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Use the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $\sqrt{-98}$

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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{-98}

Replace with imaginary unit: Now, we know that $\sqrt{-1}$ is the imaginary unit $i$ , so we can replace that part of the expression. $\newline$ $\sqrt{-1 \times 98} = i \times \sqrt{98}$
Simplify square root of $98$ : Next, we simplify $\sqrt{98}$ by finding the prime factors of $98$ and looking for pairs to take out of the square root. $\newline$ $98 = 2 \times 49 = 2 \times 7 \times 7$ $\newline$ We can take a pair of $7$ out of the square root as $7$ . $\newline$ $\sqrt{98} = 7 \times \sqrt{2}$
Combine $i$ and square root: Now we can combine the $i$ and the simplified square root to get the final complex number. $i \cdot \sqrt{98} = i \cdot 7 \cdot \sqrt{2}$
Write final complex number: Finally, we write the expression as a complex number. $\newline$ $i \times 7 \times \sqrt{2} = 7i \times \sqrt{2}$ $\newline$ But wait, we made a mistake here. We should have written $7 \times \sqrt{2}$ as $7 \times i \times \sqrt{2}$ .

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