Explanation:
Okay, so I need to find the difference between the positive square root of 64 and the cube root of 64. Hmm, let me think. First, let me recall what square roots and cube roots are.

The square root of a number is a value that, when multiplied by itself, gives the original number. So, the square root of 64 would be a number that when multiplied by itself equals 64. Similarly, the cube root of 64 is a number that when multiplied by itself three times equals 64.

Wait, the question specifies the positive square root. That makes sense because square roots can have both positive and negative values, but since they're asking for the positive one, I should focus on that.

Let me start with the square root of 64. I know that 8 times 8 is 64, so the square root of 64 is 8. Let me check that: 8*8 = 64. Yep, that's right. So the positive square root is definitely 8.

Now, the cube root of 64. Hmm, cube roots are a bit trickier. Let me think. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. So, what number multiplied by itself three times equals 64? Let me try some numbers.

Starting with 1: 1*1*1 = 1. Too small. How about 2? 2*2*2 = 8. Still too small. 3? 3*3*3 = 27. Still not 64. 4? 4*4*4 = 64. Oh, there we go! So the cube root of 64 is 4. Let me verify that: 4*4 is 16, and 16*4 is 64. Yep, that checks out.

So now, the problem asks for the difference between the positive square root of 64 and the cube root of 64. That means I need to subtract the cube root from the square root, or is it the other way around? Wait, the wording is "difference between A and B," which usually means A - B. But sometimes people might interpret it differently. Let me make sure.

The question says: "the difference between the positive square root of 64 and the cube root of 64." So, difference between A and B is typically A - B. So, positive square root (which is 8) minus cube root (which is 4). So 8 - 4. That would be 4.

But let me double-check. Maybe I should confirm the order. If it's difference between A and B, sometimes it's absolute value, but since they specify "positive square root," maybe they just want the subtraction as stated. Let me think. If they had said "the difference between the cube root and the square root," then it would be 4 - 8, which is -4, but since they specify "difference between the positive square root and the cube root," it's 8 - 4.

Alternatively, maybe they just want the absolute difference, but since both numbers are positive and 8 is larger than 4, the difference is 4.

But let me make sure I didn't mix up the roots. Let me recheck the square root and cube root.

Square root of 64: 8, because 8*8=64. Cube root of 64: 4, because 4*4*4=64. So yes, that's correct.

Therefore, subtracting the cube root from the square root: 8 - 4 = 4.

Wait, but let me check if there's any chance that the cube root could be negative? But the cube root of 64 is 4, but technically, there is also a negative cube root, which is -4, because (-4)*(-4)*(-4) = -64. Wait, no. Wait, (-4)^3 is -64, not 64. So, the cube root of 64 is 4, and the cube root of -64 is -4. So, since the question is about the cube root of 64, it's 4.

Therefore, the difference is 8 - 4 = 4.

But let me check once more. Maybe I made a mistake in calculating the roots? Let me verify:

Square root of 64: 8. Correct. Because 8 squared is 64.

Cube root of 64: 4. Because 4 cubed is 64.

Yes. So difference is 8 - 4 = 4.

Alternatively, if someone thought that the cube root of 64 is 8? Wait, no. Because 8 cubed is 512, which is way bigger than 64. So that's not.

Alternatively, maybe someone might confuse square root and cube root? But no, the square root is 8, cube root is 4.

Alternatively, maybe the question is asking for the difference between the square root and the cube root, but in reverse? Let me check the exact wording again: "the difference between the positive square root of 64 and the cube root of 64."

So, difference between A and B is A - B. So, positive square root (A) is 8, cube root (B) is 4. So 8 - 4 = 4.

Alternatively, if someone thought that difference is |A - B|, but since they are both positive and 8 > 4, it's still 4.

Therefore, the answer should be 4.

But let me check once again. Maybe I need to write down the steps more formally.

First, compute the positive square root of 64:

√64 = 8, since 8 > 0 and 8² = 64.

Then, compute the cube root of 64:

³√64 = 4, since 4³ = 64.

Then, find the difference between the positive square root and the cube root:

8 - 4 = 4.

Yes, that seems correct.

Alternatively, if someone thought that the cube root of 64 is 8, but that would be wrong. Because 8³ is 512, not 64. So that's not.

Alternatively, maybe someone might think that the cube root of 64 is 3? Because 3³ is 27, 4³ is 64. So no.

Alternatively, maybe someone might think that square root of 64 is 4? But no, square root of 64 is 8. Because square root is a number that when squared gives 64. 8 squared is 64, 4 squared is 16. So no.

Therefore, I think my answer is correct. The difference is 4.

But just to make sure, let me think if there's any other interpretation. For example, maybe the question is in some other context where "difference" is defined differently? But no, in mathematics, difference between two numbers is the result of subtracting one from the other.

Alternatively, maybe the question is a trick question? For example, if someone confuses square root with something else? But I don't think so.

Alternatively, maybe the question is in a different language, and "difference between A and B" is translated in a way that might mean something else? But assuming standard English math terminology, difference is A - B.

Alternatively, maybe the problem is expecting a negative number? But since they specify the positive square root, and cube root is positive as well, so 8 - 4 is positive 4.

Alternatively, if someone thought that cube root can be negative, but since the question says "the cube root of 64," and 64 is positive, the real cube root is positive. If they were talking about complex roots, but that's more advanced and not typically considered here.

Therefore, I think my answer is correct. The difference is 4.

**Final Answer**
The difference is \boxed{4}.

To solve the problem, we need to determine the **difference between the positive square root of 64 and the cube root of 64**.

---

### Step 1: Find the Positive Square Root of 64

The **square root** of a number is a value that, when multiplied by itself, gives the original number. Since the question specifies the **positive** square root, we are looking for the **non-negative** value.

$$
\sqrt{64} = 8
$$

Because:

$$
8 \times 8 = 64
$$

---

### Step 2: Find the Cube Root of 64

The **cube root** of a number is a value that, when multiplied by itself three times, gives the original number.

$$
\sqrt[3]{64} = 4
$$

Because:

$$
4 \times 4 \times 4 = 64
$$

---

### Step 3: Compute the Difference

The question asks for the **difference between the positive square root of 64 and the cube root of 64**. This is interpreted as:

$$
\text{Positive square root} - \text{Cube root} = 8 - 4 = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to find the difference between the positive square root of 64 and the cube root of 64. Hmm, let me think. First, let me recall what square roots and cube roots are.

The square root of a number is a value that, when multiplied by itself, gives the original number. So, the square root of 64 would be a number that when multiplied by itself equals 64. Similarly, the cube root of 64 is a number that when multiplied by itself three times equals 64.

Wait, the question specifies the positive square root. That makes sense because square roots can have both positive and negative values, but since they're asking for the positive one, I should focus on that.

Let me start with the square root of 64. I know that 8 times 8 is 64, so the square root of 64 is 8. Let me check that: 8*8 = 64. Yep, that's right. So the positive square root is definitely 8.

Now, the cube root of 64. Hmm, cube roots are a bit trickier. Let me think. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. So, what number multiplied by itself three times equals 64? Let me try some numbers.

Starting with 1: 1*1*1 = 1. Too small. How about 2? 2*2*2 = 8. Still too small. 3? 3*3*3 = 27. Still not 64. 4? 4*4*4 = 64. Oh, there we go! So the cube root of 64 is 4. Let me verify that: 4*4 is 16, and 16*4 is 64. Yep, that checks out.

So now, the problem asks for the difference between the positive square root of 64 and the cube root of 64. That means I need to subtract the cube root from the square root, or is it the other way around? Wait, the wording is "difference between A and B," which usually means A - B. But sometimes people might interpret it differently. Let me make sure.

The question says: "the difference between the positive square root of 64 and the cube root of 64." So, difference between A and B is typically A - B. So, positive square root (which is 8) minus cube root (which is 4). So 8 - 4. That would be 4.

But let me double-check. Maybe I should confirm the order. If it's difference between A and B, sometimes it's absolute value, but since they specify "positive square root," maybe they just want the subtraction as stated. Let me think. If they had said "the difference between the cube root and the square root," then it would be 4 - 8, which is -4, but since they specify "difference between the positive square root and the cube root," it's 8 - 4.

Alternatively, maybe they just want the absolute difference, but since both numbers are positive and 8 is larger than 4, the difference is 4.

But let me make sure I didn't mix up the roots. Let me recheck the square root and cube root.

Square root of 64: 8, because 8*8=64. Cube root of 64: 4, because 4*4*4=64. So yes, that's correct.

Therefore, subtracting the cube root from the square root: 8 - 4 = 4.

Wait, but let me check if there's any chance that the cube root could be negative? But the cube root of 64 is 4, but technically, there is also a negative cube root, which is -4, because (-4)*(-4)*(-4) = -64. Wait, no. Wait, (-4)^3 is -64, not 64. So, the cube root of 64 is 4, and the cube root of -64 is -4. So, since the question is about the cube root of 64, it's 4.

Therefore, the difference is 8 - 4 = 4.

But let me check once more. Maybe I made a mistake in calculating the roots? Let me verify:

Square root of 64: 8. Correct. Because 8 squared is 64.

Cube root of 64: 4. Because 4 cubed is 64.

Yes. So difference is 8 - 4 = 4.

Alternatively, if someone thought that the cube root of 64 is 8? Wait, no. Because 8 cubed is 512, which is way bigger than 64. So that's not.

Alternatively, maybe someone might confuse square root and cube root? But no, the square root is 8, cube root is 4.

Alternatively, maybe the question is asking for the difference between the square root and the cube root, but in reverse? Let me check the exact wording again: "the difference between the positive square root of 64 and the cube root of 64."

So, difference between A and B is A - B. So, positive square root (A) is 8, cube root (B) is 4. So 8 - 4 = 4.

Alternatively, if someone thought that difference is |A - B|, but since they are both positive and 8 > 4, it's still 4.

Therefore, the answer should be 4.

But let me check once again. Maybe I need to write down the steps more formally.

First, compute the positive square root of 64:

√64 = 8, since 8 > 0 and 8² = 64.

Then, compute the cube root of 64:

³√64 = 4, since 4³ = 64.

Then, find the difference between the positive square root and the cube root:

8 - 4 = 4.

Yes, that seems correct.

Alternatively, if someone thought that the cube root of 64 is 8, but that would be wrong. Because 8³ is 512, not 64. So that's not.

Alternatively, maybe someone might think that the cube root of 64 is 3? Because 3³ is 27, 4³ is 64. So no.

Alternatively, maybe someone might think that square root of 64 is 4? But no, square root of 64 is 8. Because square root is a number that when squared gives 64. 8 squared is 64, 4 squared is 16. So no.

Therefore, I think my answer is correct. The difference is 4.

But just to make sure, let me think if there's any other interpretation. For example, maybe the question is in some other context where "difference" is defined differently? But no, in mathematics, difference between two numbers is the result of subtracting one from the other.

Alternatively, maybe the question is a trick question? For example, if someone confuses square root with something else? But I don't think so.

Alternatively, maybe the question is in a different language, and "difference between A and B" is translated in a way that might mean something else? But assuming standard English math terminology, difference is A - B.

Alternatively, maybe the problem is expecting a negative number? But since they specify the positive square root, and cube root is positive as well, so 8 - 4 is positive 4.

Alternatively, if someone thought that cube root can be negative, but since the question says "the cube root of 64," and 64 is positive, the real cube root is positive. If they were talking about complex roots, but that's more advanced and not typically considered here.

Therefore, I think my answer is correct. The difference is 4.

