,673. Our goal is to find the sum $ A + B $, where "AB" represents the two-digit number formed by the digits A and B.

---

### Step 1: Understanding the Relationship

We interpret the two-digit number "AB" as a variable representing a number between 10 and 99. Let's denote this number as $ n = 10A + B $. We are told that:

$$
n^3 = 912,673
$$

This implies:

$$
n = \sqrt[3]{912,673}
$$

---

### Step 2: Estimating the Cube Root

We estimate the cube root of 912,673 by testing some values.

- $ 95^3 = 857,375 $
- $ 96^3 = 884,736 $
- $ 97^3 = 912,673 $ ✅ (matches the given cube)

Thus:

$$
n = 97
$$

This means the two-digit number "AB" is 97, where:
- $ A = 9 $
- $ B = 7 $

---

### Step 3: Calculating $ A + B $

$$
A + B = 9 + 7 = 16
$$

---

### Final Answer

$$
\boxed{16}
$$

Solution:
Since $90^3=729,\!000$, $\text{AB}$ is greater than 90.  Therefore, $\text{A}=9$.  Since the ones digit of $\text{AB}^3$ is 3, $\text{AB}$ must be odd.  The ones digit of $\text{AB}^3$ is the same as the ones digit of $\text{B}^3$, so we look at the ones digits of the cubes of the odd digits. \[
\begin{array}{c}
\text{The ones digit of }1^3 \text{ is } 1. \\ \text{The ones digit of }3^3 \text{ is } 7. \\ \text{The ones digit of }5^3 \text{ is } 5. \\ \text{The ones digit of }7^3 \text{ is } 3. \\ \text{The ones digit of }9^3 \text{ is } 9.
\end{array}
\] Only $7^3$ has a ones digit of 3, so $\text{B}=7$.  Therefore, $\text{A}+\text{B}=9+7=\boxed{16}$.


Explanation:
Okay, let me try to figure out how to solve this problem: Express $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$ as an integer. Alright, so I need to simplify this expression step by step. Let me start by recalling the order of operations, which is parentheses, exponents, multiplication and division, and then addition and subtraction. So PEMDAS, right?

First, let me write down the expression again to make sure I have it right: $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$. Hmm, no parentheses here, so I need to handle exponents first, then do the division and multiplication, and then the subtraction and addition. Wait, but division and multiplication have the same priority, so I should do them left to right. Similarly for addition and subtraction. Let me confirm that.

Yes, according to PEMDAS, after exponents, we do multiplication and division from left to right, then addition and subtraction from left to right. So let me break it down step by step.

First, let me compute each term separately. Let me see:

Starting with $5^5 \div 5^4$. I remember that when you divide exponents with the same base, you subtract the exponents. So $5^5 \div 5^4 = 5^{5-4} = 5^1 = 5$. That seems straightforward.

Next term is $-5^3$. Wait, is that negative 5 cubed or negative 5 to the power of 3? Well, according to order of operations, exponents come before multiplication, so it's negative (5^3). So $-5^3 = - (5^3) = -125$. Let me check. If it were (-5)^3, it would be -125 as well, but since there's no parentheses, it's just the negative of 5 cubed. So that's -125. Okay.

Third term is $5^2 \cdot 5^1$. Again, same base, so when multiplying, we add exponents. So $5^{2+1} = 5^3 = 125$. Wait, so that term is +125?

So putting it all together, the expression simplifies to 5 - 125 + 125. Let me write that down:

First term: 5

Second term: -125

Third term: +125

So combining them: 5 - 125 + 125. Let me compute that step by step.

Starting with 5 - 125. That would be... 5 - 125 is like subtracting 125 from 5. Which is negative 120, right? Because 125 - 5 is 120, so 5 - 125 is -120.

Then, adding 125 to that result: -120 + 125. Hmm, that would be positive 5. Because 125 - 120 is 5. So the total is 5?

Wait, that seems too simple. Let me check my steps again because sometimes when you have multiple operations, it's easy to make a mistake.

First, $5^5 \div 5^4$. Let me compute that without using exponent rules. Let me compute 5^5 and 5^4 first. 5^1 is 5, 5^2 is 25, 5^3 is 125, 5^4 is 625, 5^5 is 3125. So 3125 divided by 625. Let me do that division. 625 times 5 is 3125, right? Because 625*5 = 3125. So yes, 3125 divided by 625 is 5. So that term is indeed 5. Okay, that checks out.

Then the next term is -5^3. As we said, that is -125. Then the last term is 5^2 * 5^1. 5^2 is 25, 5^1 is 5, so 25*5 is 125. So that term is +125. Therefore, combining them: 5 - 125 + 125.

So, 5 - 125 is -120, and then -120 + 125 is indeed 5. So the entire expression simplifies to 5. Wait, so the answer is 5? That seems surprisingly simple, but maybe that's correct. Let me try another way to verify.

Alternatively, maybe I can combine the terms differently. Let me think. The original expression is $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$. Let me handle all the exponent operations first. So:

First term: $5^5 \div 5^4$ is 5^(5-4) = 5^1 = 5

Second term: -5^3 is -125

Third term: 5^2 * 5^1 is 5^(2+1) = 5^3 = 125

So altogether, 5 - 125 + 125. Which is 5 + (-125 + 125). Since -125 + 125 is 0, so 5 + 0 = 5. So that's another way to see it. So yes, that's 5.

Alternatively, if I didn't simplify the exponents first, but did the arithmetic step by step, would I get the same result?

Let me try that. Let me compute each term:

First term: 5^5 / 5^4. As above, 3125 / 625 = 5.

Second term: -5^3. Which is -125.

Third term: 5^2 * 5^1. Which is 25 * 5 = 125.

So, adding them up: 5 - 125 + 125. Which again is 5.

Alternatively, maybe someone could misinterpret the order of operations and do something wrong? For example, if someone thought that subtraction and addition are done before division and multiplication, but that's not the case. Let me check if that's possible.

Wait, according to PEMDAS, exponents are done first, then multiplication and division, then addition and subtraction. So the original expression is:

First, handle exponents:

- $5^5$ is 3125

- $5^4$ is 625

- $5^3$ is 125

- $5^2$ is 25

- $5^1$ is 5

So the expression becomes:

3125 ÷ 625 - 125 + 25 * 5

Then, handle division and multiplication from left to right.

First, 3125 ÷ 625 = 5.

Then, 25 * 5 = 125.

So now the expression is:

5 - 125 + 125

Which again is 5. So yes, same result.

Alternatively, if someone incorrectly did the multiplication before division, but that's not the case. Because division and multiplication have equal priority and are done left to right. So in the expression 3125 ÷ 625 - 125 + 25 * 5, you first do 3125 ÷ 625 = 5, then 25 * 5 = 125, so now you have 5 - 125 + 125.

Alternatively, if someone thought that multiplication should be done before division, they might have done 25 * 5 first, which is 125, and then 3125 ÷ 625 - 125 + 125. But that's still the same as before because you do the division and multiplication left to right. Wait, in the original expression, after exponents, the operations are division, subtraction, multiplication, addition. So since division and multiplication are at the same level, you do them left to right. So in the original expression, after exponents, you have division, then subtraction, then multiplication, then addition? Wait, no, the original expression is:

Original expression is:

$5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$

So after exponents are computed, it's:

(3125 ÷ 625) - 125 + (25 * 5)

So first you do 3125 ÷ 625 = 5, then you do 25 * 5 = 125. So you have 5 - 125 + 125.

So in this case, the multiplication (25*5) comes after the division (3125/625) because they are separated by operations. So since they are on different parts of the expression, you do them left to right. So first division, then multiplication.

But if the expression had been, say, 5^5 ÷ (5^4 - 5^3) + 5^2 * 5^1, then the parentheses would change things, but there are no parentheses here.

So, yeah, I think my steps are correct.

But let me check once again by calculating numerically:

Compute each term:

First term: 5^5 / 5^4 = 3125 / 625 = 5.

Second term: -5^3 = -125

Third term: 5^2 * 5^1 = 25 * 5 = 125

Then total is 5 - 125 + 125.

Calculating 5 - 125 first: 5 - 125 = -120. Then, -120 + 125 = 5.

Alternatively, if I rearrange the terms, since addition and subtraction are commutative? Wait, no, addition and subtraction are not commutative in the sense that you can rearrange them? Wait, actually, addition and subtraction can be rearranged if you consider the signs. For example, 5 - 125 + 125 is the same as 5 + (-125 + 125) = 5 + 0 = 5.

So, yes, that works.

Alternatively, if I thought of it as 5 + (-125 + 125). Since -125 + 125 is zero, so 5 + 0 is 5.

So, regardless of how I approach it, the result seems to be 5.

