Explanation:
Okay, so I need to find the dot product of two vectors. Let me recall what a dot product is. From what I remember, the dot product of two vectors is calculated by multiplying their corresponding components and then adding those products together. Let me check if that's correct.

So, if I have two vectors, say, vector **a** = [a₁, a₂, a₃] and vector **b** = [b₁, b₂, b₃], then their dot product **a** · **b** should be a₁*b₁ + a₂*b₂ + a₃*b₃. Yeah, that sounds right. I think that's the formula. Let me make sure there isn't a different formula or something else involved. Hmm, maybe there's something with angles or magnitudes? Wait, no, the dot product can also be expressed as |a||b|cosθ, where θ is the angle between them. But since I don't know the angle here, I think I should stick with the component-wise multiplication and addition method.

Alright, so the problem gives me two vectors:

First vector:
$$
\begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}
$$
Second vector:
$$
\begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

So, let me label them for clarity. Let me call the first vector **u** and the second vector **v**. So,

**u** = [-2, 0, 7]
**v** = [3, 4, -5]

Now, applying the formula for the dot product, which is u₁*v₁ + u₂*v₂ + u₃*v₃. Let me compute each term step by step.

First term: u₁*v₁. That would be (-2) multiplied by 3. Let me calculate that. -2 * 3 is... -6. Okay, that's straightforward.

Second term: u₂*v₂. Here, u₂ is 0 and v₂ is 4. So, 0 multiplied by 4. That's 0. Hmm, that term is zero. Interesting, so that term doesn't contribute anything to the sum.

Third term: u₃*v₃. u₃ is 7 and v₃ is -5. So, 7 multiplied by -5. Let me compute that. 7 * 5 is 35, so with the negative sign, it's -35.

Now, adding all three terms together: first term (-6) + second term (0) + third term (-35). Let me do the addition step by step.

Starting with -6 + 0. That's just -6. Then, adding -35 to that. So, -6 + (-35) is the same as -(6 + 35) which is -41.

Wait, let me verify that again. If I have -6 and then subtract another 35, that's definitely -41. Yeah, that seems right.

So, putting it all together, the dot product of **u** and **v** is -41.

Let me just check my calculations again to make sure I didn't make any mistakes.

First component: -2 * 3 = -6. Correct. Second component: 0 * 4 = 0. Correct. Third component: 7 * (-5) = -35. Correct. Then adding them: -6 + 0 + (-35) = -41. Yep, that seems consistent.

Is there any chance I mixed up the components? Let me check the original vectors again.

First vector is [-2, 0, 7]. Second vector is [3, 4, -5]. So, the first components are -2 and 3, second components 0 and 4, third components 7 and -5. So, multiplying each pair and adding. Yep, that's exactly what I did.

Alternatively, maybe I can think of it as:

(-2)(3) + (0)(4) + (7)(-5) = -6 + 0 -35 = -41.

Yes, same result.

I think that's solid. I don't see any errors here. So, the dot product is -41.

Just to recap, the steps were:

1. Multiply the corresponding components of the two vectors.
2. Sum the results of those multiplications.
3. The result is the dot product.

Since all steps check out, I think that's the correct answer.

Another way to think about it: if I had to visualize these vectors, the dot product gives a scalar value that can be related to the angle between them. But since we don't have the angle, the component method is the way to go.

Alternatively, if I wanted to use the formula with magnitudes and cosine, I would need to compute the magnitudes of both vectors and the angle between them. But that seems more complicated here because we don't have the angle. So, definitely, the component method is the right approach here.

Let me just compute the magnitudes for fun, just to see what they are, even though it's not necessary for the problem.

The magnitude of **u** is sqrt((-2)^2 + 0^2 + 7^2) = sqrt(4 + 0 + 49) = sqrt(53).

The magnitude of **v** is sqrt(3^2 + 4^2 + (-5)^2) = sqrt(9 + 16 + 25) = sqrt(50).

So, if I were to use the other formula, the dot product would be |u||v|cosθ. So, -41 = sqrt(53)*sqrt(50)*cosθ. But since we don't need θ, this isn't helpful here.