**Final Answer**
The difference is \boxed{4}.

To solve the problem, we need to determine the **difference between the positive square root of 64 and the cube root of 64**.

---

### Step 1: Find the Positive Square Root of 64

The **square root** of a number is a value that, when multiplied by itself, gives the original number. Since the question specifies the **positive** square root, we are looking for the **non-negative** value.

$$
\sqrt{64} = 8
$$

Because:

$$
8 \times 8 = 64
$$

---

### Step 2: Find the Cube Root of 64

The **cube root** of a number is a value that, when multiplied by itself three times, gives the original number.

$$
\sqrt[3]{64} = 4
$$

Because:

$$
4 \times 4 \times 4 = 64
$$

---

### Step 3: Compute the Difference

The question asks for the **difference between the positive square root of 64 and the cube root of 64**. This is interpreted as:

$$
\text{Positive square root} - \text{Cube root} = 8 - 4 = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

Solution: The positive square root of 64 is $\sqrt{64}=8$. The cube root of 64 is $\sqrt[3]{64}=4$. The difference is $8-4=\boxed{4}$.


Explanation:
Okay, so I need to find the value of f(-2) for the function f(x) = 5x² + 3x + 4. Let me think about how to approach this.

First, I remember that when you have a function like this, to find f(-2), you just substitute x with -2 in the equation. That means wherever there's an x in the function, I replace it with -2. Let me write that down step by step to make sure I don't make a mistake.

So the original function is f(x) = 5x² + 3x + 4. Let me substitute x with -2:

f(-2) = 5(-2)² + 3(-2) + 4.

Now, I need to calculate each term separately. Let me start with the first term, which is 5 times (-2) squared.

First, let me handle the exponent part. (-2) squared. Wait, squaring a negative number... Hmm, when you square a negative number, the result is positive because a negative times a negative is a positive. So (-2) squared is (-2) * (-2) = 4. Okay, so that term becomes 5 * 4. Let me compute that: 5 * 4 is 20. So the first term is 20.

Next, the second term is 3 times (-2). Let me calculate that. 3 multiplied by -2. Well, multiplying a positive by a negative gives a negative result. So 3 * (-2) = -6. So the second term is -6.

The third term is just 4, there's no x there, so that stays as 4.

Now, putting all those terms together: 20 (from the first term) plus (-6) (from the second term) plus 4 (from the third term). Let me write that as 20 - 6 + 4.

Let me compute that step by step. First, 20 - 6. That equals 14. Then, 14 + 4. That equals 18. So, putting it all together, f(-2) should be 18.

Wait, let me check my steps again to make sure I didn't make any mistakes. Sometimes signs can be tricky. Let me verify each term again.

First term: 5*(-2)^2. The exponent is applied before the multiplication, right? So (-2)^2 is 4, then multiplied by 5 gives 20. That seems correct.

Second term: 3*(-2). That's definitely -6. Third term is 4. So adding them up: 20 - 6 + 4.

Another way to think about 20 - 6 + 4 is to combine the numbers. Let me do 20 + (-6) + 4. Which is the same as 20 + (-6 + 4) = 20 + (-2) = 18. Yeah, that still gives me 18.

Alternatively, maybe I can compute it as 20 - 6 is 14, then 14 + 4 is 18. Same result.

Hmm, seems consistent. Let me try plugging in x = -2 into the original function again, just to make sure I didn't misread the original equation. The original function is 5x² + 3x + 4. Yes, so substituting -2 in for x gives 5*(-2)^2 + 3*(-2) + 4. Yep, that's what I did.

Wait, just to be thorough, let me check if I interpreted the original function correctly. The user wrote f(x) = 5x² + 3x + 4. So that's 5 times x squared, plus 3 times x, plus 4. No parentheses or other operations that might change the order of operations. So, yes, substitution is straightforward.

Another thing to check: did I handle the exponent correctly? (-2)^2 is 4, not -4. So that's correct. If it was -2^2, that would be different, but since it's (-2)^2, it's definitely positive 4.

So, 5*4 is 20. Then 3*(-2) is -6. Then +4. So 20 - 6 + 4.

Alternatively, maybe I can think of it as 20 + (-6) + 4. Which is 20 - 6 + 4.

Alternatively, combining the constants: -6 + 4 is -2, so 20 - 2 is 18.

Yes, that's another way to look at it. So, 20 - 2 = 18.

I think that's solid. Let me try another approach. Maybe expanding the function or something else? Wait, not sure. Maybe plugging in another value to check if my method is correct? For example, if I plug in x = 0, f(0) should be 5*(0)^2 + 3*(0) + 4 = 0 + 0 + 4 = 4. That seems right. If I plug in x = 1, f(1) = 5*1 + 3*1 + 4 = 5 + 3 + 4 = 12. Let me check that with my method. If x = 1, then f(1) = 5(1)^2 + 3(1) + 4 = 5 + 3 + 4 = 12. Correct.

So my method works for other values. Therefore, it's likely correct for x = -2.

Alternatively, maybe I can use a different order of operations. Let me compute each term again:

First term: 5*(-2)^2.

Compute (-2)^2 first: (-2)*(-2) = 4. Then multiply by 5: 5*4 = 20.

Second term: 3*(-2) = -6.

Third term: 4.

So total is 20 - 6 + 4.

Alternatively, if I do 20 + (-6) + 4, that's 20 - 6 + 4.

Alternatively, 20 + (-6 + 4) = 20 + (-2) = 18.

Yes, same result.

I think I've checked it multiple ways. Maybe I can also think of it as combining the terms:

5x² + 3x + 4 at x = -2:

5x² = 5*(4) = 20

3x = 3*(-2) = -6

Constant term = 4

So total is 20 - 6 + 4 = 18.

Alternatively, maybe I can use a calculator to verify? Well, since I don't have a calculator here, but I can do the arithmetic again.

20 - 6 is 14. 14 + 4 is 18.

Yes.

Alternatively, if I made a mistake in the sign of the second term? Let me check again. 3 times -2. 3 times 2 is 6, and since one of them is negative, the result is negative. So -6. Correct.

So, adding 20 - 6 + 4.

Another way: 20 + 4 = 24, then 24 - 6 = 18.

Yes, same answer.

So, I think I can be confident that f(-2) is 18.

Wait, just to make sure, let me write out all the steps again in order:

f(-2) = 5(-2)^2 + 3(-2) + 4

Calculate (-2)^2: (-2)*(-2) = 4

Multiply by 5: 5*4 = 20

Calculate 3*(-2): 3*(-2) = -6

Add the constant term 4:

So, 20 + (-6) + 4 = 20 - 6 + 4 = 14 + 4 = 18.

Yep, that's exactly what I did.

I think that's thorough. I don't see any mistakes here. So the answer should be 18.

**Final Answer**
The value of $ f(-2) $ is \boxed{18}.

To find the value of $ f(-2) $ for the function $ f(x) = 5x^2 + 3x + 4 $, we follow a straightforward substitution process.

---

### Step 1: Substitute $ x = -2 $ into the function

$$
f(-2) = 5(-2)^2 + 3(-2) + 4
$$

---

### Step 2: Evaluate each term individually

- **First term:**
  $$
  5(-2)^2 = 5 \cdot 4 = 20
  $$

- **Second term:**
  $$
  3(-2) = -6
  $$

- **Third term:**
  $$
  4
  $$

---

### Step 3: Add the results

$$
f(-2) = 20 - 6 + 4 = 14 + 4 = 18
$$

---

### Final Answer

$$
\boxed{18}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of f(-2) for the function f(x) = 5x² + 3x + 4. Let me think about how to approach this.

First, I remember that when you have a function like this, to find f(-2), you just substitute x with -2 in the equation. That means wherever there's an x in the function, I replace it with -2. Let me write that down step by step to make sure I don't make a mistake.

So the original function is f(x) = 5x² + 3x + 4. Let me substitute x with -2:

f(-2) = 5(-2)² + 3(-2) + 4.

Now, I need to calculate each term separately. Let me start with the first term, which is 5 times (-2) squared.

First, let me handle the exponent part. (-2) squared. Wait, squaring a negative number... Hmm, when you square a negative number, the result is positive because a negative times a negative is a positive. So (-2) squared is (-2) * (-2) = 4. Okay, so that term becomes 5 * 4. Let me compute that: 5 * 4 is 20. So the first term is 20.

Next, the second term is 3 times (-2). Let me calculate that. 3 multiplied by -2. Well, multiplying a positive by a negative gives a negative result. So 3 * (-2) = -6. So the second term is -6.

The third term is just 4, there's no x there, so that stays as 4.

Now, putting all those terms together: 20 (from the first term) plus (-6) (from the second term) plus 4 (from the third term). Let me write that as 20 - 6 + 4.

Let me compute that step by step. First, 20 - 6. That equals 14. Then, 14 + 4. That equals 18. So, putting it all together, f(-2) should be 18.

Wait, let me check my steps again to make sure I didn't make any mistakes. Sometimes signs can be tricky. Let me verify each term again.

First term: 5*(-2)^2. The exponent is applied before the multiplication, right? So (-2)^2 is 4, then multiplied by 5 gives 20. That seems correct.

Second term: 3*(-2). That's definitely -6. Third term is 4. So adding them up: 20 - 6 + 4.

Another way to think about 20 - 6 + 4 is to combine the numbers. Let me do 20 + (-6) + 4. Which is the same as 20 + (-6 + 4) = 20 + (-2) = 18. Yeah, that still gives me 18.

Alternatively, maybe I can compute it as 20 - 6 is 14, then 14 + 4 is 18. Same result.

Hmm, seems consistent. Let me try plugging in x = -2 into the original function again, just to make sure I didn't misread the original equation. The original function is 5x² + 3x + 4. Yes, so substituting -2 in for x gives 5*(-2)^2 + 3*(-2) + 4. Yep, that's what I did.

Wait, just to be thorough, let me check if I interpreted the original function correctly. The user wrote f(x) = 5x² + 3x + 4. So that's 5 times x squared, plus 3 times x, plus 4. No parentheses or other operations that might change the order of operations. So, yes, substitution is straightforward.

Another thing to check: did I handle the exponent correctly? (-2)^2 is 4, not -4. So that's correct. If it was -2^2, that would be different, but since it's (-2)^2, it's definitely positive 4.

So, 5*4 is 20. Then 3*(-2) is -6. Then +4. So 20 - 6 + 4.

Alternatively, maybe I can think of it as 20 + (-6) + 4. Which is 20 - 6 + 4.

Alternatively, combining the constants: -6 + 4 is -2, so 20 - 2 is 18.

Yes, that's another way to look at it. So, 20 - 2 = 18.

I think that's solid. Let me try another approach. Maybe expanding the function or something else? Wait, not sure. Maybe plugging in another value to check if my method is correct? For example, if I plug in x = 0, f(0) should be 5*(0)^2 + 3*(0) + 4 = 0 + 0 + 4 = 4. That seems right. If I plug in x = 1, f(1) = 5*1 + 3*1 + 4 = 5 + 3 + 4 = 12. Let me check that with my method. If x = 1, then f(1) = 5(1)^2 + 3(1) + 4 = 5 + 3 + 4 = 12. Correct.

So my method works for other values. Therefore, it's likely correct for x = -2.

Alternatively, maybe I can use a different order of operations. Let me compute each term again:

First term: 5*(-2)^2.

Compute (-2)^2 first: (-2)*(-2) = 4. Then multiply by 5: 5*4 = 20.

Second term: 3*(-2) = -6.

Third term: 4.

So total is 20 - 6 + 4.

