Question

question 1
which expression is equal to $2\sqrt{-32}$?
\bigcirc $8i\sqrt{2}$
\bigcirc $-8i\sqrt{2}$
\bigcirc $-8\sqrt{2}$
\bigcirc $8\sqrt{2}$

Explanation:

Step1: Recall the imaginary unit

Recall that \( \sqrt{-1} = i \), so we can rewrite \( \sqrt{-32} \) as \( \sqrt{32 \times (-1)} \). Using the property of square roots \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0, b\geq0 \), here we extend it to complex numbers), we have \( \sqrt{-32}=\sqrt{32}\times\sqrt{-1} \).

Step2: Simplify \( \sqrt{32} \)

Simplify \( \sqrt{32} \). We know that \( 32 = 16\times2 \), so \( \sqrt{32}=\sqrt{16\times2}=\sqrt{16}\times\sqrt{2}=4\sqrt{2} \).

Step3: Substitute back and multiply by 2

Substitute \( \sqrt{32} = 4\sqrt{2} \) and \( \sqrt{-1}=i \) into \( \sqrt{-32} \), we get \( \sqrt{-32}=4\sqrt{2}i \). Then multiply by 2: \( 2\sqrt{-32}=2\times4\sqrt{2}i = 8i\sqrt{2} \).

Answer:

A. \( 8i\sqrt{2} \)