But just to check, if I compute sqrt(53)*sqrt(50), that's sqrt(53*50) = sqrt(2650). Let me approximate that. sqrt(2650) is approximately sqrt(2500) = 50, so sqrt(2650) is a bit more, like 51.47. Then, -41 divided by 51.47 is approximately -0.797. So, cosθ ≈ -0.797, which would mean θ is about 142.7 degrees. But again, this is extra information not needed for the problem.

But just to confirm that the dot product is negative, which makes sense because the cosine of the angle is negative, meaning the angle is obtuse. So, the vectors are pointing in directions that are more than 90 degrees apart.

But again, this is just extra. The main point is that the component calculation gives -41, and all steps check out.

I think I'm confident that the answer is -41.

**Final Answer**
The dot product is \boxed{-41}.

To find the **dot product** of two vectors, we use the formula:

$$
\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3
$$

---

### Step 1: Identify the vectors

We are given the vectors:

$$
\mathbf{u} = \begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

---

### Step 2: Multiply corresponding components

We compute the product of each pair of corresponding components:

- First component: $ (-2) \cdot 3 = -6 $
- Second component: $ 0 \cdot 4 = 0 $
- Third component: $ 7 \cdot (-5) = -35 $

---

### Step 3: Add the results

Now, sum the products:

$$
-6 + 0 + (-35) = -41
$$

---

### Final Answer

$$
\boxed{-41}
$$

[Assistant]
Explanation:
Okay, so I need to find the dot product of two vectors. Let me recall what a dot product is. From what I remember, the dot product of two vectors is calculated by multiplying their corresponding components and then adding those products together. Let me check if that's correct.

So, if I have two vectors, say, vector **a** = [a₁, a₂, a₃] and vector **b** = [b₁, b₂, b₃], then their dot product **a** · **b** should be a₁*b₁ + a₂*b₂ + a₃*b₃. Yeah, that sounds right. I think that's the formula. Let me make sure there isn't a different formula or something else involved. Hmm, maybe there's something with angles or magnitudes? Wait, no, the dot product can also be expressed as |a||b|cosθ, where θ is the angle between them. But since I don't know the angle here, I think I should stick with the component-wise multiplication and addition method.

Alright, so the problem gives me two vectors:

First vector:
$$
\begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}
$$
Second vector:
$$
\begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

So, let me label them for clarity. Let me call the first vector **u** and the second vector **v**. So,

**u** = [-2, 0, 7]
**v** = [3, 4, -5]

Now, applying the formula for the dot product, which is u₁*v₁ + u₂*v₂ + u₃*v₃. Let me compute each term step by step.

First term: u₁*v₁. That would be (-2) multiplied by 3. Let me calculate that. -2 * 3 is... -6. Okay, that's straightforward.

Second term: u₂*v₂. Here, u₂ is 0 and v₂ is 4. So, 0 multiplied by 4. That's 0. Hmm, that term is zero. Interesting, so that term doesn't contribute anything to the sum.

Third term: u₃*v₃. u₃ is 7 and v₃ is -5. So, 7 multiplied by -5. Let me compute that. 7 * 5 is 35, so with the negative sign, it's -35.

Now, adding all three terms together: first term (-6) + second term (0) + third term (-35). Let me do the addition step by step.

Starting with -6 + 0. That's just -6. Then, adding -35 to that. So, -6 + (-35) is the same as -(6 + 35) which is -41.

Wait, let me verify that again. If I have -6 and then subtract another 35, that's definitely -41. Yeah, that seems right.

So, putting it all together, the dot product of **u** and **v** is -41.

Let me just check my calculations again to make sure I didn't make any mistakes.

First component: -2 * 3 = -6. Correct. Second component: 0 * 4 = 0. Correct. Third component: 7 * (-5) = -35. Correct. Then adding them: -6 + 0 + (-35) = -41. Yep, that seems consistent.

Is there any chance I mixed up the components? Let me check the original vectors again.

First vector is [-2, 0, 7]. Second vector is [3, 4, -5]. So, the first components are -2 and 3, second components 0 and 4, third components 7 and -5. So, multiplying each pair and adding. Yep, that's exactly what I did.

Alternatively, maybe I can think of it as:

(-2)(3) + (0)(4) + (7)(-5) = -6 + 0 -35 = -41.

Yes, same result.