Alternatively, if I do 20 + (-6) + 4, that's 20 - 6 + 4.

Alternatively, 20 + (-6 + 4) = 20 + (-2) = 18.

Yes, same result.

I think I've checked it multiple ways. Maybe I can also think of it as combining the terms:

5x² + 3x + 4 at x = -2:

5x² = 5*(4) = 20

3x = 3*(-2) = -6

Constant term = 4

So total is 20 - 6 + 4 = 18.

Alternatively, maybe I can use a calculator to verify? Well, since I don't have a calculator here, but I can do the arithmetic again.

20 - 6 is 14. 14 + 4 is 18.

Yes.

Alternatively, if I made a mistake in the sign of the second term? Let me check again. 3 times -2. 3 times 2 is 6, and since one of them is negative, the result is negative. So -6. Correct.

So, adding 20 - 6 + 4.

Another way: 20 + 4 = 24, then 24 - 6 = 18.

Yes, same answer.

So, I think I can be confident that f(-2) is 18.

Wait, just to make sure, let me write out all the steps again in order:

f(-2) = 5(-2)^2 + 3(-2) + 4

Calculate (-2)^2: (-2)*(-2) = 4

Multiply by 5: 5*4 = 20

Calculate 3*(-2): 3*(-2) = -6

Add the constant term 4:

So, 20 + (-6) + 4 = 20 - 6 + 4 = 14 + 4 = 18.

Yep, that's exactly what I did.

I think that's thorough. I don't see any mistakes here. So the answer should be 18.

**Final Answer**
The value of $ f(-2) $ is \boxed{18}.

To find the value of $ f(-2) $ for the function $ f(x) = 5x^2 + 3x + 4 $, we follow a straightforward substitution process.

---

### Step 1: Substitute $ x = -2 $ into the function

$$
f(-2) = 5(-2)^2 + 3(-2) + 4
$$

---

### Step 2: Evaluate each term individually

- **First term:**
  $$
  5(-2)^2 = 5 \cdot 4 = 20
  $$

- **Second term:**
  $$
  3(-2) = -6
  $$

- **Third term:**
  $$
  4
  $$

---

### Step 3: Add the results

$$
f(-2) = 20 - 6 + 4 = 14 + 4 = 18
$$

---

### Final Answer

$$
\boxed{18}
$$

Solution: We have $f(-2) = 5(-2)^2 + 3(-2) + 4 = 5(4) -6 + 4 = \boxed{18}$.


Explanation:
Okay, so I need to find the value of this expression: (16 times the cube root of a squared) all raised to the 1/3 power, and they told me that a is 8. Let me write that down step by step.

First, let me make sure I understand the expression correctly. It's written as (16∛(a²))^(1/3). So, inside the parentheses, there's 16 multiplied by the cube root of a squared, and then the whole thing is raised to the power of 1/3. Alright.

Since a is given as 8, I can substitute that in. Let me do that first. So replacing a with 8, the expression becomes (16 * ∛(8²))^(1/3).

Now, let me compute the cube root of 8 squared. Let me break that down. First, 8 squared is 8*8, which is 64. So, ∛(8²) is the cube root of 64. Wait, what's the cube root of 64? Well, 4 cubed is 64 because 4*4*4 is 64. So, ∛64 is 4.

So, substituting that back into the expression, we have (16 * 4)^(1/3). Let me compute 16 times 4. 16*4 is 64. So now the expression simplifies to 64^(1/3).

Now, 64 is a number I know the cube root of. Since 4^3 is 64, the cube root of 64 is 4. Therefore, 64^(1/3) is 4.

Wait, so putting it all together, the entire expression simplifies to 4? Let me check my steps again to make sure I didn't make a mistake.

Starting with a = 8. Then, inside the parentheses: 16 multiplied by the cube root of a squared. So, a squared is 8 squared, which is 64. The cube root of 64 is 4. Multiply that by 16: 16*4 is 64. Then, raising that to the 1/3 power: cube root of 64 is 4. Yep, that seems right.

Alternatively, maybe I can approach this using exponents instead of radicals to see if I get the same result. Let me try that.

The original expression is (16 * ∛(a²))^(1/3). Let me rewrite the cube root as an exponent. The cube root of something is the same as raising it to the 1/3 power. So, ∛(a²) is a^(2/3). Therefore, the expression becomes [16 * a^(2/3)]^(1/3).

Now, since a is 8, let me substitute that in: [16 * 8^(2/3)]^(1/3). Let me compute 8^(2/3). The exponent 2/3 can be thought of as taking the cube root first and then squaring it, or squaring first and then taking the cube root. Let me do both ways to check.

First, cube root of 8 is 2, then square it: 2 squared is 4. Alternatively, square 8 first: 8 squared is 64, then take the cube root of 64, which is 4. Either way, 8^(2/3) is 4.

So, substituting back, we have [16 * 4]^(1/3) which is [64]^(1/3), which again is 4. So that's the same result as before.

Alternatively, maybe I can combine the exponents. Let me think. The expression is [16 * a^(2/3)]^(1/3). Using exponent rules, this is equal to 16^(1/3) * [a^(2/3)]^(1/3).

Simplifying the exponents: [a^(2/3)]^(1/3) is a^(2/9). And 16^(1/3) is the cube root of 16. So, the entire expression becomes 16^(1/3) * a^(2/9).

But since a is 8, let me substitute that in: 16^(1/3) * 8^(2/9). Hmm, this seems more complicated. Maybe I should compute each part separately.

First, 16^(1/3). 16 is 2^4, so 16^(1/3) is 2^(4/3). Then, 8 is 2^3, so 8^(2/9) is (2^3)^(2/9) = 2^(6/9) = 2^(2/3).

Therefore, multiplying these together: 2^(4/3) * 2^(2/3) = 2^(4/3 + 2/3) = 2^(6/3) = 2^2 = 4.

Oh, so even when breaking it down into exponents of 2, I still get 4. So that confirms the answer again.

So, regardless of the method I use—whether substituting first or manipulating exponents—the result is 4.

Wait, but let me check once more. Maybe I made a mistake in the exponent rules? Let me verify.

Original expression: (16 * ∛(a²))^(1/3).

If I write all terms with exponents, 16 is 2^4, a is 8 which is 2^3. So substituting:

(2^4 * ( (2^3)^2 )^(1/3) )^(1/3).

Let me compute inside out. First, (2^3)^2 is 2^(3*2) = 2^6. Then, taking the cube root of that is (2^6)^(1/3) = 2^(6*(1/3)) = 2^2 = 4.

Then, multiplying by 2^4: 2^4 * 4 = 2^4 * 2^2 = 2^(4+2) = 2^6.

Then, raising that to the 1/3 power: (2^6)^(1/3) = 2^(6*(1/3)) = 2^2 = 4.

Same result again. So, that's consistent.

Alternatively, maybe I can think of the entire expression as 16^(1/3) * (a²)^(1/9). Wait, no. Let me check.

Original expression: [16 * a^(2/3)]^(1/3) = 16^(1/3) * a^(2/9). Which is what I had before.

But since a is 8, which is 2^3, then a^(2/9) is (2^3)^(2/9) = 2^(6/9) = 2^(2/3). And 16^(1/3) is 2^(4/3). Multiplying them gives 2^(4/3 + 2/3) = 2^2 = 4.

So, all methods lead to 4. Therefore, I think the answer is 4.

But just to make sure, let me compute the original expression step by step numerically.

Original expression: (16 * ∛(8²))^(1/3).

First, compute 8 squared: 8*8 = 64.

Then, cube root of 64: ∛64 = 4.

Multiply by 16: 16 * 4 = 64.

Now, take the cube root of 64: ∛64 = 4.

Yep, that's straightforward.

Alternatively, if I use decimal approximations, just to check.

Cube root of 8 squared: 8 squared is 64. Cube root of 64 is 4. So 16*4 is 64. Cube root of 64 is 4.

Alternatively, if I didn't remember that cube root of 64 is 4, I could calculate it. Let me do that.

What's the cube root of 64? Let me think. 4*4*4 is 64. 3*3*3 is 27, 4*4*4 is 64, 5*5*5 is 125. So yes, 4.

Alternatively, if I didn't remember, I could approximate. But since 64 is a perfect cube, it's easy.

So, all steps check out.

Therefore, I think confidently the answer is 4.

**Final Answer**
\boxed{4}

To evaluate the expression
$$
\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}
$$
given that $ a = 8 $, we proceed step by step.

---

### Step 1: Substitute the value of $ a $

We are told that $ a = 8 $, so we substitute this into the expression:

$$
\left(16\sqrt[3]{8^2}\right)^{\frac{1}{3}}
$$

---

### Step 2: Simplify the inner cube root

First, compute $ 8^2 $:

$$
8^2 = 64
$$

Now, take the cube root of 64:

$$
\sqrt[3]{64} = 4 \quad \text{(since } 4^3 = 64\text{)}
$$

So the expression becomes:

$$
\left(16 \cdot 4\right)^{\frac{1}{3}} = \left(64\right)^{\frac{1}{3}}
$$

---

### Step 3: Evaluate the final cube root

Now, compute the cube root of 64:

$$
\sqrt[3]{64} = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of this expression: (16 times the cube root of a squared) all raised to the 1/3 power, and they told me that a is 8. Let me write that down step by step.

First, let me make sure I understand the expression correctly. It's written as (16∛(a²))^(1/3). So, inside the parentheses, there's 16 multiplied by the cube root of a squared, and then the whole thing is raised to the power of 1/3. Alright.

Since a is given as 8, I can substitute that in. Let me do that first. So replacing a with 8, the expression becomes (16 * ∛(8²))^(1/3).

Now, let me compute the cube root of 8 squared. Let me break that down. First, 8 squared is 8*8, which is 64. So, ∛(8²) is the cube root of 64. Wait, what's the cube root of 64? Well, 4 cubed is 64 because 4*4*4 is 64. So, ∛64 is 4.

So, substituting that back into the expression, we have (16 * 4)^(1/3). Let me compute 16 times 4. 16*4 is 64. So now the expression simplifies to 64^(1/3).

Now, 64 is a number I know the cube root of. Since 4^3 is 64, the cube root of 64 is 4. Therefore, 64^(1/3) is 4.

Wait, so putting it all together, the entire expression simplifies to 4? Let me check my steps again to make sure I didn't make a mistake.

Starting with a = 8. Then, inside the parentheses: 16 multiplied by the cube root of a squared. So, a squared is 8 squared, which is 64. The cube root of 64 is 4. Multiply that by 16: 16*4 is 64. Then, raising that to the 1/3 power: cube root of 64 is 4. Yep, that seems right.

Alternatively, maybe I can approach this using exponents instead of radicals to see if I get the same result. Let me try that.

The original expression is (16 * ∛(a²))^(1/3). Let me rewrite the cube root as an exponent. The cube root of something is the same as raising it to the 1/3 power. So, ∛(a²) is a^(2/3). Therefore, the expression becomes [16 * a^(2/3)]^(1/3).

Now, since a is 8, let me substitute that in: [16 * 8^(2/3)]^(1/3). Let me compute 8^(2/3). The exponent 2/3 can be thought of as taking the cube root first and then squaring it, or squaring first and then taking the cube root. Let me do both ways to check.

First, cube root of 8 is 2, then square it: 2 squared is 4. Alternatively, square 8 first: 8 squared is 64, then take the cube root of 64, which is 4. Either way, 8^(2/3) is 4.

So, substituting back, we have [16 * 4]^(1/3) which is [64]^(1/3), which again is 4. So that's the same result as before.