Is there any possibility that I misapplied the exponent rules? Let me check:

For the first term, $5^5 \div 5^4$ is indeed 5^{5-4} = 5^1 = 5. Correct.

For the third term, $5^2 \cdot 5^1$ is 5^{2+1} = 5^3 = 125. Correct.

The second term is straightforward: -5^3 is - (5^3) = -125. Correct.

So all the exponent rules are applied correctly.

Alternatively, if someone thought that the negative sign is part of the exponent, but since it's written as -5^3, without parentheses, it's only the 5 that is raised to the power. If it were (-5)^3, that would be -125 as well, so either way, it's -125. So no issue there.

Therefore, I think my answer is correct. The entire expression simplifies to 5.

But just to make sure, let me compute the entire original expression step by step without simplifying exponents first.

Original expression: $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$

Calculate each term:

First term: 5^5 is 3125. Divide by 5^4, which is 625. 3125 ÷ 625 = 5. So first term is 5.

Second term: -5^3. 5^3 is 125, so -125.

Third term: 5^2 * 5^1. 5^2 is 25, 5^1 is 5. 25*5 = 125. So third term is 125.

Now, combine all three terms: 5 - 125 + 125.

Compute 5 - 125 = -120.

Then, -120 + 125 = 5.

Same result.

Alternatively, if I compute all terms numerically first and then combine:

First term: 5

Second term: -125

Third term: +125

So total: 5 - 125 + 125 = 5.

Therefore, regardless of the method, it seems that the answer is 5.

Therefore, I think that is the correct answer.

But let me check once again if I misread the original problem. The original problem is:

Express $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$ as an integer.

Yes, that's what I solved. So no, I didn't misread.

Wait, another thought: Maybe there's an order of operations mistake? For example, if someone thought that the division and multiplication are done after the exponents, but also if we have to consider that subtraction is done before addition? Wait, but according to PEMDAS, after exponents, you do multiplication and division left to right, then addition and subtraction left to right. So in the expression 5 - 125 + 125, it's left to right, so 5 - 125 is -120, then -120 + 125 is 5.

Alternatively, if someone thought that addition comes before subtraction, they might do 5 + 125 first, which would be 130, then subtract 125, resulting in 5. Wait, that would also give 5. Wait, but addition and subtraction have the same priority and are done left to right. So regardless of the order, if you have 5 - 125 + 125, it's the same as 5 + (-125) + 125, which is 5 + 0 = 5. So even if you rearrange them, you still get 5.

So, even if someone does it in a different order, as long as they follow the left to right rule, they get the same answer.

Therefore, I think the answer is definitely 5.

I can't find any mistakes in my reasoning, and multiple methods lead to the same result. Therefore, I feel confident that the answer is 5.

**Final Answer**
\boxed{5}

To evaluate the expression:

$$
5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1
$$

we proceed by applying the **order of operations** (PEMDAS): Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

---

### Step 1: Simplify Exponents

We’ll simplify each term with an exponent:

- $5^5 = 3125$
- $5^4 = 625$
- $5^3 = 125$
- $5^2 = 25$
- $5^1 = 5$

So the expression becomes:

$$
\frac{3125}{625} - 125 + 25 \cdot 5
$$

---

### Step 2: Perform Division and Multiplication (from left to right)

**First Term:**

$$
\frac{3125}{625} = 5
$$

**Second Term:**

$$
25 \cdot 5 = 125
$$

Now the expression is simplified to:

$$
5 - 125 + 125
$$

---

### Step 3: Perform Addition and Subtraction (from left to right)

$$
5 - 125 = -120
$$
$$
-120 + 125 = 5
$$

---

### Final Answer

$$
\boxed{5}
$$

[Assistant]
Explanation:
Okay, let me try to figure out how to solve this problem: Express $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$ as an integer. Alright, so I need to simplify this expression step by step. Let me start by recalling the order of operations, which is parentheses, exponents, multiplication and division, and then addition and subtraction. So PEMDAS, right?

First, let me write down the expression again to make sure I have it right: $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$. Hmm, no parentheses here, so I need to handle exponents first, then do the division and multiplication, and then the subtraction and addition. Wait, but division and multiplication have the same priority, so I should do them left to right. Similarly for addition and subtraction. Let me confirm that.

Yes, according to PEMDAS, after exponents, we do multiplication and division from left to right, then addition and subtraction from left to right. So let me break it down step by step.

First, let me compute each term separately. Let me see:

Starting with $5^5 \div 5^4$. I remember that when you divide exponents with the same base, you subtract the exponents. So $5^5 \div 5^4 = 5^{5-4} = 5^1 = 5$. That seems straightforward.

Next term is $-5^3$. Wait, is that negative 5 cubed or negative 5 to the power of 3? Well, according to order of operations, exponents come before multiplication, so it's negative (5^3). So $-5^3 = - (5^3) = -125$. Let me check. If it were (-5)^3, it would be -125 as well, but since there's no parentheses, it's just the negative of 5 cubed. So that's -125. Okay.

Third term is $5^2 \cdot 5^1$. Again, same base, so when multiplying, we add exponents. So $5^{2+1} = 5^3 = 125$. Wait, so that term is +125?

So putting it all together, the expression simplifies to 5 - 125 + 125. Let me write that down:

First term: 5

Second term: -125

Third term: +125

So combining them: 5 - 125 + 125. Let me compute that step by step.

Starting with 5 - 125. That would be... 5 - 125 is like subtracting 125 from 5. Which is negative 120, right? Because 125 - 5 is 120, so 5 - 125 is -120.

Then, adding 125 to that result: -120 + 125. Hmm, that would be positive 5. Because 125 - 120 is 5. So the total is 5?

Wait, that seems too simple. Let me check my steps again because sometimes when you have multiple operations, it's easy to make a mistake.

First, $5^5 \div 5^4$. Let me compute that without using exponent rules. Let me compute 5^5 and 5^4 first. 5^1 is 5, 5^2 is 25, 5^3 is 125, 5^4 is 625, 5^5 is 3125. So 3125 divided by 625. Let me do that division. 625 times 5 is 3125, right? Because 625*5 = 3125. So yes, 3125 divided by 625 is 5. So that term is indeed 5. Okay, that checks out.

Then the next term is -5^3. As we said, that is -125. Then the last term is 5^2 * 5^1. 5^2 is 25, 5^1 is 5, so 25*5 is 125. So that term is +125. Therefore, combining them: 5 - 125 + 125.

So, 5 - 125 is -120, and then -120 + 125 is indeed 5. So the entire expression simplifies to 5. Wait, so the answer is 5? That seems surprisingly simple, but maybe that's correct. Let me try another way to verify.

Alternatively, maybe I can combine the terms differently. Let me think. The original expression is $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$. Let me handle all the exponent operations first. So:

First term: $5^5 \div 5^4$ is 5^(5-4) = 5^1 = 5

Second term: -5^3 is -125

Third term: 5^2 * 5^1 is 5^(2+1) = 5^3 = 125

So altogether, 5 - 125 + 125. Which is 5 + (-125 + 125). Since -125 + 125 is 0, so 5 + 0 = 5. So that's another way to see it. So yes, that's 5.

Alternatively, if I didn't simplify the exponents first, but did the arithmetic step by step, would I get the same result?

Let me try that. Let me compute each term:

First term: 5^5 / 5^4. As above, 3125 / 625 = 5.

Second term: -5^3. Which is -125.

Third term: 5^2 * 5^1. Which is 25 * 5 = 125.

So, adding them up: 5 - 125 + 125. Which again is 5.

Alternatively, maybe someone could misinterpret the order of operations and do something wrong? For example, if someone thought that subtraction and addition are done before division and multiplication, but that's not the case. Let me check if that's possible.

Wait, according to PEMDAS, exponents are done first, then multiplication and division, then addition and subtraction. So the original expression is:

First, handle exponents:

- $5^5$ is 3125

- $5^4$ is 625

- $5^3$ is 125

- $5^2$ is 25

- $5^1$ is 5

So the expression becomes:

3125 ÷ 625 - 125 + 25 * 5

Then, handle division and multiplication from left to right.

First, 3125 ÷ 625 = 5.

Then, 25 * 5 = 125.

So now the expression is:

5 - 125 + 125

Which again is 5. So yes, same result.

Alternatively, if someone incorrectly did the multiplication before division, but that's not the case. Because division and multiplication have equal priority and are done left to right. So in the expression 3125 ÷ 625 - 125 + 25 * 5, you first do 3125 ÷ 625 = 5, then 25 * 5 = 125, so now you have 5 - 125 + 125.