I think that's solid. I don't see any errors here. So, the dot product is -41.

Just to recap, the steps were:

1. Multiply the corresponding components of the two vectors.
2. Sum the results of those multiplications.
3. The result is the dot product.

Since all steps check out, I think that's the correct answer.

Another way to think about it: if I had to visualize these vectors, the dot product gives a scalar value that can be related to the angle between them. But since we don't have the angle, the component method is the way to go.

Alternatively, if I wanted to use the formula with magnitudes and cosine, I would need to compute the magnitudes of both vectors and the angle between them. But that seems more complicated here because we don't have the angle. So, definitely, the component method is the right approach here.

Let me just compute the magnitudes for fun, just to see what they are, even though it's not necessary for the problem.

The magnitude of **u** is sqrt((-2)^2 + 0^2 + 7^2) = sqrt(4 + 0 + 49) = sqrt(53).

The magnitude of **v** is sqrt(3^2 + 4^2 + (-5)^2) = sqrt(9 + 16 + 25) = sqrt(50).

So, if I were to use the other formula, the dot product would be |u||v|cosθ. So, -41 = sqrt(53)*sqrt(50)*cosθ. But since we don't need θ, this isn't helpful here.

But just to check, if I compute sqrt(53)*sqrt(50), that's sqrt(53*50) = sqrt(2650). Let me approximate that. sqrt(2650) is approximately sqrt(2500) = 50, so sqrt(2650) is a bit more, like 51.47. Then, -41 divided by 51.47 is approximately -0.797. So, cosθ ≈ -0.797, which would mean θ is about 142.7 degrees. But again, this is extra information not needed for the problem.

But just to confirm that the dot product is negative, which makes sense because the cosine of the angle is negative, meaning the angle is obtuse. So, the vectors are pointing in directions that are more than 90 degrees apart.

But again, this is just extra. The main point is that the component calculation gives -41, and all steps check out.

I think I'm confident that the answer is -41.

**Final Answer**
The dot product is \boxed{-41}.

To find the **dot product** of two vectors, we use the formula:

$$
\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3
$$

---

### Step 1: Identify the vectors

We are given the vectors:

$$
\mathbf{u} = \begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

---

### Step 2: Multiply corresponding components

We compute the product of each pair of corresponding components:

- First component: $ (-2) \cdot 3 = -6 $
- Second component: $ 0 \cdot 4 = 0 $
- Third component: $ 7 \cdot (-5) = -35 $

---

### Step 3: Add the results

Now, sum the products:

$$
-6 + 0 + (-35) = -41
$$

---

### Final Answer

$$
\boxed{-41}
$$

Solution:
The dot product of $\begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}$ is
\[(-2)(3) + (0)(4) + (7)(-5) = \boxed{-41}.\]


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.

Wait, so does that mean f(-2) is 18? Let me check my calculations again to make sure I didn't make any mistakes.

Starting over:

f(-2) = 5(-2)² + 3(-2) + 4.

Calculating (-2)²: (-2) * (-2) = 4. Then 5 * 4 = 20. Correct.

Then 3 * (-2) = -6. Correct.

Then adding 4. So 20 - 6 + 4.

20 - 6 is 14, and 14 + 4 is 18. Hmm, seems right.

Wait, but let me think again. Sometimes when you have multiple operations, order of operations can be tricky. Let me verify each step once more.

First term: 5 times (-2) squared. The exponent comes before multiplication, so yes, first square -2, then multiply by 5. So that's 5*(4) = 20. Correct.

Second term: 3 times (-2). That's straightforward, -6. Correct.

Third term: +4. So adding all together: 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. Yep, same result.

Alternatively, maybe I can compute it as 20 - 6 = 14, then 14 + 4 = 18. Either way, same answer.

Is there a chance I messed up the signs? Let me check again.

Original function: 5x² + 3x + 4. Substituting x = -2:

5*(-2)^2 + 3*(-2) + 4.

Yes, so the first term is positive because it's squared. The second term is negative because of the 3 times negative. The third term is positive.

So 20 - 6 + 4.

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

Alternatively, maybe I can combine the constants first. Let me see.

But I think my calculation is correct.

Wait, but just to be thorough, let me compute the entire expression again:

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

(-2)^2 = 4. 5*4 = 20.