Alternatively, maybe I can combine the exponents. Let me think. The expression is [16 * a^(2/3)]^(1/3). Using exponent rules, this is equal to 16^(1/3) * [a^(2/3)]^(1/3).

Simplifying the exponents: [a^(2/3)]^(1/3) is a^(2/9). And 16^(1/3) is the cube root of 16. So, the entire expression becomes 16^(1/3) * a^(2/9).

But since a is 8, let me substitute that in: 16^(1/3) * 8^(2/9). Hmm, this seems more complicated. Maybe I should compute each part separately.

First, 16^(1/3). 16 is 2^4, so 16^(1/3) is 2^(4/3). Then, 8 is 2^3, so 8^(2/9) is (2^3)^(2/9) = 2^(6/9) = 2^(2/3).

Therefore, multiplying these together: 2^(4/3) * 2^(2/3) = 2^(4/3 + 2/3) = 2^(6/3) = 2^2 = 4.

Oh, so even when breaking it down into exponents of 2, I still get 4. So that confirms the answer again.

So, regardless of the method I use—whether substituting first or manipulating exponents—the result is 4.

Wait, but let me check once more. Maybe I made a mistake in the exponent rules? Let me verify.

Original expression: (16 * ∛(a²))^(1/3).

If I write all terms with exponents, 16 is 2^4, a is 8 which is 2^3. So substituting:

(2^4 * ( (2^3)^2 )^(1/3) )^(1/3).

Let me compute inside out. First, (2^3)^2 is 2^(3*2) = 2^6. Then, taking the cube root of that is (2^6)^(1/3) = 2^(6*(1/3)) = 2^2 = 4.

Then, multiplying by 2^4: 2^4 * 4 = 2^4 * 2^2 = 2^(4+2) = 2^6.

Then, raising that to the 1/3 power: (2^6)^(1/3) = 2^(6*(1/3)) = 2^2 = 4.

Same result again. So, that's consistent.

Alternatively, maybe I can think of the entire expression as 16^(1/3) * (a²)^(1/9). Wait, no. Let me check.

Original expression: [16 * a^(2/3)]^(1/3) = 16^(1/3) * a^(2/9). Which is what I had before.

But since a is 8, which is 2^3, then a^(2/9) is (2^3)^(2/9) = 2^(6/9) = 2^(2/3). And 16^(1/3) is 2^(4/3). Multiplying them gives 2^(4/3 + 2/3) = 2^2 = 4.

So, all methods lead to 4. Therefore, I think the answer is 4.

But just to make sure, let me compute the original expression step by step numerically.

Original expression: (16 * ∛(8²))^(1/3).

First, compute 8 squared: 8*8 = 64.

Then, cube root of 64: ∛64 = 4.

Multiply by 16: 16 * 4 = 64.

Now, take the cube root of 64: ∛64 = 4.

Yep, that's straightforward.

Alternatively, if I use decimal approximations, just to check.

Cube root of 8 squared: 8 squared is 64. Cube root of 64 is 4. So 16*4 is 64. Cube root of 64 is 4.

Alternatively, if I didn't remember that cube root of 64 is 4, I could calculate it. Let me do that.

What's the cube root of 64? Let me think. 4*4*4 is 64. 3*3*3 is 27, 4*4*4 is 64, 5*5*5 is 125. So yes, 4.

Alternatively, if I didn't remember, I could approximate. But since 64 is a perfect cube, it's easy.

So, all steps check out.

Therefore, I think confidently the answer is 4.

**Final Answer**
\boxed{4}

To evaluate the expression
$$
\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}
$$
given that $ a = 8 $, we proceed step by step.

---

### Step 1: Substitute the value of $ a $

We are told that $ a = 8 $, so we substitute this into the expression:

$$
\left(16\sqrt[3]{8^2}\right)^{\frac{1}{3}}
$$

---

### Step 2: Simplify the inner cube root

First, compute $ 8^2 $:

$$
8^2 = 64
$$

Now, take the cube root of 64:

$$
\sqrt[3]{64} = 4 \quad \text{(since } 4^3 = 64\text{)}
$$

So the expression becomes:

$$
\left(16 \cdot 4\right)^{\frac{1}{3}} = \left(64\right)^{\frac{1}{3}}
$$

---

### Step 3: Evaluate the final cube root

Now, compute the cube root of 64:

$$
\sqrt[3]{64} = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

Solution: Note that $a^2 = 64$ and $\sqrt[3]{64} = 4$. Therefore, $$\left(16\sqrt[3]{a^2}\right)^{\frac {1}{3}} = \left(16 \times 4\right)^{\frac{1}{3}} = 64^\frac{1}{3} = \boxed{4}.$$


Explanation:
Okay, so Roslyn has ten boxes total. Let me try to figure out how many boxes contain neither pens nor pencils. The problem says five boxes have pencils, four have pens, and two have both. Hmm, okay. Let me think about this step by step.

First, I remember something about sets and maybe the principle of inclusion-exclusion. Let me recall. If I have two sets, the total number of elements in either set is the sum of the sizes of the sets minus the size of their intersection. So, in this case, the boxes with pencils and the boxes with pens.

So, if I let P be the set of boxes with pencils and Q be the set of boxes with pens, then the number of boxes that have either pencils or pens is |P ∪ Q| = |P| + |Q| - |P ∩ Q|.

Given that, the problem states that there are 5 boxes with pencils, so |P| = 5. Four boxes with pens, so |Q| = 4. And two boxes contain both pens and pencils, so |P ∩ Q| = 2.

Plugging those numbers into the formula: |P ∪ Q| = 5 + 4 - 2 = 7. So, there are 7 boxes that have either pencils or pens.

But wait, the total number of boxes is 10. So, if 7 boxes have pencils or pens, then the remaining boxes must have neither. So, 10 - 7 = 3. Therefore, the number of boxes that contain neither pens nor pencils is 3?

Let me check if that makes sense. Let me visualize it. Suppose there are 10 boxes. Five have pencils. But two of those also have pens. So, the number of boxes with only pencils would be 5 - 2 = 3. Similarly, the number of boxes with only pens would be 4 - 2 = 2. Then, adding the ones with only pencils, only pens, and both: 3 + 2 + 2 = 7. Which matches the previous calculation. So, 10 - 7 = 3 boxes with neither.

Wait, but let me make sure I didn't make a mistake here. Let me think again. If there are 5 boxes with pencils, and 2 of them also have pens, then yes, 3 have only pencils. Then, there are 4 boxes with pens. But 2 of those are already counted in the pencils. So, the remaining pens are 4 - 2 = 2 boxes with only pens. So, total boxes with pencils or pens: 3 (only pencils) + 2 (only pens) + 2 (both) = 7. So, 10 - 7 = 3.

Alternatively, maybe I can use a Venn diagram. Imagine two overlapping circles. One circle for pencils (P) with 5, one for pens (Q) with 4. The overlapping part is 2. So, the total in the union is 5 + 4 - 2 = 7. So, the rest is 10 - 7 = 3.

Another way to think about it: if there are two boxes that have both, then those two are counted in both the pencil boxes and the pen boxes. So, if I just added 5 + 4, that would be 9, but that counts the two boxes with both twice. So, to get the correct number of boxes that have either, we subtract the overlap once. So, 9 - 2 = 7.

Therefore, the number of boxes with neither is 10 - 7 = 3.

Wait, but let me check if there's another way someone might approach this problem and maybe get confused. For example, someone might think that since there are 5 pencils and 4 pens, the total is 5 + 4 = 9, and then subtract the overlap? Wait, but that's exactly what we did. So, 9 - 2 = 7.

Alternatively, someone might think that the overlap is 2, so subtract that from the total? But that would be 10 - 2 = 8, which is wrong. So, that's a common mistake. But no, the correct way is to use inclusion-exclusion.

Alternatively, maybe someone might think that the boxes with both are separate? But no, the problem says two boxes contain both pens and pencils. So, those two are part of both the pencil boxes and the pen boxes.

So, I think my reasoning is correct. So, the answer is 3.

But let me just make sure that there isn't some trick in the question. The problem says "five of the boxes contain pencils, four of the boxes contain pens, and two of the boxes contain both pens and pencils." So, the numbers are given as counts of boxes, not the number of items. So, each box is counted once, even if it contains multiple items. So, for example, the two boxes that contain both pens and pencils are counted in both the pencil count and the pen count.

Therefore, the inclusion-exclusion principle applies here.

So, total boxes with pencils or pens: 5 + 4 - 2 = 7. Therefore, boxes with neither: 10 - 7 = 3.

Yes, that seems right.

Alternatively, if I think about it in terms of sets:

Number of boxes with pencils only: 5 - 2 = 3

Number of boxes with pens only: 4 - 2 = 2

Number of boxes with both: 2

Number of boxes with neither: Total - (pencils only + pens only + both) = 10 - (3 + 2 + 2) = 10 - 7 = 3.

Same result.

Therefore, I think the answer is 3.

**Final Answer**
The number of boxes that contain neither pens nor pencils is \boxed{3}.

To determine how many boxes contain **neither pens nor pencils**, we begin by analyzing the information provided:

- **Total number of boxes**: 10
- **Boxes with pencils**: 5
- **Boxes with pens**: 4
- **Boxes with both pens and pencils**: 2

---

### Step 1: Apply the Principle of Inclusion-Exclusion

We are given the number of boxes that contain **pencils**, **pens**, and **both**. To find the number of boxes that contain **either pencils or pens**, we use the inclusion-exclusion principle:

$$
|P \cup Q| = |P| + |Q| - |P \cap Q|
$$

Where:
- $ |P| = 5 $ (boxes with pencils)
- $ |Q| = 4 $ (boxes with pens)
- $ |P \cap Q| = 2 $ (boxes with both)

$$
|P \cup Q| = 5 + 4 - 2 = 7
$$

So, **7 boxes** contain **either pencils or pens** (or both).

---

### Step 2: Determine the Number of Boxes with Neither

Since there are **10 boxes** in total, the number of boxes that contain **neither pens nor pencils** is:

$$
10 - |P \cup Q| = 10 - 7 = 3
$$

---

### Final Answer

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so Roslyn has ten boxes total. Let me try to figure out how many boxes contain neither pens nor pencils. The problem says five boxes have pencils, four have pens, and two have both. Hmm, okay. Let me think about this step by step.

First, I remember something about sets and maybe the principle of inclusion-exclusion. Let me recall. If I have two sets, the total number of elements in either set is the sum of the sizes of the sets minus the size of their intersection. So, in this case, the boxes with pencils and the boxes with pens.

So, if I let P be the set of boxes with pencils and Q be the set of boxes with pens, then the number of boxes that have either pencils or pens is |P ∪ Q| = |P| + |Q| - |P ∩ Q|.

Given that, the problem states that there are 5 boxes with pencils, so |P| = 5. Four boxes with pens, so |Q| = 4. And two boxes contain both pens and pencils, so |P ∩ Q| = 2.

Plugging those numbers into the formula: |P ∪ Q| = 5 + 4 - 2 = 7. So, there are 7 boxes that have either pencils or pens.

But wait, the total number of boxes is 10. So, if 7 boxes have pencils or pens, then the remaining boxes must have neither. So, 10 - 7 = 3. Therefore, the number of boxes that contain neither pens nor pencils is 3?

Let me check if that makes sense. Let me visualize it. Suppose there are 10 boxes. Five have pencils. But two of those also have pens. So, the number of boxes with only pencils would be 5 - 2 = 3. Similarly, the number of boxes with only pens would be 4 - 2 = 2. Then, adding the ones with only pencils, only pens, and both: 3 + 2 + 2 = 7. Which matches the previous calculation. So, 10 - 7 = 3 boxes with neither.