Alternatively, if someone thought that multiplication should be done before division, they might have done 25 * 5 first, which is 125, and then 3125 ÷ 625 - 125 + 125. But that's still the same as before because you do the division and multiplication left to right. Wait, in the original expression, after exponents, the operations are division, subtraction, multiplication, addition. So since division and multiplication are at the same level, you do them left to right. So in the original expression, after exponents, you have division, then subtraction, then multiplication, then addition? Wait, no, the original expression is:

Original expression is:

$5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$

So after exponents are computed, it's:

(3125 ÷ 625) - 125 + (25 * 5)

So first you do 3125 ÷ 625 = 5, then you do 25 * 5 = 125. So you have 5 - 125 + 125.

So in this case, the multiplication (25*5) comes after the division (3125/625) because they are separated by operations. So since they are on different parts of the expression, you do them left to right. So first division, then multiplication.

But if the expression had been, say, 5^5 ÷ (5^4 - 5^3) + 5^2 * 5^1, then the parentheses would change things, but there are no parentheses here.

So, yeah, I think my steps are correct.

But let me check once again by calculating numerically:

Compute each term:

First term: 5^5 / 5^4 = 3125 / 625 = 5.

Second term: -5^3 = -125

Third term: 5^2 * 5^1 = 25 * 5 = 125

Then total is 5 - 125 + 125.

Calculating 5 - 125 first: 5 - 125 = -120. Then, -120 + 125 = 5.

Alternatively, if I rearrange the terms, since addition and subtraction are commutative? Wait, no, addition and subtraction are not commutative in the sense that you can rearrange them? Wait, actually, addition and subtraction can be rearranged if you consider the signs. For example, 5 - 125 + 125 is the same as 5 + (-125 + 125) = 5 + 0 = 5.

So, yes, that works.

Alternatively, if I thought of it as 5 + (-125 + 125). Since -125 + 125 is zero, so 5 + 0 is 5.

So, regardless of how I approach it, the result seems to be 5.

Is there any possibility that I misapplied the exponent rules? Let me check:

For the first term, $5^5 \div 5^4$ is indeed 5^{5-4} = 5^1 = 5. Correct.

For the third term, $5^2 \cdot 5^1$ is 5^{2+1} = 5^3 = 125. Correct.

The second term is straightforward: -5^3 is - (5^3) = -125. Correct.

So all the exponent rules are applied correctly.

Alternatively, if someone thought that the negative sign is part of the exponent, but since it's written as -5^3, without parentheses, it's only the 5 that is raised to the power. If it were (-5)^3, that would be -125 as well, so either way, it's -125. So no issue there.

Therefore, I think my answer is correct. The entire expression simplifies to 5.

But just to make sure, let me compute the entire original expression step by step without simplifying exponents first.

Original expression: $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$

Calculate each term:

First term: 5^5 is 3125. Divide by 5^4, which is 625. 3125 ÷ 625 = 5. So first term is 5.

Second term: -5^3. 5^3 is 125, so -125.

Third term: 5^2 * 5^1. 5^2 is 25, 5^1 is 5. 25*5 = 125. So third term is 125.

Now, combine all three terms: 5 - 125 + 125.

Compute 5 - 125 = -120.

Then, -120 + 125 = 5.

Same result.

Alternatively, if I compute all terms numerically first and then combine:

First term: 5

Second term: -125

Third term: +125

So total: 5 - 125 + 125 = 5.

Therefore, regardless of the method, it seems that the answer is 5.

Therefore, I think that is the correct answer.

But let me check once again if I misread the original problem. The original problem is:

Express $5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1$ as an integer.

Yes, that's what I solved. So no, I didn't misread.

Wait, another thought: Maybe there's an order of operations mistake? For example, if someone thought that the division and multiplication are done after the exponents, but also if we have to consider that subtraction is done before addition? Wait, but according to PEMDAS, after exponents, you do multiplication and division left to right, then addition and subtraction left to right. So in the expression 5 - 125 + 125, it's left to right, so 5 - 125 is -120, then -120 + 125 is 5.

Alternatively, if someone thought that addition comes before subtraction, they might do 5 + 125 first, which would be 130, then subtract 125, resulting in 5. Wait, that would also give 5. Wait, but addition and subtraction have the same priority and are done left to right. So regardless of the order, if you have 5 - 125 + 125, it's the same as 5 + (-125) + 125, which is 5 + 0 = 5. So even if you rearrange them, you still get 5.

So, even if someone does it in a different order, as long as they follow the left to right rule, they get the same answer.

Therefore, I think the answer is definitely 5.

I can't find any mistakes in my reasoning, and multiple methods lead to the same result. Therefore, I feel confident that the answer is 5.

**Final Answer**
\boxed{5}

To evaluate the expression:

$$
5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1
$$

we proceed by applying the **order of operations** (PEMDAS): Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

---

### Step 1: Simplify Exponents

We’ll simplify each term with an exponent:

- $5^5 = 3125$
- $5^4 = 625$
- $5^3 = 125$
- $5^2 = 25$
- $5^1 = 5$

So the expression becomes:

$$
\frac{3125}{625} - 125 + 25 \cdot 5
$$

---

### Step 2: Perform Division and Multiplication (from left to right)

**First Term:**

$$
\frac{3125}{625} = 5
$$

**Second Term:**

$$
25 \cdot 5 = 125
$$

Now the expression is simplified to:

$$
5 - 125 + 125
$$

---

### Step 3: Perform Addition and Subtraction (from left to right)

$$
5 - 125 = -120
$$
$$
-120 + 125 = 5
$$

---

### Final Answer

$$
\boxed{5}
$$

Solution:
Recall that $a^m\div a^n=a^{m-n}$ for positive integers $m>n$ and $a^m\cdot a^n=a^{m+n}$. Now we can write $5^5\div5^4-5^3+5^2\cdot5^1$ as $5^1-5^3+5^3$. Using the definition of subtraction and the associative property of addition, we get \begin{align*}
5^1-5^3+5^3&=5^1+-5^3+5^3\\
&=5^1+(-5^3+5^3)\\
&=5^1+0\\
&=\boxed{5}.
\end{align*}


Explanation:
Okay, so I need to find the midpoint between two points, (-5, 5) and (3, 7). Hmm, let me remember how to do this. I think there's a formula for the midpoint of a line segment. Let me recall... Is it something like averaging the x-coordinates and the y-coordinates? Yeah, that sounds right.

Wait, let me make sure. If you have two points, say (x₁, y₁) and (x₂, y₂), then the midpoint should be ((x₁ + x₂)/2, (y₁ + y₂)/2). Yeah, that seems familiar from what I studied before. So basically, you take the average of the x-values and the average of the y-values.

Alright, let me apply that to the given points. The first point is (-5, 5), so x₁ is -5 and y₁ is 5. The second point is (3, 7), so x₂ is 3 and y₂ is 7.

First, let me calculate the x-coordinate of the midpoint. That would be (-5 + 3)/2. Let me do the addition first: -5 + 3 is... let me think. If I have a debt of 5 and then I get 3, I still owe 2, right? So that's -2. Then divide by 2. So -2 divided by 2 is -1. Okay, so the x-coordinate is -1.

Now for the y-coordinate. That would be (5 + 7)/2. Adding those together: 5 + 7 is 12. Then divide by 2, so 12 divided by 2 is 6. So the y-coordinate is 6.

Putting those together, the midpoint should be (-1, 6). Let me just check my calculations again to make sure I didn't make a mistake.

Starting with the x-coordinate: (-5 + 3)/2. Yes, that's (-2)/2 which is -1. For the y-coordinate: 5 + 7 is 12, divided by 2 is 6. Yep, that seems right.

Wait, maybe I should visualize it. If I plot the points (-5,5) and (3,7), the midpoint should be exactly halfway between them. Let me think about the horizontal distance. From x = -5 to x = 3. The distance between them is 3 - (-5) = 8 units. Half of that is 4 units. So starting at -5 and moving 4 units to the right would be -5 + 4 = -1. That matches the x-coordinate I found earlier.

Similarly for the vertical distance. From y = 5 to y = 7, that's 2 units. Half of that is 1 unit. Starting at 5 and moving up 1 unit gets me to 6. Which matches the y-coordinate.

Okay, so both methods give me the same result. That makes me more confident that (-1, 6) is correct.

Alternatively, maybe I can use the midpoint formula step by step again. Let me write down the formula again: midpoint M is ((x₁ + x₂)/2, (y₁ + y₂)/2). Plugging in the values:

x₁ = -5, x₂ = 3. So (-5 + 3)/2 = (-2)/2 = -1.

y₁ = 5, y₂ = 7. So (5 + 7)/2 = 12/2 = 6.

Yep, same result.

I think that's solid. There's no way I could have messed up the arithmetic here. The numbers aren't too bad. Let me check once more.

