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Use the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $-\sqrt{-75}$

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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{-75}

Use Imaginary Unit: Now, we know that $\sqrt{-1}$ is the imaginary unit $i$ . So we can write $-\sqrt{-75}$ as $-i \times \sqrt{75}$ .
Simplify $\sqrt{75}$ : Next, we simplify $\sqrt{75}$ . The number $75$ can be factored into $3 \times 25$ , and $\sqrt{25}$ is $5$ . So $\sqrt{75}$ is $\sqrt{3 \times 25}$ which is $5 \times \sqrt{3}$ .
Combine Terms: Now we can combine the $-i$ and $5 \times \sqrt{3}$ to get the final answer. So $-i \times \sqrt{75}$ becomes $-i \times 5 \times \sqrt{3}$ .
Correct Multiplication: Finally, we multiply $-i$ by $5 \times \sqrt{3}$ to get $-5i \times \sqrt{3}$ . But wait, there's a mistake here. The correct multiplication should be $-i \times 5 \times \sqrt{3}$ , which simplifies to $-5i \times \sqrt{3}$ . So the mistake doesn't affect the final result.

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