Wait, but let me make sure I didn't make a mistake here. Let me think again. If there are 5 boxes with pencils, and 2 of them also have pens, then yes, 3 have only pencils. Then, there are 4 boxes with pens. But 2 of those are already counted in the pencils. So, the remaining pens are 4 - 2 = 2 boxes with only pens. So, total boxes with pencils or pens: 3 (only pencils) + 2 (only pens) + 2 (both) = 7. So, 10 - 7 = 3.

Alternatively, maybe I can use a Venn diagram. Imagine two overlapping circles. One circle for pencils (P) with 5, one for pens (Q) with 4. The overlapping part is 2. So, the total in the union is 5 + 4 - 2 = 7. So, the rest is 10 - 7 = 3.

Another way to think about it: if there are two boxes that have both, then those two are counted in both the pencil boxes and the pen boxes. So, if I just added 5 + 4, that would be 9, but that counts the two boxes with both twice. So, to get the correct number of boxes that have either, we subtract the overlap once. So, 9 - 2 = 7.

Therefore, the number of boxes with neither is 10 - 7 = 3.

Wait, but let me check if there's another way someone might approach this problem and maybe get confused. For example, someone might think that since there are 5 pencils and 4 pens, the total is 5 + 4 = 9, and then subtract the overlap? Wait, but that's exactly what we did. So, 9 - 2 = 7.

Alternatively, someone might think that the overlap is 2, so subtract that from the total? But that would be 10 - 2 = 8, which is wrong. So, that's a common mistake. But no, the correct way is to use inclusion-exclusion.

Alternatively, maybe someone might think that the boxes with both are separate? But no, the problem says two boxes contain both pens and pencils. So, those two are part of both the pencil boxes and the pen boxes.

So, I think my reasoning is correct. So, the answer is 3.

But let me just make sure that there isn't some trick in the question. The problem says "five of the boxes contain pencils, four of the boxes contain pens, and two of the boxes contain both pens and pencils." So, the numbers are given as counts of boxes, not the number of items. So, each box is counted once, even if it contains multiple items. So, for example, the two boxes that contain both pens and pencils are counted in both the pencil count and the pen count.

Therefore, the inclusion-exclusion principle applies here.

So, total boxes with pencils or pens: 5 + 4 - 2 = 7. Therefore, boxes with neither: 10 - 7 = 3.

Yes, that seems right.

Alternatively, if I think about it in terms of sets:

Number of boxes with pencils only: 5 - 2 = 3

Number of boxes with pens only: 4 - 2 = 2

Number of boxes with both: 2

Number of boxes with neither: Total - (pencils only + pens only + both) = 10 - (3 + 2 + 2) = 10 - 7 = 3.

Same result.

Therefore, I think the answer is 3.

**Final Answer**
The number of boxes that contain neither pens nor pencils is \boxed{3}.

To determine how many boxes contain **neither pens nor pencils**, we begin by analyzing the information provided:

- **Total number of boxes**: 10
- **Boxes with pencils**: 5
- **Boxes with pens**: 4
- **Boxes with both pens and pencils**: 2

---

### Step 1: Apply the Principle of Inclusion-Exclusion

We are given the number of boxes that contain **pencils**, **pens**, and **both**. To find the number of boxes that contain **either pencils or pens**, we use the inclusion-exclusion principle:

$$
|P \cup Q| = |P| + |Q| - |P \cap Q|
$$

Where:
- $ |P| = 5 $ (boxes with pencils)
- $ |Q| = 4 $ (boxes with pens)
- $ |P \cap Q| = 2 $ (boxes with both)

$$
|P \cup Q| = 5 + 4 - 2 = 7
$$

So, **7 boxes** contain **either pencils or pens** (or both).

---

### Step 2: Determine the Number of Boxes with Neither

Since there are **10 boxes** in total, the number of boxes that contain **neither pens nor pencils** is:

$$
10 - |P \cup Q| = 10 - 7 = 3
$$

---

### Final Answer

$$
\boxed{3}
$$

Solution:
Of the 5 boxes with pencils, 2 have pens also, so $5-2=3$ have pencils only.  Similarly, $4-2 =2$ of the boxes have pens only:

[asy]
unitsize(0.05cm);
label("Pencils", (2,74));
label("Pens", (80,74));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
label("$2$", (44, 45));
label(scale(0.8)*"$3$",(28,58));
label(scale(0.8)*"$2$",(63,58));
[/asy]

That gives us $3+2+2=7$ boxes with pens, pencils, or both.  This leaves $10-7 = \boxed{3}$ with neither.


Explanation:
Okay, so I need to find the remainder when 2004 is divided by 12. Hmm, let me think. I remember that when you divide a number by another number, the remainder is what's left over after dividing as much as possible. So, for example, if I divide 10 by 3, the quotient is 3 and the remainder is 1 because 3 times 3 is 9, and 10 minus 9 is 1.

Alright, so applying that to 2004 divided by 12. Let me try to figure out how many times 12 goes into 2004. Maybe I can do this step by step. First, I know that 12 times 100 is 1200. So if I subtract 1200 from 2004, that would leave me with 804. Then, how many times does 12 go into 804? Let me see. 12 times 60 is 720. So subtracting 720 from 804 gives me 84. Now, how many times does 12 go into 84? Well, 12 times 7 is 84. So that's 7.

So adding up the numbers I multiplied: 100 + 60 + 7 = 167. So 12 times 167 is 2004? Wait, let me check that. 12 times 167. Let me compute 12*160 first, which is 1920, and then 12*7 is 84. Adding those together, 1920 + 84 = 2004. Oh, so that means 12*167 is exactly 2004. Therefore, there is no remainder. Wait, so the remainder is zero?

But let me verify this another way to make sure I didn't make a mistake. Maybe using division algorithm. The division algorithm says that for any integers a and b (with b > 0), there exist unique integers q and r such that a = bq + r and 0 ≤ r < b. Here, a is 2004, b is 12. So we need to find q and r such that 2004 = 12q + r, where 0 ≤ r < 12.

Earlier, I found that q is 167 and r is 0. Let me check if that's correct. 12*167 is indeed 2004, so r is 0. Therefore, the remainder is 0.

But maybe I can check using another method. For example, maybe using modulo operations. 2004 divided by 12. Let me think about how to compute 2004 mod 12.

Alternatively, I can break down 2004 into smaller parts. Let me think about 2004 divided by 12. Since 12 is 3*4, maybe I can check divisibility by 3 and 4. If a number is divisible by both 3 and 4, then it's divisible by 12.

First, check divisibility by 3. The sum of the digits of 2004 is 2 + 0 + 0 + 4 = 6. Since 6 is divisible by 3, 2004 is divisible by 3.

Next, check divisibility by 4. The last two digits of 2004 are 04, which is 4. Since 4 is divisible by 4, 2004 is divisible by 4. Therefore, since it's divisible by both 3 and 4, it's divisible by 12. Therefore, the remainder is 0.

Another way: Let me divide 2004 by 12 step by step. Let me do the division process.

How many times does 12 go into 20? Once, because 12*1=12. Subtract 12 from 20, get 8. Bring down the 0, making it 80. How many times does 12 go into 80? Let me calculate 12*6=72, 12*7=84. 84 is too big, so 6 times. Subtract 72 from 80, get 8. Bring down the 4, making it 84. How many times does 12 go into 84? Exactly 7 times. Subtract 84, get 0. So the division ends here.

So the quotient is 167 (from 1, 6, 7) and the remainder is 0.

Wait, so all methods are pointing to the remainder being zero. But just to be thorough, let me try another approach. Let me subtract multiples of 12 from 2004 until I get a number less than 12.

But that might take too long, but maybe for a smaller number. Let me think. Alternatively, I can use the fact that 12*167=2004, so remainder is zero.

Alternatively, think about 2004 divided by 12. Let me divide 2004 by 12.

Well, 12*100=1200. 2004-1200=804.

Then 12*60=720. 804-720=84.

Then 12*7=84. 84-84=0.

So total is 100+60+7=167. So remainder is 0.

Alternatively, maybe using modular arithmetic properties. Let me compute 2004 mod 12.

Note that 2004 divided by 12. Let me think about 2004 divided by 12.

Alternatively, since 12*167=2004, then 2004 mod 12 is 0.

Alternatively, think about 2004 divided by 12. Let me think of 2004 as 2000 + 4.

So 2000 divided by 12. Let me compute 2000 / 12.

Well, 12*166= 12*(160 + 6) = 12*160 + 12*6 = 1920 + 72 = 1992.

So 2000 - 1992 = 8. Therefore, 2000 divided by 12 is 166 with a remainder of 8. Then add the 4 from the original 2004. So total remainder would be 8 + 4 = 12. But 12 is equal to 12, so the remainder would be 12 - 12 = 0.

Wait, that's another way to think about it. So breaking down 2004 into 2000 + 4. 2000 divided by 12 gives remainder 8, and 4 divided by 12 gives remainder 4. Then adding those remainders: 8 + 4 = 12. But since 12 is equal to the divisor, we subtract 12 to get the remainder 0.

Therefore, the remainder is 0.

Alternatively, maybe using negative numbers? Let me try.

But I think all methods are leading me to the same conclusion. The remainder is zero.

But just to make sure, let me check with a calculator. Wait, I don't have a calculator here, but maybe I can do a quick check.

Let me compute 12*167.

First, 12*160 = 1920.

Then 12*7 = 84.

Adding them together: 1920 + 84.

1920 + 80 = 2000, then +4 = 2004.

Yes, exactly. So 12*167 is 2004. Therefore, 2004 divided by 12 is exactly 167 with no remainder.

Therefore, the remainder is 0.

I think that's solid. All different methods confirm that. So I can be confident that the remainder is zero.

**Final Answer**
The remainder of 2004 divided by 12 is \boxed{0}.

To determine the remainder when 2004 is divided by 12, we can approach the problem using the **division algorithm**, which states that for any integers $ a $ and $ b $ (with $ b > 0 $), there exist unique integers $ q $ (the quotient) and $ r $ (the remainder) such that:

$$
a = bq + r \quad \text{where } 0 \leq r < b
$$

---

### Step 1: Apply the Division Algorithm

Let:
- $ a = 2004 $
- $ b = 12 $

We want to find $ q $ and $ r $ such that:

$$
2004 = 12q + r \quad \text{with } 0 \leq r < 12
$$

---

### Step 2: Perform the Division

We can compute the quotient $ q $ by dividing 2004 by 12:

$$
2004 \div 12 = 167
$$

Now, multiply back to check:

$$
12 \times 167 = 2004
$$

This means that the remainder $ r $ is:

$$
r = 2004 - (12 \times 167) = 2004 - 2004 = 0
$$

---

### Step 3: Confirm with Alternative Methods

- **Divisibility by 3 and 4**:
  - Sum of digits of 2004: $ 2 + 0 + 0 + 4 = 6 $, which is divisible by 3.
  - Last two digits: 04, which is divisible by 4.
  - Since 2004 is divisible by both 3 and 4, it is divisible by 12.
  - Therefore, the remainder is 0.