-5 + 3: negative five plus three. Since 5 is bigger than 3, it's negative. 5 - 3 is 2, so -2. Divided by 2 is -1.

5 + 7: that's 12, divided by 2 is 6.

Perfect.

Another way to think about it: if I move from the first point to the second, I change x by 3 - (-5) = 8 and y by 7 - 5 = 2. Then the midpoint is halfway, so half of 8 is 4, half of 2 is 1. So adding 4 to x-coordinate of the first point: -5 + 4 = -1, and adding 1 to y-coordinate: 5 + 1 = 6. Yep, same answer.

So all methods converge on (-1, 6). Therefore, I think that is the correct midpoint.

I don't think there's any other way this could go wrong. Unless maybe I mixed up the coordinates? Let me confirm that the points are given as (-5,5) and (3,7). The problem says "the line segment between (-5,5) and (3,7)", so yeah, that's correct.

No, I think that's all. I can't see any mistakes here. The answer should be (-1, 6).

**Final Answer**
The midpoint of the line segment is \boxed{(-1, 6)}.

To find the **midpoint** of a line segment between two points on a coordinate plane, we use the **midpoint formula**:

$$
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

Where:
- $(x_1, y_1)$ are the coordinates of the first point,
- $(x_2, y_2)$ are the coordinates of the second point.

---

### Step 1: Identify the Coordinates

We are given the two endpoints of the line segment:
- Point A: $(-5, 5)$
- Point B: $(3, 7)$

So,
- $x_1 = -5$, $y_1 = 5$
- $x_2 = 3$, $y_2 = 7$

---

### Step 2: Compute the Midpoint

#### a) Calculate the x-coordinate of the midpoint:
$$
\frac{-5 + 3}{2} = \frac{-2}{2} = -1
$$

#### b) Calculate the y-coordinate of the midpoint:
$$
\frac{5 + 7}{2} = \frac{12}{2} = 6
$$

---

### Step 3: Combine the Coordinates

Thus, the midpoint is:
$$
(-1, 6)
$$

---

### Final Answer

$$
\boxed{(-1, 6)}
$$

[Assistant]
Explanation:
Okay, so I need to find the midpoint between two points, (-5, 5) and (3, 7). Hmm, let me remember how to do this. I think there's a formula for the midpoint of a line segment. Let me recall... Is it something like averaging the x-coordinates and the y-coordinates? Yeah, that sounds right.

Wait, let me make sure. If you have two points, say (x₁, y₁) and (x₂, y₂), then the midpoint should be ((x₁ + x₂)/2, (y₁ + y₂)/2). Yeah, that seems familiar from what I studied before. So basically, you take the average of the x-values and the average of the y-values.

Alright, let me apply that to the given points. The first point is (-5, 5), so x₁ is -5 and y₁ is 5. The second point is (3, 7), so x₂ is 3 and y₂ is 7.

First, let me calculate the x-coordinate of the midpoint. That would be (-5 + 3)/2. Let me do the addition first: -5 + 3 is... let me think. If I have a debt of 5 and then I get 3, I still owe 2, right? So that's -2. Then divide by 2. So -2 divided by 2 is -1. Okay, so the x-coordinate is -1.

Now for the y-coordinate. That would be (5 + 7)/2. Adding those together: 5 + 7 is 12. Then divide by 2, so 12 divided by 2 is 6. So the y-coordinate is 6.

Putting those together, the midpoint should be (-1, 6). Let me just check my calculations again to make sure I didn't make a mistake.

Starting with the x-coordinate: (-5 + 3)/2. Yes, that's (-2)/2 which is -1. For the y-coordinate: 5 + 7 is 12, divided by 2 is 6. Yep, that seems right.

Wait, maybe I should visualize it. If I plot the points (-5,5) and (3,7), the midpoint should be exactly halfway between them. Let me think about the horizontal distance. From x = -5 to x = 3. The distance between them is 3 - (-5) = 8 units. Half of that is 4 units. So starting at -5 and moving 4 units to the right would be -5 + 4 = -1. That matches the x-coordinate I found earlier.

Similarly for the vertical distance. From y = 5 to y = 7, that's 2 units. Half of that is 1 unit. Starting at 5 and moving up 1 unit gets me to 6. Which matches the y-coordinate.

Okay, so both methods give me the same result. That makes me more confident that (-1, 6) is correct.

Alternatively, maybe I can use the midpoint formula step by step again. Let me write down the formula again: midpoint M is ((x₁ + x₂)/2, (y₁ + y₂)/2). Plugging in the values:

x₁ = -5, x₂ = 3. So (-5 + 3)/2 = (-2)/2 = -1.

y₁ = 5, y₂ = 7. So (5 + 7)/2 = 12/2 = 6.

Yep, same result.

I think that's solid. There's no way I could have messed up the arithmetic here. The numbers aren't too bad. Let me check once more.

-5 + 3: negative five plus three. Since 5 is bigger than 3, it's negative. 5 - 3 is 2, so -2. Divided by 2 is -1.

5 + 7: that's 12, divided by 2 is 6.

Perfect.

Another way to think about it: if I move from the first point to the second, I change x by 3 - (-5) = 8 and y by 7 - 5 = 2. Then the midpoint is halfway, so half of 8 is 4, half of 2 is 1. So adding 4 to x-coordinate of the first point: -5 + 4 = -1, and adding 1 to y-coordinate: 5 + 1 = 6. Yep, same answer.

So all methods converge on (-1, 6). Therefore, I think that is the correct midpoint.

I don't think there's any other way this could go wrong. Unless maybe I mixed up the coordinates? Let me confirm that the points are given as (-5,5) and (3,7). The problem says "the line segment between (-5,5) and (3,7)", so yeah, that's correct.

No, I think that's all. I can't see any mistakes here. The answer should be (-1, 6).

**Final Answer**
The midpoint of the line segment is \boxed{(-1, 6)}.

To find the **midpoint** of a line segment between two points on a coordinate plane, we use the **midpoint formula**:

$$
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

Where:
- $(x_1, y_1)$ are the coordinates of the first point,
- $(x_2, y_2)$ are the coordinates of the second point.

---

### Step 1: Identify the Coordinates

We are given the two endpoints of the line segment:
- Point A: $(-5, 5)$
- Point B: $(3, 7)$

So,
- $x_1 = -5$, $y_1 = 5$
- $x_2 = 3$, $y_2 = 7$

---

### Step 2: Compute the Midpoint

#### a) Calculate the x-coordinate of the midpoint:
$$
\frac{-5 + 3}{2} = \frac{-2}{2} = -1
$$

#### b) Calculate the y-coordinate of the midpoint:
$$
\frac{5 + 7}{2} = \frac{12}{2} = 6
$$

---

### Step 3: Combine the Coordinates

Thus, the midpoint is:
$$
(-1, 6)
$$

---

### Final Answer

$$
\boxed{(-1, 6)}
$$

Solution: Applying the midpoint formula gives us $$\left(\frac{-5+3}{2},\frac{5+7}{2}\right)=\boxed{(-1,6)}.$$


Explanation:
Okay, so I need to find the smallest possible angle between vectors a and b, given their magnitudes and the cross product. Let me recall what I know about cross products and angles between vectors.

First, the magnitude of the cross product of two vectors a and b is given by the formula:

$$
\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin(\theta)
$$

where θ is the angle between the vectors a and b.

So, if I can find sin(θ), then I can find θ itself. The problem gives me the cross product vector, which is (3, 2, 6). Therefore, its magnitude should be the square root of (3² + 2² + 6²). Let me calculate that:

First, square each component:
3² = 9,
2² = 4,
6² = 36.

Adding them up: 9 + 4 + 36 = 49.

So the magnitude of the cross product is sqrt(49) = 7.

Wait, so according to the formula:

$$
\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin(\theta)
$$

We have:

7 = 2 * 7 * sin(θ)

Let me check that. The magnitude of a is 2, magnitude of b is 7. So 2*7 is 14. Then 14*sin(theta) = 7. Therefore, sin(theta) = 7 / 14 = 1/2.

So sin(theta) = 1/2. Then theta is arcsin(1/2). The angles that satisfy this are 30 degrees and 150 degrees, since sine is positive in the first and second quadrants.

But the question is asking for the smallest possible angle between a and b. So between 30 degrees and 150 degrees, the smaller one is 30 degrees. Therefore, the smallest possible angle is 30 degrees.

Wait, but hold on. Is there any possibility that the cross product's magnitude could be different? Let me double-check my calculation.

Given the cross product vector is (3, 2, 6), so magnitude squared is 3² + 2² + 6² = 9 + 4 + 36 = 49, so magnitude is 7. That's correct. Then 7 equals ||a|| ||b|| sin(theta), which is 2*7*sin(theta) = 14 sin(theta). Therefore sin(theta) = 7/14 = 1/2.

