https://file.aiquickdraw.com/mathbot/file/1730476225214nkfauflr.jpg

To solve this problem, we need to simplify the expression \( 2 \sqrt{-32} \) and match it to one of the given answer choices.

### Step 1: Simplify the Square Root of a Negative Number
The expression inside the square root is \(-32\). Since this is a negative number, we can rewrite it using \( i \), where \( i = \sqrt{-1} \):
\[
\sqrt{-32} = \sqrt{-1 \cdot 32} = \sqrt{-1} \cdot \sqrt{32} = i \cdot \sqrt{32}
\]

### Step 2: Simplify \( \sqrt{32} \)
We can simplify \( \sqrt{32} \) by factoring out a perfect square:
\[
\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4 \sqrt{2}
\]

Therefore,
\[
\sqrt{-32} = i \cdot 4 \sqrt{2} = 4i \sqrt{2}
\]

### Step 3: Multiply by 2
Now, substitute back into the original expression:
\[
2 \sqrt{-32} = 2 \cdot (4i \sqrt{2}) = 8i \sqrt{2}
\]

### Answer
The correct answer is:
\[
\boxed{8i \sqrt{2}}
\]

Would you like further details or have any questions?

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Here are five related questions for further practice:
1. What is the value of \( \sqrt{-49} \) in terms of \( i \)?
2. How would you simplify \( 3 \sqrt{-18} \)?
3. What is the principal square root of \(-9\)?
4. How do you simplify \( \sqrt{50} \)?
5. How does \( i \) affect the simplification of square roots of negative numbers?

**Tip:** Remember that \( i = \sqrt{-1} \), which is useful for handling square roots of negative numbers in complex numbers.

Learn how to simplify expressions with square roots of negative numbers, specifically 2√(-32). This problem covers key concepts in complex numbers and imaginary numbers, showing step-by-step calculations to find the correct solution. Suitable for high school students in Grades 10-12.

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