- **Breakdown of 2004**:
  - $ 2004 = 2000 + 4 $
  - $ 2000 \div 12 = 166 $ remainder 8
  - $ 4 \div 12 = 0 $ remainder 4
  - Total remainder: $ 8 + 4 = 12 $, which is equal to the divisor, so the actual remainder is $ 12 - 12 = 0 $

---

### Final Answer

$$
\boxed{0}
$$

[Assistant]
Explanation:
Okay, so I need to find the remainder when 2004 is divided by 12. Hmm, let me think. I remember that when you divide a number by another number, the remainder is what's left over after dividing as much as possible. So, for example, if I divide 10 by 3, the quotient is 3 and the remainder is 1 because 3 times 3 is 9, and 10 minus 9 is 1.

Alright, so applying that to 2004 divided by 12. Let me try to figure out how many times 12 goes into 2004. Maybe I can do this step by step. First, I know that 12 times 100 is 1200. So if I subtract 1200 from 2004, that would leave me with 804. Then, how many times does 12 go into 804? Let me see. 12 times 60 is 720. So subtracting 720 from 804 gives me 84. Now, how many times does 12 go into 84? Well, 12 times 7 is 84. So that's 7.

So adding up the numbers I multiplied: 100 + 60 + 7 = 167. So 12 times 167 is 2004? Wait, let me check that. 12 times 167. Let me compute 12*160 first, which is 1920, and then 12*7 is 84. Adding those together, 1920 + 84 = 2004. Oh, so that means 12*167 is exactly 2004. Therefore, there is no remainder. Wait, so the remainder is zero?

But let me verify this another way to make sure I didn't make a mistake. Maybe using division algorithm. The division algorithm says that for any integers a and b (with b > 0), there exist unique integers q and r such that a = bq + r and 0 ≤ r < b. Here, a is 2004, b is 12. So we need to find q and r such that 2004 = 12q + r, where 0 ≤ r < 12.

Earlier, I found that q is 167 and r is 0. Let me check if that's correct. 12*167 is indeed 2004, so r is 0. Therefore, the remainder is 0.

But maybe I can check using another method. For example, maybe using modulo operations. 2004 divided by 12. Let me think about how to compute 2004 mod 12.

Alternatively, I can break down 2004 into smaller parts. Let me think about 2004 divided by 12. Since 12 is 3*4, maybe I can check divisibility by 3 and 4. If a number is divisible by both 3 and 4, then it's divisible by 12.

First, check divisibility by 3. The sum of the digits of 2004 is 2 + 0 + 0 + 4 = 6. Since 6 is divisible by 3, 2004 is divisible by 3.

Next, check divisibility by 4. The last two digits of 2004 are 04, which is 4. Since 4 is divisible by 4, 2004 is divisible by 4. Therefore, since it's divisible by both 3 and 4, it's divisible by 12. Therefore, the remainder is 0.

Another way: Let me divide 2004 by 12 step by step. Let me do the division process.

How many times does 12 go into 20? Once, because 12*1=12. Subtract 12 from 20, get 8. Bring down the 0, making it 80. How many times does 12 go into 80? Let me calculate 12*6=72, 12*7=84. 84 is too big, so 6 times. Subtract 72 from 80, get 8. Bring down the 4, making it 84. How many times does 12 go into 84? Exactly 7 times. Subtract 84, get 0. So the division ends here.

So the quotient is 167 (from 1, 6, 7) and the remainder is 0.

Wait, so all methods are pointing to the remainder being zero. But just to be thorough, let me try another approach. Let me subtract multiples of 12 from 2004 until I get a number less than 12.

But that might take too long, but maybe for a smaller number. Let me think. Alternatively, I can use the fact that 12*167=2004, so remainder is zero.

Alternatively, think about 2004 divided by 12. Let me divide 2004 by 12.

Well, 12*100=1200. 2004-1200=804.

Then 12*60=720. 804-720=84.

Then 12*7=84. 84-84=0.

So total is 100+60+7=167. So remainder is 0.

Alternatively, maybe using modular arithmetic properties. Let me compute 2004 mod 12.

Note that 2004 divided by 12. Let me think about 2004 divided by 12.

Alternatively, since 12*167=2004, then 2004 mod 12 is 0.

Alternatively, think about 2004 divided by 12. Let me think of 2004 as 2000 + 4.

So 2000 divided by 12. Let me compute 2000 / 12.

Well, 12*166= 12*(160 + 6) = 12*160 + 12*6 = 1920 + 72 = 1992.

So 2000 - 1992 = 8. Therefore, 2000 divided by 12 is 166 with a remainder of 8. Then add the 4 from the original 2004. So total remainder would be 8 + 4 = 12. But 12 is equal to 12, so the remainder would be 12 - 12 = 0.

Wait, that's another way to think about it. So breaking down 2004 into 2000 + 4. 2000 divided by 12 gives remainder 8, and 4 divided by 12 gives remainder 4. Then adding those remainders: 8 + 4 = 12. But since 12 is equal to the divisor, we subtract 12 to get the remainder 0.

Therefore, the remainder is 0.

Alternatively, maybe using negative numbers? Let me try.

But I think all methods are leading me to the same conclusion. The remainder is zero.

But just to make sure, let me check with a calculator. Wait, I don't have a calculator here, but maybe I can do a quick check.

Let me compute 12*167.

First, 12*160 = 1920.

Then 12*7 = 84.

Adding them together: 1920 + 84.

1920 + 80 = 2000, then +4 = 2004.

Yes, exactly. So 12*167 is 2004. Therefore, 2004 divided by 12 is exactly 167 with no remainder.

Therefore, the remainder is 0.

I think that's solid. All different methods confirm that. So I can be confident that the remainder is zero.

**Final Answer**
The remainder of 2004 divided by 12 is \boxed{0}.

To determine the remainder when 2004 is divided by 12, we can approach the problem using the **division algorithm**, which states that for any integers $ a $ and $ b $ (with $ b > 0 $), there exist unique integers $ q $ (the quotient) and $ r $ (the remainder) such that:

$$
a = bq + r \quad \text{where } 0 \leq r < b
$$

---

### Step 1: Apply the Division Algorithm

Let:
- $ a = 2004 $
- $ b = 12 $

We want to find $ q $ and $ r $ such that:

$$
2004 = 12q + r \quad \text{with } 0 \leq r < 12
$$

---

### Step 2: Perform the Division

We can compute the quotient $ q $ by dividing 2004 by 12:

$$
2004 \div 12 = 167
$$

Now, multiply back to check:

$$
12 \times 167 = 2004
$$

This means that the remainder $ r $ is:

$$
r = 2004 - (12 \times 167) = 2004 - 2004 = 0
$$

---

### Step 3: Confirm with Alternative Methods

- **Divisibility by 3 and 4**:
  - Sum of digits of 2004: $ 2 + 0 + 0 + 4 = 6 $, which is divisible by 3.
  - Last two digits: 04, which is divisible by 4.
  - Since 2004 is divisible by both 3 and 4, it is divisible by 12.
  - Therefore, the remainder is 0.

- **Breakdown of 2004**:
  - $ 2004 = 2000 + 4 $
  - $ 2000 \div 12 = 166 $ remainder 8
  - $ 4 \div 12 = 0 $ remainder 4
  - Total remainder: $ 8 + 4 = 12 $, which is equal to the divisor, so the actual remainder is $ 12 - 12 = 0 $

---

### Final Answer

$$
\boxed{0}
$$

Solution: An integer is divisible by 12 if and only if it is divisible by both 3 and 4.  Because $2+0+0+4=6$ is divisible by 3, 2004 is divisible by 3.  Also, the last two digits of 2004 form a multiple of 4, so 2004 is divisible by 4 as well. Therefore, 2004 is divisible by 12 and hence leaves a remainder of $\boxed{0}$ when divided by 12.


Explanation:
Okay, so there's this problem about thirty-five students attending a math club meeting. The number of girls is a multiple of 13, and there are more girls than boys. I need to find out how many boys are there. Let me think through this step by step.

First, let me note down the given information:

- Total number of students = 35
- Number of girls is a multiple of 13.
- Number of girls > number of boys.

Let me denote the number of girls as G and the number of boys as B. So, we have:

G + B = 35

And also, G is a multiple of 13. So, G can be 13, 26, 39, etc. But since the total number of students is 35, G can't be more than 35. So possible values for G are 13 and 26. Because 13*1=13, 13*2=26, 13*3=39 which is more than 35. So only 13 and 26 are possible.

Now, the problem also says that there are more girls than boys. So, G > B. Since G + B = 35, we can write B = 35 - G. Therefore, the condition G > B becomes G > 35 - G. Let me solve that inequality:

G > 35 - G
Adding G to both sides:
2G > 35
Dividing both sides by 2:
G > 17.5

Since the number of girls has to be an integer, G must be at least 18. But G is a multiple of 13. The multiples of 13 less than 35 are 13 and 26. So, which of these are greater than 17.5?

Well, 13 is less than 17.5, so that doesn't satisfy the condition. 26 is greater than 17.5, so that works. Therefore, G must be 26. Then, the number of boys is 35 - 26 = 9.

Wait, let me check again. If G is 26, then B is 9. So girls are 26, boys are 9. 26 is indeed more than 9, so that satisfies the condition. Also, 26 is a multiple of 13. So that seems to fit.

But let me check if there are any other possibilities. The next multiple of 13 is 39, which is more than 35, so that's not possible. The previous one is 13, but 13 is less than 17.5, so that would mean B = 35 - 13 = 22. Then, girls are 13 and boys are 22. But then girls are less than boys, which contradicts the given condition. Therefore, only G=26 is possible.

Therefore, the number of boys is 9. Let me just make sure there's no mistake here.

Total students: 26 girls + 9 boys = 35. Correct. Girls are 26, which is 2*13, so multiple of 13. And 26 > 9, so more girls than boys. All conditions are satisfied. So the answer should be 9.

But wait, let me think again. Is there a possibility that the problem is in another way? For example, maybe the number of girls is a multiple of 13, but maybe there's another interpretation? Like, maybe the number of girls is a multiple of 13, but not necessarily exactly 13 or 26? But since total is 35, the maximum number of girls can be 35, so multiples of 13 up to 35 are 13 and 26. So no, there are no other possibilities. So yeah, 26 is the only one that works.

Alternatively, maybe the problem is in some other way? For example, maybe the number of girls is a multiple of 13, but maybe there are more than one multiple? But no, as we saw, only 13 and 26. So yeah, 26 is the only one that satisfies G > B. Therefore, boys are 9. So I think that's the answer.

But let me check once more. Suppose someone might think that maybe the number of girls is 0? But 0 is a multiple of every number, including 13. But then, if girls are 0, boys are 35. But then girls are not more than boys. So that's invalid. Similarly, if girls are 13, boys are 22. Then girls are less than boys. So that's invalid. If girls are 26, boys are 9. Girls more than boys. So that's valid. If girls are 39, which is more than 35, impossible. So yeah, only 26. Therefore, boys are 9. So answer is 9.

I think that's solid. Let me check if there's any other possible way. For example, maybe the problem is in the translation? The original problem says "the number of girls at the meeting is a multiple of 13". So that's straightforward. So yeah, seems like 9 is the answer.

**Final Answer**
The number of boys at the meeting is \boxed{9}.

To solve the problem, we are given the following:

- A total of **35 students** attend the meeting.
- The **number of girls** is a **multiple of 13**.
- There are **more girls than boys**.

---

### Step 1: Define Variables

Let:
- $ G $ = number of girls
- $ B $ = number of boys

From the total number of students:
$$
G + B = 35
$$

We are also told that $ G $ is a **multiple of 13**, and that $ G > B $.

