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Use the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $\sqrt{-32}$

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Introduction to complex numbers

Full solution

Q. Use the imaginary number

i

to rewrite the expression below as a complex number. Simplify all radicals.

\sqrt{-32}

Break Down $-32$ : First, let's break down $-32$ into $-1$ and $32$ to separate the negative sign which will become the imaginary unit $i$ . $\sqrt{-32} = \sqrt{-1 \times 32}$
Replace $\sqrt{-1}$ : Now, we know that $\sqrt{-1}$ is the imaginary unit $i$ , so we can replace that part. $\newline$ $\sqrt{-1 \times 32} = i \times \sqrt{32}$
Simplify $\sqrt{32}$ : Next, we simplify $\sqrt{32}$ by finding the prime factors of $32$ which are all $2$ s. $\sqrt{32} = \sqrt{2^5} = \sqrt{2^4 \times 2} = \sqrt{16 \times 2} = 4 \times \sqrt{2}$
Combine $i$ and $\sqrt{32}$ : Now we can combine the $i$ from the imaginary unit with the simplified square root of $32$ . $i \times \sqrt{32} = i \times 4 \times \sqrt{2}$
Multiply $i$ by $4$ : Finally, we multiply the $i$ by $4$ to get the complex number in standard form. $i \times 4 \times \sqrt{2} = 4i \times \sqrt{2}$

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