Yes, that's correct. So theta is arcsin(1/2). Which is 30 degrees or 150 degrees. Since we need the smallest angle between two vectors, it's always between 0 and 180 degrees, so the smallest one is 30 degrees.

Wait, but let me confirm if there's any other factors here. For example, does the cross product direction affect the angle? Well, the magnitude of the cross product only depends on the sine of the angle between them. The direction is perpendicular to both vectors, but since we're only given the magnitude, we don't need to consider direction for this problem.

Therefore, the answer should be 30 degrees.

But let me think again. The cross product's magnitude is dependent on the sine of the angle. So if sin(theta) = 1/2, theta can be 30° or 150°, but the angle between two vectors is defined as the smallest angle between them when they are placed tail to tail, so it's always between 0° and 180°, so the smallest possible angle is 30°.

Therefore, the answer should be 30 degrees.

Wait, but is there a possibility that the cross product could be different? For example, if the cross product was a different vector, but here the cross product is given as (3,2,6). So the magnitude is fixed. Therefore, the calculation is correct.

Therefore, the smallest angle is 30 degrees.

But just to be thorough, let me check if there are any restrictions on the angle. For example, if the cross product's magnitude was zero, the angle would be zero, but here it's non-zero. Also, since sin(theta) is positive, the angle is between 0 and 180, which is consistent.

Therefore, I think my reasoning is correct. The minimal angle is 30 degrees.

Wait, but maybe there is a trick here. Let me check if the cross product vector being (3,2,6) is possible with the given magnitudes of a and b.

For example, the cross product's magnitude is 7, which is equal to ||a|| ||b|| sin(theta). But also, the cross product vector must satisfy the property that it's perpendicular to both a and b. However, since we are not given specific vectors a and b, just their magnitudes and the cross product vector, we can assume that such vectors exist.

Alternatively, maybe we can check if the cross product vector's magnitude is compatible with the given magnitudes. Since we have ||a|| = 2, ||b|| = 7, then the maximum possible magnitude of cross product is ||a|| ||b|| = 14, when the angle is 90 degrees. Since 7 is less than 14, that's okay. The minimum possible magnitude is zero, which occurs when the vectors are parallel. So 7 is in between, so it's possible.

Therefore, sin(theta) = 1/2, which gives theta as 30° or 150°, so the smallest angle is 30°, which is the answer.

Therefore, I think the answer is 30 degrees.

**Final Answer**
The smallest possible angle between $\mathbf{a}$ and $\mathbf{b}$ is $\boxed{30}$ degrees.

To find the **smallest possible angle** between two vectors $\mathbf{a}$ and $\mathbf{b}$, we are given the following:

- $\|\mathbf{a}\| = 2$
- $\|\mathbf{b}\| = 7$
- $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}$

---

### Step 1: Use the magnitude of the cross product

The magnitude of the cross product of two vectors is a well-known formula:

$$
\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin(\theta)
$$

where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

We are given the resultant vector of the cross product, $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}$. Its magnitude is:

$$
\|\mathbf{a} \times \mathbf{b}\| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7
$$

Substitute into the cross product magnitude formula:

$$
7 = 2 \cdot 7 \cdot \sin(\theta)
\Rightarrow \sin(\theta) = \frac{7}{14} = \frac{1}{2}
$$

---

### Step 2: Determine the angle(s) satisfying $\sin(\theta) = \frac{1}{2}$

Recall that:

- $\sin(\theta) = \frac{1}{2}$ when $\theta = 30^\circ$ or $150^\circ$, within the interval $[0^\circ, 180^\circ]$ (which is the typical range for the angle between vectors).

Since we're asked for the **smallest possible** angle between the vectors, the answer is:

$$
\theta = 30^\circ
$$

---

### Final Answer

$$
\boxed{30}
$$

[Assistant]
Explanation:
Okay, so I need to find the smallest possible angle between vectors a and b, given their magnitudes and the cross product. Let me recall what I know about cross products and angles between vectors.

First, the magnitude of the cross product of two vectors a and b is given by the formula:

$$
\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin(\theta)
$$

where θ is the angle between the vectors a and b.

So, if I can find sin(θ), then I can find θ itself. The problem gives me the cross product vector, which is (3, 2, 6). Therefore, its magnitude should be the square root of (3² + 2² + 6²). Let me calculate that:

First, square each component:
3² = 9,
2² = 4,
6² = 36.

Adding them up: 9 + 4 + 36 = 49.

So the magnitude of the cross product is sqrt(49) = 7.

Wait, so according to the formula:

$$
\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin(\theta)
$$

We have:

7 = 2 * 7 * sin(θ)

Let me check that. The magnitude of a is 2, magnitude of b is 7. So 2*7 is 14. Then 14*sin(theta) = 7. Therefore, sin(theta) = 7 / 14 = 1/2.

So sin(theta) = 1/2. Then theta is arcsin(1/2). The angles that satisfy this are 30 degrees and 150 degrees, since sine is positive in the first and second quadrants.

But the question is asking for the smallest possible angle between a and b. So between 30 degrees and 150 degrees, the smaller one is 30 degrees. Therefore, the smallest possible angle is 30 degrees.

Wait, but hold on. Is there any possibility that the cross product's magnitude could be different? Let me double-check my calculation.

Given the cross product vector is (3, 2, 6), so magnitude squared is 3² + 2² + 6² = 9 + 4 + 36 = 49, so magnitude is 7. That's correct. Then 7 equals ||a|| ||b|| sin(theta), which is 2*7*sin(theta) = 14 sin(theta). Therefore sin(theta) = 7/14 = 1/2.

Yes, that's correct. So theta is arcsin(1/2). Which is 30 degrees or 150 degrees. Since we need the smallest angle between two vectors, it's always between 0 and 180 degrees, so the smallest one is 30 degrees.

Wait, but let me confirm if there's any other factors here. For example, does the cross product direction affect the angle? Well, the magnitude of the cross product only depends on the sine of the angle between them. The direction is perpendicular to both vectors, but since we're only given the magnitude, we don't need to consider direction for this problem.

Therefore, the answer should be 30 degrees.

But let me think again. The cross product's magnitude is dependent on the sine of the angle. So if sin(theta) = 1/2, theta can be 30° or 150°, but the angle between two vectors is defined as the smallest angle between them when they are placed tail to tail, so it's always between 0° and 180°, so the smallest possible angle is 30°.

Therefore, the answer should be 30 degrees.

Wait, but is there a possibility that the cross product could be different? For example, if the cross product was a different vector, but here the cross product is given as (3,2,6). So the magnitude is fixed. Therefore, the calculation is correct.

Therefore, the smallest angle is 30 degrees.

But just to be thorough, let me check if there are any restrictions on the angle. For example, if the cross product's magnitude was zero, the angle would be zero, but here it's non-zero. Also, since sin(theta) is positive, the angle is between 0 and 180, which is consistent.

Therefore, I think my reasoning is correct. The minimal angle is 30 degrees.

Wait, but maybe there is a trick here. Let me check if the cross product vector being (3,2,6) is possible with the given magnitudes of a and b.

For example, the cross product's magnitude is 7, which is equal to ||a|| ||b|| sin(theta). But also, the cross product vector must satisfy the property that it's perpendicular to both a and b. However, since we are not given specific vectors a and b, just their magnitudes and the cross product vector, we can assume that such vectors exist.

Alternatively, maybe we can check if the cross product vector's magnitude is compatible with the given magnitudes. Since we have ||a|| = 2, ||b|| = 7, then the maximum possible magnitude of cross product is ||a|| ||b|| = 14, when the angle is 90 degrees. Since 7 is less than 14, that's okay. The minimum possible magnitude is zero, which occurs when the vectors are parallel. So 7 is in between, so it's possible.

Therefore, sin(theta) = 1/2, which gives theta as 30° or 150°, so the smallest angle is 30°, which is the answer.

Therefore, I think the answer is 30 degrees.

**Final Answer**
The smallest possible angle between $\mathbf{a}$ and $\mathbf{b}$ is $\boxed{30}$ degrees.

To find the **smallest possible angle** between two vectors $\mathbf{a}$ and $\mathbf{b}$, we are given the following:

- $\|\mathbf{a}\| = 2$
- $\|\mathbf{b}\| = 7$
- $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}$

---

### Step 1: Use the magnitude of the cross product

The magnitude of the cross product of two vectors is a well-known formula:

$$
\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin(\theta)
$$

where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

We are given the resultant vector of the cross product, $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}$. Its magnitude is:

$$
\|\mathbf{a} \times \mathbf{b}\| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7
$$

Substitute into the cross product magnitude formula:

$$
7 = 2 \cdot 7 \cdot \sin(\theta)
\Rightarrow \sin(\theta) = \frac{7}{14} = \frac{1}{2}
$$

---

### Step 2: Determine the angle(s) satisfying $\sin(\theta) = \frac{1}{2}$

Recall that:

- $\sin(\theta) = \frac{1}{2}$ when $\theta = 30^\circ$ or $150^\circ$, within the interval $[0^\circ, 180^\circ]$ (which is the typical range for the angle between vectors).