---

### Step 2: Determine Possible Values for $ G $

Since $ G $ is a multiple of 13 and $ G + B = 35 $, the possible values for $ G $ are limited by the total number of students. The multiples of 13 less than or equal to 35 are:

$$
13, 26
$$

(Next multiple is 39, which is greater than 35, so it's not valid.)

So, the possible values for $ G $ are **13** and **26**.

---

### Step 3: Apply the Condition $ G > B $

We also know that the number of girls must be **greater than** the number of boys. Since $ B = 35 - G $, the condition becomes:

$$
G > 35 - G
$$

Solving this inequality:

$$
2G > 35 \Rightarrow G > 17.5
$$

So, $ G $ must be **greater than 17.5**, and since $ G $ is an integer, the smallest possible value is **18**.

Now, among the possible values of $ G $ (13 and 26), only **26** satisfies $ G > 17.5 $.

---

### Step 4: Calculate the Number of Boys

If $ G = 26 $, then:

$$
B = 35 - G = 35 - 26 = 9
$$

Check the conditions:
- Total students: $ 26 + 9 = 35 $ ✅
- $ G = 26 $ is a multiple of 13 ✅
- $ G > B $ (26 > 9) ✅

All conditions are satisfied.

---

### Final Answer

$$
\boxed{9}
$$

[Assistant]
Explanation:
Okay, so there's this problem about thirty-five students attending a math club meeting. The number of girls is a multiple of 13, and there are more girls than boys. I need to find out how many boys are there. Let me think through this step by step.

First, let me note down the given information:

- Total number of students = 35
- Number of girls is a multiple of 13.
- Number of girls > number of boys.

Let me denote the number of girls as G and the number of boys as B. So, we have:

G + B = 35

And also, G is a multiple of 13. So, G can be 13, 26, 39, etc. But since the total number of students is 35, G can't be more than 35. So possible values for G are 13 and 26. Because 13*1=13, 13*2=26, 13*3=39 which is more than 35. So only 13 and 26 are possible.

Now, the problem also says that there are more girls than boys. So, G > B. Since G + B = 35, we can write B = 35 - G. Therefore, the condition G > B becomes G > 35 - G. Let me solve that inequality:

G > 35 - G
Adding G to both sides:
2G > 35
Dividing both sides by 2:
G > 17.5

Since the number of girls has to be an integer, G must be at least 18. But G is a multiple of 13. The multiples of 13 less than 35 are 13 and 26. So, which of these are greater than 17.5?

Well, 13 is less than 17.5, so that doesn't satisfy the condition. 26 is greater than 17.5, so that works. Therefore, G must be 26. Then, the number of boys is 35 - 26 = 9.

Wait, let me check again. If G is 26, then B is 9. So girls are 26, boys are 9. 26 is indeed more than 9, so that satisfies the condition. Also, 26 is a multiple of 13. So that seems to fit.

But let me check if there are any other possibilities. The next multiple of 13 is 39, which is more than 35, so that's not possible. The previous one is 13, but 13 is less than 17.5, so that would mean B = 35 - 13 = 22. Then, girls are 13 and boys are 22. But then girls are less than boys, which contradicts the given condition. Therefore, only G=26 is possible.

Therefore, the number of boys is 9. Let me just make sure there's no mistake here.

Total students: 26 girls + 9 boys = 35. Correct. Girls are 26, which is 2*13, so multiple of 13. And 26 > 9, so more girls than boys. All conditions are satisfied. So the answer should be 9.

But wait, let me think again. Is there a possibility that the problem is in another way? For example, maybe the number of girls is a multiple of 13, but maybe there's another interpretation? Like, maybe the number of girls is a multiple of 13, but not necessarily exactly 13 or 26? But since total is 35, the maximum number of girls can be 35, so multiples of 13 up to 35 are 13 and 26. So no, there are no other possibilities. So yeah, 26 is the only one that works.

Alternatively, maybe the problem is in some other way? For example, maybe the number of girls is a multiple of 13, but maybe there are more than one multiple? But no, as we saw, only 13 and 26. So yeah, 26 is the only one that satisfies G > B. Therefore, boys are 9. So I think that's the answer.

But let me check once more. Suppose someone might think that maybe the number of girls is 0? But 0 is a multiple of every number, including 13. But then, if girls are 0, boys are 35. But then girls are not more than boys. So that's invalid. Similarly, if girls are 13, boys are 22. Then girls are less than boys. So that's invalid. If girls are 26, boys are 9. Girls more than boys. So that's valid. If girls are 39, which is more than 35, impossible. So yeah, only 26. Therefore, boys are 9. So answer is 9.

I think that's solid. Let me check if there's any other possible way. For example, maybe the problem is in the translation? The original problem says "the number of girls at the meeting is a multiple of 13". So that's straightforward. So yeah, seems like 9 is the answer.

**Final Answer**
The number of boys at the meeting is \boxed{9}.

To solve the problem, we are given the following:

- A total of **35 students** attend the meeting.
- The **number of girls** is a **multiple of 13**.
- There are **more girls than boys**.

---

### Step 1: Define Variables

Let:
- $ G $ = number of girls
- $ B $ = number of boys

From the total number of students:
$$
G + B = 35
$$

We are also told that $ G $ is a **multiple of 13**, and that $ G > B $.

---

### Step 2: Determine Possible Values for $ G $

Since $ G $ is a multiple of 13 and $ G + B = 35 $, the possible values for $ G $ are limited by the total number of students. The multiples of 13 less than or equal to 35 are:

$$
13, 26
$$

(Next multiple is 39, which is greater than 35, so it's not valid.)

So, the possible values for $ G $ are **13** and **26**.

---

### Step 3: Apply the Condition $ G > B $

We also know that the number of girls must be **greater than** the number of boys. Since $ B = 35 - G $, the condition becomes:

$$
G > 35 - G
$$

Solving this inequality:

$$
2G > 35 \Rightarrow G > 17.5
$$

So, $ G $ must be **greater than 17.5**, and since $ G $ is an integer, the smallest possible value is **18**.

Now, among the possible values of $ G $ (13 and 26), only **26** satisfies $ G > 17.5 $.

---

### Step 4: Calculate the Number of Boys

If $ G = 26 $, then:

$$
B = 35 - G = 35 - 26 = 9
$$

Check the conditions:
- Total students: $ 26 + 9 = 35 $ ✅
- $ G = 26 $ is a multiple of 13 ✅
- $ G > B $ (26 > 9) ✅

All conditions are satisfied.

---

### Final Answer

$$
\boxed{9}
$$

Solution: We assume the number of girls is a nonnegative integer and no greater than 35. The nonnegative multiples of 13 less than 35 are 0, 13 and 26. As the number of girls is greater than the number of boys, the only valid choice is for there to be 26 girls. That leaves $35-26 = \boxed{9}$ boys at the meeting.


Explanation:
Okay, so I need to figure out the value of 5 star 1, where the operation star is defined as a star b equals 9a plus 2b minus ab plus 5. Let me write that down to make sure I have it right.

The operation is given by:
a ⋆ b = 9a + 2b - ab + 5

And I need to compute 5 ⋆ 1. Alright, so that means I need to substitute a with 5 and b with 1 in the formula. Let me do that step by step.

First, let me rewrite the formula with the substitutions. So replacing a with 5 and b with 1:

5 ⋆ 1 = 9*(5) + 2*(1) - (5)*(1) + 5

Now let me compute each term one by one.

Starting with 9 times 5. 9*5 is 45. Then 2 times 1 is 2. Then the next term is minus 5 times 1, which is 5. And then plus 5 at the end.

So putting those numbers together:

45 + 2 - 5 + 5

Let me compute that step by step.

First, 45 + 2 is 47. Then, 47 - 5 is 42. Then, 42 + 5 is 47.

Wait, so does that mean 5 ⋆ 1 equals 47? Let me check my calculations again to make sure I didn't make a mistake.

Original expression: 9a + 2b - ab + 5. Substituting a=5, b=1:

9*5 = 45, 2*1 = 2, ab = 5*1 = 5. So the expression becomes 45 + 2 - 5 + 5.

Yes, that's correct. Then 45 + 2 is 47. Then subtract 5: 47 - 5 = 42. Then add 5: 42 + 5 = 47.

Hmm, so the result is 47. Let me think if there's another way to approach this or if I might have misread the problem.

Wait, the operation is defined as 9a + 2b - ab + 5. Is there a possibility that the operation is meant to be grouped differently? For example, maybe it's 9a + (2b - ab) + 5? But that's how I interpreted it already.

Alternatively, could there be parentheses that I missed? Let me check the original problem again.

"If a ⋆ b = 9a + 2b - ab + 5, what is the value of 5 ⋆ 1?"

No, there are no parentheses in the definition, so it's just 9a + 2b - ab + 5. So my calculation seems right.

Alternatively, maybe I should check if there's a different interpretation of the operation. For example, sometimes operations can have different precedence, but in this case, the formula is written as 9a + 2b - ab + 5. So multiplication comes before addition and subtraction, so ab is 5*1, which is 5.

So 9a is 45, 2b is 2, then subtract ab which is 5, then add 5. So 45 + 2 is 47, minus 5 is 42, plus 5 is 47.

Alternatively, maybe the problem is written with some different grouping? Let me think. If the operation was 9a + 2b - (ab + 5), that would be different. But the original problem says 9a + 2b - ab + 5. So the way it's written, the terms are 9a, +2b, -ab, +5.

So the order of operations is correct. So I think my calculation is correct.

But just to be thorough, let me try another approach. Maybe factor the expression? Let me see if there's a way to simplify the operation.

Original operation: a ⋆ b = 9a + 2b - ab + 5. Let me rearrange the terms:

= 9a - ab + 2b + 5

Factor terms with a:

= a(9 - b) + 2b + 5

Alternatively, maybe factor something else? Let me see.

Alternatively, maybe factor terms with b:

= -ab + 2b + 9a + 5 = b(-a + 2) + 9a + 5

But I don't know if that helps. Let me try plugging in a=5 and b=1 again with this factored form:

b(-a + 2) + 9a + 5. So with a=5, b=1:

1*(-5 + 2) + 9*5 + 5 = 1*(-3) + 45 + 5 = -3 + 45 + 5 = 47.

Same result. So that's consistent.

Alternatively, maybe there's a different way to think about this operation? For example, maybe it's related to some kind of function or transformation? But I don't think that's necessary here.

Alternatively, maybe check if there's a typo in my calculation? Let me do the arithmetic again:

9*5 = 45. 2*1 = 2. ab = 5*1 = 5. So 45 + 2 = 47. Then subtract 5: 47 - 5 = 42. Then add 5: 42 + 5 = 47.

Yes, that's correct.

Alternatively, maybe the problem is written with a different order? For example, if the operation was 9a + 2b - (ab + 5), then it would be 45 + 2 - (5 + 5) = 47 - 10 = 37. But that's not the case here. The original problem says 9a + 2b - ab + 5, so the +5 is separate.

Alternatively, maybe the problem is written with a different operator precedence? For example, if the operation was 9a + 2b - ab + 5, but if someone misreads it as 9a + 2b - (ab + 5), but that would be a misinterpretation.

Alternatively, maybe the original problem had parentheses that were omitted? But as per the given problem, there are no parentheses.

Alternatively, maybe I need to check if the operation is commutative or something? But that's not necessary here.

Alternatively, maybe I can test with another example? For instance, if I compute 1 ⋆ 1, what would that be? Let's see:

1 ⋆ 1 = 9*1 + 2*1 - 1*1 + 5 = 9 + 2 - 1 + 5 = 15.

Alternatively, if I use the factored form: a(9 - b) + 2b + 5. For a=1, b=1: 1*(9 - 1) + 2*1 + 5 = 8 + 2 + 5 = 15. Same result.

