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Use the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $\sqrt{-84}$

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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{-84}

Use Imaginary Unit: Now, we know that $\sqrt{-1}$ is the imaginary unit $i$ . So, $\sqrt{-84}$ becomes $i \times \sqrt{84}$ .
Factor and Simplify: Next, we simplify $\sqrt{84}$ . The number $84$ can be factored into $4 \times 21$ , and $4$ is a perfect square. $\newline$ $\sqrt{84} = \sqrt{4 \times 21} = \sqrt{4} \times \sqrt{21} = 2 \times \sqrt{21}$
Combine Imaginary Unit: Now we can combine the imaginary unit with the simplified radical. $i \cdot \sqrt{84} = i \cdot 2 \cdot \sqrt{21}$
Multiply to Get Complex Number: Finally, we multiply the $i$ by $2$ to get the complex number. $i \times 2 \times \sqrt{21} = 2i \times \sqrt{21}$

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