Since we're asked for the **smallest possible** angle between the vectors, the answer is:

$$
\theta = 30^\circ
$$

---

### Final Answer

$$
\boxed{30}
$$

Solution:
Let $\theta$ be the angle between $\mathbf{a}$ and $\mathbf{b}.$  Then
\[\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin \theta.\]From the given information, $7 = 14 \sin \theta,$ so $\sin \theta = \frac{1}{2}.$  Thus, the smallest possible value of $\theta$ is $\boxed{30^\circ}.$


Explanation:
Okay, so I need to find the length of the third side of a triangle when two sides are given as 1 unit and 3 units. The third side has to be an integer. Hmm, let me think.

First, I remember something about the triangle inequality theorem. It says that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. Right? So, if I have three sides, let's call them a, b, and c, then:

a + b > c
a + c > b
b + c > a

So in this case, the sides are 1, 3, and let's call the third side x. Since x is an integer, I need to find all integers x such that the three sides satisfy the triangle inequalities.

Let me write down the inequalities specifically for this problem.

First inequality: 1 + 3 > x
Which simplifies to 4 > x, or x < 4.

Second inequality: 1 + x > 3
Which simplifies to x > 3 - 1
So x > 2.

Third inequality: 3 + x > 1
Which simplifies to x > 1 - 3
So x > -2.

But since lengths can't be negative, the third inequality doesn't really restrict us because x has to be positive anyway. So combining the first two inequalities:

x must be greater than 2 and less than 4.

But x has to be an integer. So integers between 2 and 4. Let me think. If x must be greater than 2, then possible integers are 3, 4, 5,... but also less than 4. So less than 4 means x can be up to 3.999... so the integer less than 4 is 3.

Wait, so x has to be greater than 2 and less than 4. So integers in this range are only 3. Because 2 is not greater than 2, and 4 is not less than 4. So x must be 3.

But let me check again to make sure. Let me plug x=3 into the triangle inequalities.

First: 1 + 3 > 3? 4 > 3? Yes.
Second: 1 + 3 > 3? Same as first.
Third: 3 + 3 > 1? 6 > 1? Yes.
Wait, but the third inequality is 3 + x > 1. If x is 3, then 3 + 3 = 6 > 1, which is true.

But wait, maybe I should check all three inequalities.

So if x is 3, the three sides are 1, 3, 3.

Check all three inequalities:

1 + 3 > 3: 4 > 3 ✔
1 + 3 > 3: same as above ✔
3 + 3 > 1: 6 > 1 ✔

So 3 works.

But wait, is there another possible integer? Let me check.

If x was 2, then sides would be 1, 3, 2. Let's check the inequalities.

1 + 3 > 2: 4 > 2 ✔
1 + 2 > 3: 3 > 3? No, 3 is not greater than 3. So that's not valid. So 2 doesn't work.

If x was 4, sides would be 1, 3, 4. Check inequalities.

1 + 3 > 4? 4 > 4? No, 4 is not greater than 4. So that's invalid.

If x was 5, even more so, 1 + 3 = 4 < 5. Definitely invalid.

What if x is 0? Well, negative numbers are invalid. So only x=3 is possible.

Wait, but maybe I made a mistake with the inequalities. Let me re-examine them.

Original inequalities:

For sides a, b, c, each side must be less than the sum of the other two.

So for the sides 1, 3, x:

x < 1 + 3 → x < 4
1 < 3 + x → 1 - 3 < x → -2 < x
3 < 1 + x → 3 - 1 < x → 2 < x

So combining the two inequalities: 2 < x < 4.

Since x must be an integer, then x is 3.

Yes, that's the only integer between 2 and 4. So x=3.

Therefore, the length of the third side is 3 units.

But wait, just to be thorough, let me think if there's any other possibilities or if I made a mistake here.

Another thought: sometimes people get confused with the triangle inequality, thinking that it's about the sum being equal or something. But no, it's strictly greater.

So for example, if you have sides 1, 3, and 3, it is a valid triangle. If you have sides 1, 3, and 2, it's not a valid triangle because 1 + 2 is not greater than 3.

Similarly, 1,3,4 would have 1 + 3 = 4, which is not greater, so it's a degenerate triangle.

But the problem says "a triangle has sides measuring...", so it must be a valid triangle, not degenerate. So x must be strictly less than 4, and strictly greater than 2. Therefore, integer value 3.

So yeah, the answer is 3.

But let me check once more with another approach.

Alternatively, in a triangle, the length of the third side must be such that it's the difference between the other two sides and the sum.

Wait, more accurately, the third side must be greater than the difference of the other two sides and less than the sum.

So if we have two sides, say, a and b, then the third side c must satisfy |a - b| < c < a + b.

So in this case, a = 1, b = 3.

Then |1 - 3| = 2. So 2 < c < 4.

Again, c is between 2 and 4, so the integer value is 3.

So this method also gives 3.

Therefore, seems like 3 is the answer.

But just to think, is there any chance that the problem is referring to a different type of triangle? Like maybe a right triangle? But the problem doesn't specify that. It just says a triangle. So unless there's more information, we can't assume it's a right triangle.

Wait, but the problem says "the length of the third side is an integer value of units". So maybe there's only one possible integer.

So yes, seems like the answer is 3.

Therefore, I think the length of the third side is 3 units.

**Final Answer**
The length of the third side is \boxed{3}.

To determine the length of the third side of a triangle where the other two sides are 1 unit and 3 units, we apply the **triangle inequality theorem**, which states that the sum of the lengths of any two sides of a triangle must be **greater than** the length of the third side. This leads to the following three inequalities for sides $ a $, $ b $, and $ c $:

$$
a + b > c \\
a + c > b \\
b + c > a
$$

In our case, the two known side lengths are 1 and 3, and the third side is an unknown integer $ x $. Applying the inequalities:

1. $ 1 + 3 > x \Rightarrow 4 > x $ or $ x < 4 $
2. $ 1 + x > 3 \Rightarrow x > 2 $
3. $ 3 + x > 1 \Rightarrow x > -2 $

Note that the third inequality ($ x > -2 $) is always true since side lengths are positive. Therefore, combining the valid inequalities:

$$
2 < x < 4
$$

Since $ x $ must be an **integer**, the only integer that satisfies this inequality is:

$$
x = 3
$$

---

To confirm, if the sides of the triangle were 1, 3, and 3:

- $ 1 + 3 > 3 \Rightarrow 4 > 3 $ ✔
- $ 1 + 3 > 3 \Rightarrow 4 > 3 $ ✔
- $ 3 + 3 > 1 \Rightarrow 6 > 1 $ ✔

All conditions are satisfied, confirming that the third side is valid.

---

Thus, the length of the third side is:

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to find the length of the third side of a triangle when two sides are given as 1 unit and 3 units. The third side has to be an integer. Hmm, let me think.

First, I remember something about the triangle inequality theorem. It says that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. Right? So, if I have three sides, let's call them a, b, and c, then:

a + b > c
a + c > b
b + c > a

So in this case, the sides are 1, 3, and let's call the third side x. Since x is an integer, I need to find all integers x such that the three sides satisfy the triangle inequalities.

Let me write down the inequalities specifically for this problem.

First inequality: 1 + 3 > x
Which simplifies to 4 > x, or x < 4.

Second inequality: 1 + x > 3
Which simplifies to x > 3 - 1
So x > 2.

Third inequality: 3 + x > 1
Which simplifies to x > 1 - 3
So x > -2.

But since lengths can't be negative, the third inequality doesn't really restrict us because x has to be positive anyway. So combining the first two inequalities:

x must be greater than 2 and less than 4.

But x has to be an integer. So integers between 2 and 4. Let me think. If x must be greater than 2, then possible integers are 3, 4, 5,... but also less than 4. So less than 4 means x can be up to 3.999... so the integer less than 4 is 3.

Wait, so x has to be greater than 2 and less than 4. So integers in this range are only 3. Because 2 is not greater than 2, and 4 is not less than 4. So x must be 3.

But let me check again to make sure. Let me plug x=3 into the triangle inequalities.

First: 1 + 3 > 3? 4 > 3? Yes.
Second: 1 + 3 > 3? Same as first.
Third: 3 + 3 > 1? 6 > 1? Yes.
Wait, but the third inequality is 3 + x > 1. If x is 3, then 3 + 3 = 6 > 1, which is true.

