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Use the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $\sqrt{-52}$

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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{-52}

Breakdown of $-52$ : First, let's break down $-52$ into $-1$ and $52$ to separate the negative sign which is associated with the imaginary unit $i$ . $\sqrt{-52} = \sqrt{-1 \times 52}$
Rewrite using i: Now, we know that $\sqrt{-1}$ is the imaginary unit $i$ , so we can rewrite the expression using $i$ . $\sqrt{-1 \times 52} = \sqrt{-1} \times \sqrt{52} = i \times \sqrt{52}$
Simplify $\sqrt{52}$ : Next, we simplify $\sqrt{52}$ . The number $52$ can be factored into $4 \times 13$ , where $4$ is a perfect square. $\newline$ $\sqrt{52} = \sqrt{4 \times 13} = \sqrt{4} \times \sqrt{13} = 2 \times \sqrt{13}$
Combine i with radical: Now, we can combine the $i$ from the imaginary unit with the simplified radical. $i \times \sqrt{52} = i \times 2 \times \sqrt{13} = 2i \times \sqrt{13}$
Write as complex number: Finally, we write the expression as a complex number, which is in the form $a + bi$ , where $a$ is the real part and $b$ is the imaginary part. Since there is no real part in this expression, it's just the imaginary part. $\newline$ $2i \times \sqrt{13} = 2i \sqrt{13}$

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