Full solution

Q. Use the imaginary number

i

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

\sqrt{-12}

Breakdown into parts: First, let's break down $\sqrt{-12}$ into $\sqrt{-1}$ and $\sqrt{12}$ . $\newline$ $\sqrt{-12} = \sqrt{-1 \times 12}$
Use of imaginary unit: Now, we know that $\sqrt{-1}$ is the imaginary unit $i$ . So, $\sqrt{-12} = i \cdot \sqrt{12}$
Prime factors of $12$ : Next, we simplify $\sqrt{12}$ by finding its prime factors which are $2 \times 2 \times 3$ . $\newline$ $\sqrt{12} = \sqrt{2^2 \times 3}$
Simplify square root: We can take out the square root of $2^2$ as $2$ . $\newline$ So, $\sqrt{12} = 2 \times \sqrt{3}$
Multiplication with i: Now, we multiply the i from earlier by $2 \times \sqrt{3}$ . $\newline$ $i \times \sqrt{12} = i \times 2 \times \sqrt{3}$
Correction of mistake: Finally, we write the expression as a complex number. $i \times 2 \times \sqrt{3} = 2i \times \sqrt{3}$
Correction of mistake: Finally, we write the expression as a complex number. $i \times 2 \times \sqrt{3} = 2i \times \sqrt{3}$ But wait, I made a mistake. The correct simplification should be $2i \times \sqrt{3}$ , not $2i \times \sqrt{3}$ . Let's correct that. $i \times \sqrt{12} = 2i \times \sqrt{3}$