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

Third term: +4.

So total is 20 - 6 + 4.

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

Yes, that seems consistent.

Alternatively, maybe I can plug in x = -2 into the original function using another method. Let me try expanding the function step by step.

Alternatively, maybe I can use a different order of operations. Let me check if I have any parentheses that I might have misread.

The original function is 5x² + 3x + 4. So when substituting x = -2, it's 5*(-2)^2 + 3*(-2) + 4.

No parentheses around the -2 except what's necessary for the exponent. So the exponent applies only to the -2, not to the 3 or anything else.

So, no, I think that's correct.

Alternatively, if I had parentheses like 5(-2)^2, which is the same as 5*( (-2)^2 ).

Alternatively, if there were parentheses around the 3x, but no, the original function is 5x² + 3x + 4.

So, I think my calculation is correct.

Wait, but just to make sure, let me compute the entire expression using another approach. Let me compute each term again:

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

(-2)^2 is 4. 5*4 is 20.

Second term: 3*(-2).

3*(-2) is -6.

Third term: 4.

So adding them up: 20 + (-6) + 4.

Which is 20 - 6 + 4.

Let me compute 20 - 6 first. 20 - 6 is 14. Then 14 + 4 is 18.

Alternatively, if I do 20 + 4 first, that's 24, then 24 - 6 is 18. Same result.

So regardless of the order, it's 18.

Therefore, I think f(-2) is 18.

But just to be 100% sure, maybe I can plug in x = -2 into the original function using another method. Let me try expanding the function.

Alternatively, maybe I can use a different order of operations. Let me think.

Wait, another way to check is to compute the value step by step.

Let me write down the function again:

f(x) = 5x² + 3x + 4.

So, f(-2) = 5*(-2)^2 + 3*(-2) + 4.

Let me compute each term:

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

Compute (-2)^2: (-2)*(-2) = 4.

Multiply by 5: 5*4 = 20.

Second term: 3*(-2).

Multiply 3 and -2: 3*(-2) = -6.

Third term: 4.

So adding them up: 20 + (-6) + 4.

Which is 20 - 6 + 4.

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

Same result.

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.

Alternatively, maybe I can think of the function as a quadratic and plug in the value.

Alternatively, maybe I can check if there's a possibility of a sign error. For example, if I had forgotten that (-2)^2 is positive, but I think I accounted for that.

Wait, if someone mistakenly thought that (-2)^2 is -4, then they would get 5*(-4) = -20, then -20 + 3*(-2) + 4 = -20 -6 +4 = -22. But that's wrong. But I know that (-2)^2 is positive 4, so that's not the case here.

Alternatively, if someone thought that the exponent applies only to the 2, not the negative sign, but in standard order of operations, the exponent applies to the entire -2, so (-2)^2 is 4.

So, I think my calculation is correct.

Therefore, after checking multiple times, I believe that f(-2) is 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 the standard process of function evaluation by substituting $ x = -2 $ into the expression.

---

### 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.

Wait, so does that mean f(-2) is 18? Let me check my calculations again to make sure I didn't make any mistakes.

Starting over:

f(-2) = 5(-2)² + 3(-2) + 4.

Calculating (-2)²: (-2) * (-2) = 4. Then 5 * 4 = 20. Correct.

Then 3 * (-2) = -6. Correct.

Then adding 4. So 20 - 6 + 4.

20 - 6 is 14, and 14 + 4 is 18. Hmm, seems right.

Wait, but let me think again. Sometimes when you have multiple operations, order of operations can be tricky. Let me verify each step once more.

First term: 5 times (-2) squared. The exponent comes before multiplication, so yes, first square -2, then multiply by 5. So that's 5*(4) = 20. Correct.

Second term: 3 times (-2). That's straightforward, -6. Correct.

Third term: +4. So adding all together: 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. Yep, same result.

Alternatively, maybe I can compute it as 20 - 6 = 14, then 14 + 4 = 18. Either way, same answer.

Is there a chance I messed up the signs? Let me check again.

Original function: 5x² + 3x + 4. Substituting x = -2:

5*(-2)^2 + 3*(-2) + 4.

Yes, so the first term is positive because it's squared. The second term is negative because of the 3 times negative. The third term is positive.