But wait, maybe I should check all three inequalities.

So if x is 3, the three sides are 1, 3, 3.

Check all three inequalities:

1 + 3 > 3: 4 > 3 ✔
1 + 3 > 3: same as above ✔
3 + 3 > 1: 6 > 1 ✔

So 3 works.

But wait, is there another possible integer? Let me check.

If x was 2, then sides would be 1, 3, 2. Let's check the inequalities.

1 + 3 > 2: 4 > 2 ✔
1 + 2 > 3: 3 > 3? No, 3 is not greater than 3. So that's not valid. So 2 doesn't work.

If x was 4, sides would be 1, 3, 4. Check inequalities.

1 + 3 > 4? 4 > 4? No, 4 is not greater than 4. So that's invalid.

If x was 5, even more so, 1 + 3 = 4 < 5. Definitely invalid.

What if x is 0? Well, negative numbers are invalid. So only x=3 is possible.

Wait, but maybe I made a mistake with the inequalities. Let me re-examine them.

Original inequalities:

For sides a, b, c, each side must be less than the sum of the other two.

So for the sides 1, 3, x:

x < 1 + 3 → x < 4
1 < 3 + x → 1 - 3 < x → -2 < x
3 < 1 + x → 3 - 1 < x → 2 < x

So combining the two inequalities: 2 < x < 4.

Since x must be an integer, then x is 3.

Yes, that's the only integer between 2 and 4. So x=3.

Therefore, the length of the third side is 3 units.

But wait, just to be thorough, let me think if there's any other possibilities or if I made a mistake here.

Another thought: sometimes people get confused with the triangle inequality, thinking that it's about the sum being equal or something. But no, it's strictly greater.

So for example, if you have sides 1, 3, and 3, it is a valid triangle. If you have sides 1, 3, and 2, it's not a valid triangle because 1 + 2 is not greater than 3.

Similarly, 1,3,4 would have 1 + 3 = 4, which is not greater, so it's a degenerate triangle.

But the problem says "a triangle has sides measuring...", so it must be a valid triangle, not degenerate. So x must be strictly less than 4, and strictly greater than 2. Therefore, integer value 3.

So yeah, the answer is 3.

But let me check once more with another approach.

Alternatively, in a triangle, the length of the third side must be such that it's the difference between the other two sides and the sum.

Wait, more accurately, the third side must be greater than the difference of the other two sides and less than the sum.

So if we have two sides, say, a and b, then the third side c must satisfy |a - b| < c < a + b.

So in this case, a = 1, b = 3.

Then |1 - 3| = 2. So 2 < c < 4.

Again, c is between 2 and 4, so the integer value is 3.

So this method also gives 3.

Therefore, seems like 3 is the answer.

But just to think, is there any chance that the problem is referring to a different type of triangle? Like maybe a right triangle? But the problem doesn't specify that. It just says a triangle. So unless there's more information, we can't assume it's a right triangle.

Wait, but the problem says "the length of the third side is an integer value of units". So maybe there's only one possible integer.

So yes, seems like the answer is 3.

Therefore, I think the length of the third side is 3 units.

**Final Answer**
The length of the third side is \boxed{3}.

To determine the length of the third side of a triangle where the other two sides are 1 unit and 3 units, we apply the **triangle inequality theorem**, which states that the sum of the lengths of any two sides of a triangle must be **greater than** the length of the third side. This leads to the following three inequalities for sides $ a $, $ b $, and $ c $:

$$
a + b > c \\
a + c > b \\
b + c > a
$$

In our case, the two known side lengths are 1 and 3, and the third side is an unknown integer $ x $. Applying the inequalities:

1. $ 1 + 3 > x \Rightarrow 4 > x $ or $ x < 4 $
2. $ 1 + x > 3 \Rightarrow x > 2 $
3. $ 3 + x > 1 \Rightarrow x > -2 $

Note that the third inequality ($ x > -2 $) is always true since side lengths are positive. Therefore, combining the valid inequalities:

$$
2 < x < 4
$$

Since $ x $ must be an **integer**, the only integer that satisfies this inequality is:

$$
x = 3
$$

---

To confirm, if the sides of the triangle were 1, 3, and 3:

- $ 1 + 3 > 3 \Rightarrow 4 > 3 $ ✔
- $ 1 + 3 > 3 \Rightarrow 4 > 3 $ ✔
- $ 3 + 3 > 1 \Rightarrow 6 > 1 $ ✔

All conditions are satisfied, confirming that the third side is valid.

---

Thus, the length of the third side is:

$$
\boxed{3}
$$

Solution: Call the length of the third side $n$. By the triangle inequality, $1+3>n$ and $1+n>3$, or $2<n<4$. The only integer $n$ which satisfies this is $\boxed{3}$.


Explanation:
Okay, let's see. So there are twelve friends who went to dinner, and each ordered one meal. But the portions were so large that there was enough food for 18 people. Now, the question is, if they share, how many meals should they have ordered to have just enough food for the 12 of them?

Hmm. Let me try to break this down. So originally, twelve people ordered meals, but the food was enough for 18. That means each meal they ordered was actually enough for more than one person, right? So maybe each meal is like a portion that can serve multiple people?

So if twelve meals were ordered, and that's enough for 18 people, then we can figure out how much food each meal provides. Let me think. If twelve meals provide enough for 18 people, then each meal is providing food for 18 divided by 12 people. Let me write that as a fraction. So 18 divided by 12 is... let's see, 18/12 simplifies to 3/2, which is 1.5. So each meal is equivalent to 1.5 servings? Or maybe each meal can feed 1.5 people? That seems like the way to think about it.

So if each meal can feed 1.5 people, then to have enough food for 12 people, how many meals do they need? Well, if one meal feeds 1.5 people, then the number of meals needed for 12 people would be 12 divided by 1.5. Let me calculate that. 12 divided by 1.5. Hmm. 1.5 times 8 is 12, right? Because 1.5 times 2 is 3, times 4 is 6, times 8 is 12. So that would be 8 meals.

Wait, so originally they ordered 12 meals, which were enough for 18 people. But if they wanted to have just enough for 12 people, they should have ordered 8 meals? That seems like a lot less. Let me check my logic again.

Alternatively, maybe we can think of it as a ratio. The original number of meals is 12, and the original number of people is 12, but the food was enough for 18. So the amount of food is proportional to the number of meals. So if they want to have enough food for 12 people, how many meals should they order?

Wait, perhaps we can set up a proportion. Let me try that. Let me think. If 12 meals provide enough food for 18 people, then how many meals (let's call it x) would provide enough food for 12 people? So the ratio of meals to people should be the same. So 12 meals / 18 people = x meals / 12 people. Wait, is that correct?

Wait, actually, maybe it's the other way around. If more meals mean more food, so if you have more people, you need more meals. But if you want to have enough food for fewer people, you need fewer meals. So maybe the ratio is meals / people = constant? Let me think. Let's say that each person needs a certain amount of food. So if 12 meals are enough for 18 people, then each person needs 12 meals / 18 people = 2/3 of a meal per person. Therefore, for 12 people, they need 12 people * (2/3 meal/person) = 8 meals. So that matches my previous answer. So that seems correct.

Alternatively, maybe think in terms of how much food they had. If twelve meals were enough for 18 people, then the total amount of food is equivalent to 18 servings. But they only ordered 12 meals. Therefore, each meal is equivalent to 18/12 = 1.5 servings. So if they want to have 12 servings, they need 12 / 1.5 = 8 meals. Yep, same answer.

So regardless of how I approach it, it seems like 8 meals would be sufficient. So the answer is 8?

But let me just make sure I didn't flip any numbers. Let me check with another approach. Suppose each meal is designed for a certain number of people. Let's say that each meal is intended for 'n' people. Then, twelve meals would be intended for 12n people. However, they were enough for 18 people. Therefore, 12n = 18. Solving for n gives n = 18 / 12 = 1.5. So each meal is intended for 1.5 people. Therefore, if they want to have enough food for 12 people, they need to order 12 / 1.5 = 8 meals. Yep, same answer again.

Alternatively, maybe think about how much food each person gets. If there are twelve people, each person gets a certain amount of food. But the total food was enough for 18 people, meaning that each person only ate 12/18 = 2/3 of the meal. Wait, no, that might be another way to think about it. If twelve people ate, but there was enough food for 18, then each person ate 12/18 = 2/3 of a meal? Wait, maybe not.

Wait, let's think of the total amount of food as being equivalent to 18 meals. Because if each meal is considered as a unit, then twelve meals were ordered, but that's enough for 18 people. So the total food is 18 meal-units. Therefore, if they have 12 people, they need 12 meal-units. Therefore, the number of meals they should order is 12 meal-units divided by the amount of food per meal. Wait, but how much food is in each meal?

