Math Problem Statement

Which expression is equal to 2√(-32)?

Solution

To solve this problem, we need to simplify the expression 2322 \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 32-32. Since this is a negative number, we can rewrite it using ii, where i=1i = \sqrt{-1}: 32=132=132=i32\sqrt{-32} = \sqrt{-1 \cdot 32} = \sqrt{-1} \cdot \sqrt{32} = i \cdot \sqrt{32}

Step 2: Simplify 32\sqrt{32}

We can simplify 32\sqrt{32} by factoring out a perfect square: 32=162=162=42\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4 \sqrt{2}

Therefore, 32=i42=4i2\sqrt{-32} = i \cdot 4 \sqrt{2} = 4i \sqrt{2}

Step 3: Multiply by 2

Now, substitute back into the original expression: 232=2(4i2)=8i22 \sqrt{-32} = 2 \cdot (4i \sqrt{2}) = 8i \sqrt{2}

Answer

The correct answer is: 8i2\boxed{8i \sqrt{2}}

Would you like further details or have any questions?


Here are five related questions for further practice:

  1. What is the value of 49\sqrt{-49} in terms of ii?
  2. How would you simplify 3183 \sqrt{-18}?
  3. What is the principal square root of 9-9?
  4. How do you simplify 50\sqrt{50}?
  5. How does ii affect the simplification of square roots of negative numbers?

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

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Square Roots
Imaginary Numbers

Formulas

√(-a) = i√(a), where i = √(-1)
Simplification of square roots of negative numbers

Theorems

Properties of Imaginary Numbers

Suitable Grade Level

Grades 10-12