So 20 - 6 + 4.

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

Alternatively, maybe I can combine the constants first. Let me see.

But I think my calculation is correct.

Wait, but just to be thorough, let me compute the entire expression again:

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

(-2)^2 = 4. 5*4 = 20.

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

Third term: +4.

So total is 20 - 6 + 4.

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

Yes, that seems consistent.

Alternatively, maybe I can plug in x = -2 into the original function using another method. Let me try expanding the function step by step.

Alternatively, maybe I can use a different order of operations. Let me check if I have any parentheses that I might have misread.

The original function is 5x² + 3x + 4. So when substituting x = -2, it's 5*(-2)^2 + 3*(-2) + 4.

No parentheses around the -2 except what's necessary for the exponent. So the exponent applies only to the -2, not to the 3 or anything else.

So, no, I think that's correct.

Alternatively, if I had parentheses like 5(-2)^2, which is the same as 5*( (-2)^2 ).

Alternatively, if there were parentheses around the 3x, but no, the original function is 5x² + 3x + 4.

So, I think my calculation is correct.

Wait, but just to make sure, let me compute the entire expression using another approach. Let me compute each term again:

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

(-2)^2 is 4. 5*4 is 20.

Second term: 3*(-2).

3*(-2) is -6.

Third term: 4.

So adding them up: 20 + (-6) + 4.

Which is 20 - 6 + 4.

Let me compute 20 - 6 first. 20 - 6 is 14. Then 14 + 4 is 18.

Alternatively, if I do 20 + 4 first, that's 24, then 24 - 6 is 18. Same result.

So regardless of the order, it's 18.

Therefore, I think f(-2) is 18.

But just to be 100% sure, maybe I can plug in x = -2 into the original function using another method. Let me try expanding the function.

Alternatively, maybe I can use a different order of operations. Let me think.

Wait, another way to check is to compute the value step by step.

Let me write down the function again:

f(x) = 5x² + 3x + 4.

So, f(-2) = 5*(-2)^2 + 3*(-2) + 4.

Let me compute each term:

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

Compute (-2)^2: (-2)*(-2) = 4.

Multiply by 5: 5*4 = 20.

Second term: 3*(-2).

Multiply 3 and -2: 3*(-2) = -6.

Third term: 4.

So adding them up: 20 + (-6) + 4.

Which is 20 - 6 + 4.

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

Same result.

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.

Alternatively, maybe I can think of the function as a quadratic and plug in the value.

Alternatively, maybe I can check if there's a possibility of a sign error. For example, if I had forgotten that (-2)^2 is positive, but I think I accounted for that.

Wait, if someone mistakenly thought that (-2)^2 is -4, then they would get 5*(-4) = -20, then -20 + 3*(-2) + 4 = -20 -6 +4 = -22. But that's wrong. But I know that (-2)^2 is positive 4, so that's not the case here.

Alternatively, if someone thought that the exponent applies only to the 2, not the negative sign, but in standard order of operations, the exponent applies to the entire -2, so (-2)^2 is 4.

So, I think my calculation is correct.

Therefore, after checking multiple times, I believe that f(-2) is 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 the standard process of function evaluation by substituting $ x = -2 $ into the expression.

---

### 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, 4 boxes with pens, 2 of which are already counted in the pencils, so 4 - 2 = 2 boxes with only pens. Then, adding those: 3 (only pencils) + 2 (only pens) + 2 (both) = 7. So, total boxes with at least one of the items is 7. Therefore, 10 - 7 = 3 boxes with neither.

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 part of P not overlapping is 5 - 2 = 3, the part of Q not overlapping is 4 - 2 = 2. Then, total in the Venn diagram is 3 + 2 + 2 = 7. So, outside the Venn diagram, which is the boxes with neither, is 10 - 7 = 3.

So, seems like 3 is the answer. But let me check if there's another way to think about it. Maybe using a table or something.

Let me think of the boxes as categories:

- Only pencils
- Only pens
- Both pencils and pens
- Neither pencils nor pens

We need to find the number of boxes in the last category.

We know that:

Number of boxes with pencils (including both) is 5. So, only pencils + both = 5.

Similarly, number of boxes with pens (including both) is 4. So, only pens + both = 4.

We also know that both is 2.

