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Use the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $\sqrt{-80}$

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Introduction to complex numbers

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Q. Use the imaginary number

i

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

\sqrt{-80}

Replace Imaginary Unit: Now, we know that $\sqrt{-1}$ is the imaginary unit $i$ , so we can replace that part of the expression. $\sqrt{-1 \times 80} = i \times \sqrt{80}$
Simplify Square Root: Next, we simplify $\sqrt{80}$ by finding the prime factors of $80$ and looking for pairs to take out of the square root. $80 = 2 \times 2 \times 2 \times 2 \times 5 = (2 \times 2) \times (2 \times 2) \times 5 = 4 \times 4 \times 5$ We can take out the $4$ as it is a perfect square. $\sqrt{80} = \sqrt{4 \times 4 \times 5} = 4 \times \sqrt{5}$
Combine Imaginary Unit: Now, we can combine the $i$ from the imaginary unit with the simplified square root. $i \times \sqrt{80} = i \times 4 \times \sqrt{5}$
Multiply to Standard Form: Finally, we multiply the $i$ by $4$ to get the complex number in standard form. $i \times 4 \times \sqrt{5} = 4i \times \sqrt{5}$

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