Wait, if twelve meals provided 18 meal-units, then each meal is 18 / 12 = 1.5 meal-units. Therefore, to get 12 meal-units, you need 12 / 1.5 = 8 meals. So again, same answer. Okay, so all these different approaches lead me to the same conclusion of 8 meals.

Wait, but let me think if there is a chance that the question is trying to trick me. For example, maybe the portions were so large that there was enough food for 18 people, meaning that if they had ordered 12 meals, they had enough for 18 people. Therefore, if they want to have just enough for 12 people, they need to order fewer meals. So the question is similar to: if 12 meals are enough for 18 people, how many meals are needed for 12 people?

Which is a classic proportion problem. So if 12 meals : 18 people = x meals : 12 people. So cross-multiplying, 12 * 12 = 18 * x. So 144 = 18x, so x = 144 / 18 = 8. Yep, same answer again.

Okay, so I think this is solid. The answer is 8. Therefore, they should have ordered 8 meals.

**Final Answer**
The friends should have ordered \boxed{8} meals.

To determine how many meals twelve friends should have ordered to have just enough food for the 12 of them, we need to understand how much food each meal actually provides, based on the initial scenario.

---

### Step 1: Understand the Original Situation

- **12 meals** were ordered.
- These meals were **sufficient for 18 people**.

This tells us that the **total food quantity is equivalent to 18 servings**, but the food originated from **12 meals**, meaning that the meals are portioned to feed more than one person each.

---

### Step 2: Determine the Portion per Meal

To find out how much food is in **each meal**, we divide the total food (for 18 people) by the number of meals:

$$
\frac{18 \text{ servings}}{12 \text{ meals}} = 1.5 \text{ servings per meal}
$$

So, **each meal can provide food for 1.5 people**.

---

### Step 3: Calculate How Many Meals Are Needed for 12 People

Now, we want to know how many meals are needed to feed **12 people**, given that each meal can serve **1.5 people**.

$$
\text{Number of meals needed} = \frac{12 \text{ people}}{1.5 \text{ servings per meal}} = 8 \text{ meals}
$$

---

### Final Answer

The twelve friends should have ordered:

$$
\boxed{8}
$$

meals to have just enough food for all of them.

[Assistant]
Explanation:
Okay, let's see. So there are twelve friends who went to dinner, and each ordered one meal. But the portions were so large that there was enough food for 18 people. Now, the question is, if they share, how many meals should they have ordered to have just enough food for the 12 of them?

Hmm. Let me try to break this down. So originally, twelve people ordered meals, but the food was enough for 18. That means each meal they ordered was actually enough for more than one person, right? So maybe each meal is like a portion that can serve multiple people?

So if twelve meals were ordered, and that's enough for 18 people, then we can figure out how much food each meal provides. Let me think. If twelve meals provide enough for 18 people, then each meal is providing food for 18 divided by 12 people. Let me write that as a fraction. So 18 divided by 12 is... let's see, 18/12 simplifies to 3/2, which is 1.5. So each meal is equivalent to 1.5 servings? Or maybe each meal can feed 1.5 people? That seems like the way to think about it.

So if each meal can feed 1.5 people, then to have enough food for 12 people, how many meals do they need? Well, if one meal feeds 1.5 people, then the number of meals needed for 12 people would be 12 divided by 1.5. Let me calculate that. 12 divided by 1.5. Hmm. 1.5 times 8 is 12, right? Because 1.5 times 2 is 3, times 4 is 6, times 8 is 12. So that would be 8 meals.

Wait, so originally they ordered 12 meals, which were enough for 18 people. But if they wanted to have just enough for 12 people, they should have ordered 8 meals? That seems like a lot less. Let me check my logic again.

Alternatively, maybe we can think of it as a ratio. The original number of meals is 12, and the original number of people is 12, but the food was enough for 18. So the amount of food is proportional to the number of meals. So if they want to have enough food for 12 people, how many meals should they order?

Wait, perhaps we can set up a proportion. Let me try that. Let me think. If 12 meals provide enough food for 18 people, then how many meals (let's call it x) would provide enough food for 12 people? So the ratio of meals to people should be the same. So 12 meals / 18 people = x meals / 12 people. Wait, is that correct?

Wait, actually, maybe it's the other way around. If more meals mean more food, so if you have more people, you need more meals. But if you want to have enough food for fewer people, you need fewer meals. So maybe the ratio is meals / people = constant? Let me think. Let's say that each person needs a certain amount of food. So if 12 meals are enough for 18 people, then each person needs 12 meals / 18 people = 2/3 of a meal per person. Therefore, for 12 people, they need 12 people * (2/3 meal/person) = 8 meals. So that matches my previous answer. So that seems correct.

Alternatively, maybe think in terms of how much food they had. If twelve meals were enough for 18 people, then the total amount of food is equivalent to 18 servings. But they only ordered 12 meals. Therefore, each meal is equivalent to 18/12 = 1.5 servings. So if they want to have 12 servings, they need 12 / 1.5 = 8 meals. Yep, same answer.

So regardless of how I approach it, it seems like 8 meals would be sufficient. So the answer is 8?

But let me just make sure I didn't flip any numbers. Let me check with another approach. Suppose each meal is designed for a certain number of people. Let's say that each meal is intended for 'n' people. Then, twelve meals would be intended for 12n people. However, they were enough for 18 people. Therefore, 12n = 18. Solving for n gives n = 18 / 12 = 1.5. So each meal is intended for 1.5 people. Therefore, if they want to have enough food for 12 people, they need to order 12 / 1.5 = 8 meals. Yep, same answer again.

Alternatively, maybe think about how much food each person gets. If there are twelve people, each person gets a certain amount of food. But the total food was enough for 18 people, meaning that each person only ate 12/18 = 2/3 of the meal. Wait, no, that might be another way to think about it. If twelve people ate, but there was enough food for 18, then each person ate 12/18 = 2/3 of a meal? Wait, maybe not.

Wait, let's think of the total amount of food as being equivalent to 18 meals. Because if each meal is considered as a unit, then twelve meals were ordered, but that's enough for 18 people. So the total food is 18 meal-units. Therefore, if they have 12 people, they need 12 meal-units. Therefore, the number of meals they should order is 12 meal-units divided by the amount of food per meal. Wait, but how much food is in each meal?

Wait, if twelve meals provided 18 meal-units, then each meal is 18 / 12 = 1.5 meal-units. Therefore, to get 12 meal-units, you need 12 / 1.5 = 8 meals. So again, same answer. Okay, so all these different approaches lead me to the same conclusion of 8 meals.

Wait, but let me think if there is a chance that the question is trying to trick me. For example, maybe the portions were so large that there was enough food for 18 people, meaning that if they had ordered 12 meals, they had enough for 18 people. Therefore, if they want to have just enough for 12 people, they need to order fewer meals. So the question is similar to: if 12 meals are enough for 18 people, how many meals are needed for 12 people?

Which is a classic proportion problem. So if 12 meals : 18 people = x meals : 12 people. So cross-multiplying, 12 * 12 = 18 * x. So 144 = 18x, so x = 144 / 18 = 8. Yep, same answer again.

Okay, so I think this is solid. The answer is 8. Therefore, they should have ordered 8 meals.

**Final Answer**
The friends should have ordered \boxed{8} meals.

To determine how many meals twelve friends should have ordered to have just enough food for the 12 of them, we need to understand how much food each meal actually provides, based on the initial scenario.

---

### Step 1: Understand the Original Situation

- **12 meals** were ordered.
- These meals were **sufficient for 18 people**.

This tells us that the **total food quantity is equivalent to 18 servings**, but the food originated from **12 meals**, meaning that the meals are portioned to feed more than one person each.

---

### Step 2: Determine the Portion per Meal

To find out how much food is in **each meal**, we divide the total food (for 18 people) by the number of meals:

$$
\frac{18 \text{ servings}}{12 \text{ meals}} = 1.5 \text{ servings per meal}
$$

So, **each meal can provide food for 1.5 people**.

---

### Step 3: Calculate How Many Meals Are Needed for 12 People

Now, we want to know how many meals are needed to feed **12 people**, given that each meal can serve **1.5 people**.

$$
\text{Number of meals needed} = \frac{12 \text{ people}}{1.5 \text{ servings per meal}} = 8 \text{ meals}
$$

---

### Final Answer

The twelve friends should have ordered:

$$
\boxed{8}
$$

meals to have just enough food for all of them.

Solution: If 12 people order $\frac{18}{12}=1\frac{1}{2}$ times too much food, they should have ordered $\frac{12}{\frac{3}{2}}=\frac{2}{3}\times 12=\boxed{8}$ meals.