Therefore, only pencils = 5 - 2 = 3.

Only pens = 4 - 2 = 2.

Therefore, total boxes with pencils or pens is only pencils + only pens + both = 3 + 2 + 2 = 7.

Total boxes is 10, so neither is 10 - 7 = 3.

Same answer. So, seems like 3 is correct.

But let me check if there's a possibility that the two boxes that contain both pens and pencils are counted in both the pencil and pen counts. So, when they say five boxes contain pencils, does that mean exactly pencils or pencils and pens? 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 wording is important here. If the two boxes that contain both pens and pencils are included in the counts of pencils and pens. So, the five boxes with pencils include the two that also have pens, and the four boxes with pens include the two that also have pencils. Therefore, when we do the inclusion-exclusion principle, we subtract the overlap once.

So, that's exactly what I did earlier. So, the calculation is correct.

Alternatively, if someone thought that the two boxes with both pens and pencils are separate, but that would be incorrect. Because if you have two boxes that have both, they are already part of the five pencils and four pens.

So, for example, if you have 5 boxes with pencils, and two of them also have pens, then the rest of the boxes with pencils don't have pens. Similarly for pens.

So, yeah, the calculation seems right.

Another way to think about it: total boxes = boxes with pencils only + boxes with pens only + boxes with both + boxes with neither.

So, boxes with pencils only = 5 - 2 = 3.

Boxes with pens only = 4 - 2 = 2.

Boxes with both = 2.

Boxes with neither = ?

So, total boxes = 3 + 2 + 2 + x = 7 + x = 10. Therefore, x = 3.

Yep, same answer.

I think that's solid. So, 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: Use 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 apply 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 pencils and pens)

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

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

---

### Step 2: Subtract from Total to Find Boxes with Neither

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

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

---

### Step 3: Verification (Optional)

We can also verify this by breaking down the categories:

- **Only pencils**: $ 5 - 2 = 3 $
- **Only pens**: $ 4 - 2 = 2 $
- **Both pencils and pens**: 2
- **Neither**: $ 10 - (3 + 2 + 2) = 3 $

This confirms the result.

---

### 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, 4 boxes with pens, 2 of which are already counted in the pencils, so 4 - 2 = 2 boxes with only pens. Then, adding those: 3 (only pencils) + 2 (only pens) + 2 (both) = 7. So, total boxes with at least one of the items is 7. Therefore, 10 - 7 = 3 boxes with neither.

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 part of P not overlapping is 5 - 2 = 3, the part of Q not overlapping is 4 - 2 = 2. Then, total in the Venn diagram is 3 + 2 + 2 = 7. So, outside the Venn diagram, which is the boxes with neither, is 10 - 7 = 3.

So, seems like 3 is the answer. But let me check if there's another way to think about it. Maybe using a table or something.

Let me think of the boxes as categories:

- Only pencils
- Only pens
- Both pencils and pens
- Neither pencils nor pens

We need to find the number of boxes in the last category.

We know that:

Number of boxes with pencils (including both) is 5. So, only pencils + both = 5.

Similarly, number of boxes with pens (including both) is 4. So, only pens + both = 4.

We also know that both is 2.

Therefore, only pencils = 5 - 2 = 3.

Only pens = 4 - 2 = 2.

Therefore, total boxes with pencils or pens is only pencils + only pens + both = 3 + 2 + 2 = 7.

Total boxes is 10, so neither is 10 - 7 = 3.

Same answer. So, seems like 3 is correct.

But let me check if there's a possibility that the two boxes that contain both pens and pencils are counted in both the pencil and pen counts. So, when they say five boxes contain pencils, does that mean exactly pencils or pencils and pens? 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 wording is important here. If the two boxes that contain both pens and pencils are included in the counts of pencils and pens. So, the five boxes with pencils include the two that also have pens, and the four boxes with pens include the two that also have pencils. Therefore, when we do the inclusion-exclusion principle, we subtract the overlap once.

So, that's exactly what I did earlier. So, the calculation is correct.

Alternatively, if someone thought that the two boxes with both pens and pencils are separate, but that would be incorrect. Because if you have two boxes that have both, they are already part of the five pencils and four pens.

