Chapter I, Problem 14E

Evaluate the expression and write your answer in the form a + bi.

14. $\sqrt{-3}\sqrt{-12}$

To determine

To evaluate: The expression $\sqrt{-3}\sqrt{-12}$ in the form of $a+bi$.

Answer to Problem 14E

The value of the expression $\sqrt{-3}\sqrt{-12}$ is $5i$.

Explanation of Solution

Simplify the expression $\sqrt{-3}\sqrt{-12}$ as follows.

$\begin{array}{c}\sqrt{-3}\sqrt{-12}=\sqrt{3\left(-1\right)}\sqrt{12\left(-1\right)}\\ =\left[\sqrt{3}\left(\sqrt{-1}\right)\right]\left[\sqrt{12}\left(\sqrt{-1}\right)\right]\\ =\left[\sqrt{3}\left(i\right)\right]\left[\sqrt{12}i\right]\left[\because i=\sqrt{-1}\right]\\ =\sqrt{36}{i}^2\\ =-6\end{array}$

Thus, the value of the expression $\sqrt{-3}\sqrt{-12}$ is $-6$.