So that seems okay.

Alternatively, maybe check another value? Let's say a=0, b=0. Then 0 ⋆ 0 = 9*0 + 2*0 - 0*0 + 5 = 0 + 0 - 0 + 5 = 5.

Alternatively, if I use the factored form: 0*(9 - 0) + 2*0 + 5 = 0 + 0 + 5 = 5. Correct.

So seems like the formula is working.

Therefore, going back to the original problem, 5 ⋆ 1 = 47.

Wait, but let me check once more. Maybe I made a mistake in the sign? Let me check the original operation again.

It says: a ⋆ b = 9a + 2b - ab + 5. So the terms are 9a, plus 2b, minus ab, plus 5. So the signs are +, +, -, +.

So substituting a=5, b=1:

9*5 = 45, + 2*1 = +2, - (5*1) = -5, +5.

So 45 + 2 is 47, minus 5 is 42, plus 5 is 47.

Yes, that's correct.

Alternatively, maybe the problem is written with a different order? For example, if the operation was 9a + 2b - ab + 5, but if someone misreads it as 9a + 2b - (ab + 5), but that would be a different expression.

But given the original problem, I think the correct interpretation is 9a + 2b - ab + 5.

Alternatively, maybe the problem is written in a different notation where the operator has different precedence? For example, if the star operation is defined with some precedence, but in standard math notation, multiplication comes before addition and subtraction.

So, in the expression 9a + 2b - ab + 5, the term ab is multiplication, so it's done first. So the expression is evaluated as 9a + 2b - (ab) + 5.

Therefore, I think my calculation is correct.

Therefore, the value of 5 ⋆ 1 is 47.

But just to be 100% sure, let me write out the entire calculation again:

5 ⋆ 1 = 9*(5) + 2*(1) - (5)*(1) + 5

Calculate each term:

9*5 = 45

2*1 = 2

5*1 = 5

So substituting back:

45 + 2 - 5 + 5

Now compute left to right:

45 + 2 = 47

47 - 5 = 42

42 + 5 = 47

Yes, that's correct.

Alternatively, if I group the terms differently:

(45 + 2) + (-5 + 5) = 47 + 0 = 47.

Same result.

Therefore, I think I can be confident that the answer is 47.

**Final Answer**
The value of $5 \star 1$ is \boxed{47}.

To evaluate the expression $ 5 \star 1 $, we are given a custom binary operation defined as:

$$
a \star b = 9a + 2b - ab + 5
$$

---

### Step 1: Substitute the values

We are asked to compute $ 5 \star 1 $, so we substitute $ a = 5 $ and $ b = 1 $ into the formula:

$$
5 \star 1 = 9(5) + 2(1) - (5)(1) + 5
$$

---

### Step 2: Compute each term

- $ 9(5) = 45 $
- $ 2(1) = 2 $
- $ (5)(1) = 5 $
- The constant term is $ +5 $

Now substitute these values back into the expression:

$$
5 \star 1 = 45 + 2 - 5 + 5
$$

---

### Step 3: Simplify the expression

We simplify step by step:

- $ 45 + 2 = 47 $
- $ 47 - 5 = 42 $
- $ 42 + 5 = 47 $

---

### Final Answer

$$
\boxed{47}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the value of 5 star 1, where the operation star is defined as a star b equals 9a plus 2b minus ab plus 5. Let me write that down to make sure I have it right.

The operation is given by:
a ⋆ b = 9a + 2b - ab + 5

And I need to compute 5 ⋆ 1. Alright, so that means I need to substitute a with 5 and b with 1 in the formula. Let me do that step by step.

First, let me rewrite the formula with the substitutions. So replacing a with 5 and b with 1:

5 ⋆ 1 = 9*(5) + 2*(1) - (5)*(1) + 5

Now let me compute each term one by one.

Starting with 9 times 5. 9*5 is 45. Then 2 times 1 is 2. Then the next term is minus 5 times 1, which is 5. And then plus 5 at the end.

So putting those numbers together:

45 + 2 - 5 + 5

Let me compute that step by step.

First, 45 + 2 is 47. Then, 47 - 5 is 42. Then, 42 + 5 is 47.

Wait, so does that mean 5 ⋆ 1 equals 47? Let me check my calculations again to make sure I didn't make a mistake.

Original expression: 9a + 2b - ab + 5. Substituting a=5, b=1:

9*5 = 45, 2*1 = 2, ab = 5*1 = 5. So the expression becomes 45 + 2 - 5 + 5.

Yes, that's correct. Then 45 + 2 is 47. Then subtract 5: 47 - 5 = 42. Then add 5: 42 + 5 = 47.

Hmm, so the result is 47. Let me think if there's another way to approach this or if I might have misread the problem.

Wait, the operation is defined as 9a + 2b - ab + 5. Is there a possibility that the operation is meant to be grouped differently? For example, maybe it's 9a + (2b - ab) + 5? But that's how I interpreted it already.

Alternatively, could there be parentheses that I missed? Let me check the original problem again.

"If a ⋆ b = 9a + 2b - ab + 5, what is the value of 5 ⋆ 1?"

No, there are no parentheses in the definition, so it's just 9a + 2b - ab + 5. So my calculation seems right.

Alternatively, maybe I should check if there's a different interpretation of the operation. For example, sometimes operations can have different precedence, but in this case, the formula is written as 9a + 2b - ab + 5. So multiplication comes before addition and subtraction, so ab is 5*1, which is 5.

So 9a is 45, 2b is 2, then subtract ab which is 5, then add 5. So 45 + 2 is 47, minus 5 is 42, plus 5 is 47.

Alternatively, maybe the problem is written with some different grouping? Let me think. If the operation was 9a + 2b - (ab + 5), that would be different. But the original problem says 9a + 2b - ab + 5. So the way it's written, the terms are 9a, +2b, -ab, +5.

So the order of operations is correct. So I think my calculation is correct.

But just to be thorough, let me try another approach. Maybe factor the expression? Let me see if there's a way to simplify the operation.

Original operation: a ⋆ b = 9a + 2b - ab + 5. Let me rearrange the terms:

= 9a - ab + 2b + 5

Factor terms with a:

= a(9 - b) + 2b + 5

Alternatively, maybe factor something else? Let me see.

Alternatively, maybe factor terms with b:

= -ab + 2b + 9a + 5 = b(-a + 2) + 9a + 5

But I don't know if that helps. Let me try plugging in a=5 and b=1 again with this factored form:

b(-a + 2) + 9a + 5. So with a=5, b=1:

1*(-5 + 2) + 9*5 + 5 = 1*(-3) + 45 + 5 = -3 + 45 + 5 = 47.

Same result. So that's consistent.

Alternatively, maybe there's a different way to think about this operation? For example, maybe it's related to some kind of function or transformation? But I don't think that's necessary here.

Alternatively, maybe check if there's a typo in my calculation? Let me do the arithmetic again:

9*5 = 45. 2*1 = 2. ab = 5*1 = 5. So 45 + 2 = 47. Then subtract 5: 47 - 5 = 42. Then add 5: 42 + 5 = 47.

Yes, that's correct.

Alternatively, maybe the problem is written with a different order? For example, if the operation was 9a + 2b - (ab + 5), then it would be 45 + 2 - (5 + 5) = 47 - 10 = 37. But that's not the case here. The original problem says 9a + 2b - ab + 5, so the +5 is separate.

Alternatively, maybe the problem is written with a different operator precedence? For example, if the operation was 9a + 2b - ab + 5, but if someone misreads it as 9a + 2b - (ab + 5), but that would be a misinterpretation.

Alternatively, maybe the original problem had parentheses that were omitted? But as per the given problem, there are no parentheses.

Alternatively, maybe I need to check if the operation is commutative or something? But that's not necessary here.

Alternatively, maybe I can test with another example? For instance, if I compute 1 ⋆ 1, what would that be? Let's see:

1 ⋆ 1 = 9*1 + 2*1 - 1*1 + 5 = 9 + 2 - 1 + 5 = 15.

Alternatively, if I use the factored form: a(9 - b) + 2b + 5. For a=1, b=1: 1*(9 - 1) + 2*1 + 5 = 8 + 2 + 5 = 15. Same result.

So that seems okay.

Alternatively, maybe check another value? Let's say a=0, b=0. Then 0 ⋆ 0 = 9*0 + 2*0 - 0*0 + 5 = 0 + 0 - 0 + 5 = 5.

Alternatively, if I use the factored form: 0*(9 - 0) + 2*0 + 5 = 0 + 0 + 5 = 5. Correct.

So seems like the formula is working.

Therefore, going back to the original problem, 5 ⋆ 1 = 47.

Wait, but let me check once more. Maybe I made a mistake in the sign? Let me check the original operation again.

It says: a ⋆ b = 9a + 2b - ab + 5. So the terms are 9a, plus 2b, minus ab, plus 5. So the signs are +, +, -, +.

So substituting a=5, b=1:

9*5 = 45, + 2*1 = +2, - (5*1) = -5, +5.

So 45 + 2 is 47, minus 5 is 42, plus 5 is 47.

Yes, that's correct.

Alternatively, maybe the problem is written with a different order? For example, if the operation was 9a + 2b - ab + 5, but if someone misreads it as 9a + 2b - (ab + 5), but that would be a different expression.

But given the original problem, I think the correct interpretation is 9a + 2b - ab + 5.

Alternatively, maybe the problem is written in a different notation where the operator has different precedence? For example, if the star operation is defined with some precedence, but in standard math notation, multiplication comes before addition and subtraction.

So, in the expression 9a + 2b - ab + 5, the term ab is multiplication, so it's done first. So the expression is evaluated as 9a + 2b - (ab) + 5.

Therefore, I think my calculation is correct.

Therefore, the value of 5 ⋆ 1 is 47.

But just to be 100% sure, let me write out the entire calculation again:

5 ⋆ 1 = 9*(5) + 2*(1) - (5)*(1) + 5

Calculate each term:

9*5 = 45

2*1 = 2

5*1 = 5

So substituting back:

45 + 2 - 5 + 5

Now compute left to right:

45 + 2 = 47

47 - 5 = 42

42 + 5 = 47

Yes, that's correct.

Alternatively, if I group the terms differently:

(45 + 2) + (-5 + 5) = 47 + 0 = 47.

Same result.

Therefore, I think I can be confident that the answer is 47.

**Final Answer**
The value of $5 \star 1$ is \boxed{47}.

To evaluate the expression $ 5 \star 1 $, we are given a custom binary operation defined as:

$$
a \star b = 9a + 2b - ab + 5
$$

---

### Step 1: Substitute the values

We are asked to compute $ 5 \star 1 $, so we substitute $ a = 5 $ and $ b = 1 $ into the formula:

$$
5 \star 1 = 9(5) + 2(1) - (5)(1) + 5
$$

---

### Step 2: Compute each term

- $ 9(5) = 45 $
- $ 2(1) = 2 $
- $ (5)(1) = 5 $
- The constant term is $ +5 $

Now substitute these values back into the expression:

$$
5 \star 1 = 45 + 2 - 5 + 5
$$

---

### Step 3: Simplify the expression

We simplify step by step:

- $ 45 + 2 = 47 $
- $ 47 - 5 = 42 $
- $ 42 + 5 = 47 $

---

### Final Answer

$$
\boxed{47}
$$

Solution: From the defined function, we know that $5\star 1 = 9(5)+2(1)-(5)(1)+5= 45+2-5+5=\boxed{47}$.