So, for example, if you have 5 boxes with pencils, and two of them also have pens, then the rest of the boxes with pencils don't have pens. Similarly for pens.

So, yeah, the calculation seems right.

Another way to think about it: total boxes = boxes with pencils only + boxes with pens only + boxes with both + boxes with neither.

So, boxes with pencils only = 5 - 2 = 3.

Boxes with pens only = 4 - 2 = 2.

Boxes with both = 2.

Boxes with neither = ?

So, total boxes = 3 + 2 + 2 + x = 7 + x = 10. Therefore, x = 3.

Yep, same answer.

I think that's solid. So, 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: Use 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 apply 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 pencils and pens)

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

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

---

### Step 2: Subtract from Total to Find Boxes with Neither

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

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

---

### Step 3: Verification (Optional)

We can also verify this by breaking down the categories:

- **Only pencils**: $ 5 - 2 = 3 $
- **Only pens**: $ 4 - 2 = 2 $
- **Both pencils and pens**: 2
- **Neither**: $ 10 - (3 + 2 + 2) = 3 $

This confirms the result.

---

### 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 only multiples of 13 possible are 13 and 26. So no, there's no other possibility.

Alternatively, could there be a mistake in the inequality? Let me check again. The condition is G > B. Since B = 35 - G, then G > 35 - G. So 2G > 35, so G > 17.5. Therefore, G must be at least 18. Since G is a multiple of 13, the next multiple after 13 is 26. So yes, 26 is the only possible value. Therefore, boys are 35 - 26 = 9. So I think that's correct.

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 as we saw, only 13 and 26. So no. So I think the answer is 9. Let me check once more.

If there are 26 girls, then boys are 9. 26 is a multiple of 13. 26 > 9. All conditions met. If there were 13 girls, then boys would be 22, but 13 < 22, which violates the condition. So 26 is the only possible. Therefore, the answer is 9. So I think that's correct.

**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:
$$
13, 26
$$

So, $ G \in \{13, 26\} $

---

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

We can express $ B $ in terms of $ G $:
$$
B = 35 - G
$$

Now, apply the condition $ G > B $:
$$
G > 35 - G
$$
$$
2G > 35
$$
$$
G > 17.5
$$

So, $ G $ must be **greater than 17.5**. The only multiple of 13 that satisfies this is:
$$
G = 26
$$

---

### Step 4: Calculate the Number of Boys

Now that we know $ G = 26 $, we can find $ B $:
$$
B = 35 - G = 35 - 26 = 9
$$

---

### Step 5: Verify the Solution

- Total students: $ 26 + 9 = 35 $ ✅
- $ G = 26 $ is a multiple of 13 ✅
- $ G = 26 > B = 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 only multiples of 13 possible are 13 and 26. So no, there's no other possibility.

Alternatively, could there be a mistake in the inequality? Let me check again. The condition is G > B. Since B = 35 - G, then G > 35 - G. So 2G > 35, so G > 17.5. Therefore, G must be at least 18. Since G is a multiple of 13, the next multiple after 13 is 26. So yes, 26 is the only possible value. Therefore, boys are 35 - 26 = 9. So I think that's correct.

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 as we saw, only 13 and 26. So no. So I think the answer is 9. Let me check once more.

If there are 26 girls, then boys are 9. 26 is a multiple of 13. 26 > 9. All conditions met. If there were 13 girls, then boys would be 22, but 13 < 22, which violates the condition. So 26 is the only possible. Therefore, the answer is 9. So I think that's correct.

**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:
$$
13, 26
$$

So, $ G \in \{13, 26\} $

---

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

We can express $ B $ in terms of $ G $:
$$
B = 35 - G
$$

Now, apply the condition $ G > B $:
$$
G > 35 - G
$$
$$
2G > 35
$$
$$
G > 17.5
$$

So, $ G $ must be **greater than 17.5**. The only multiple of 13 that satisfies this is:
$$
G = 26
$$

---

### Step 4: Calculate the Number of Boys

Now that we know $ G = 26 $, we can find $ B $:
$$
B = 35 - G = 35 - 26 = 9
$$

---

### Step 5: Verify the Solution

- Total students: $ 26 + 9 = 35 $ ✅
- $ G = 26 $ is a multiple of 13 ✅
- $ G = 26 > B = 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.