Chapter 12, Problem 1RE

To determine

Whether the given statement “${\left(\sqrt{-9}\right)}^2=9$” is true or false.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

The imaginary unit $\sqrt{-1}$ is denoted by the symbol j.

In other words, $j=\sqrt{-1}$ and $j^2=-1$.

Calculation:

The given expression ${\left(\sqrt{-9}\right)}^2$ simplified as follows.

$\begin{array}{c}{\left(\sqrt{-9}\right)}^2={\left(\sqrt{\left(9\right)\left(-1\right)}\right)}^2\\ ={\left(\sqrt{9}\sqrt{-1}\right)}^2\\ ={\left(\sqrt{9}\cdot j\right)}^2\\ ={\left(\sqrt{9}\right)}^2\left(j^2\right)\end{array}$

On further simplifications,

$\begin{array}{c}{\left(\sqrt{-9}\right)}^2=9\left(-1\right)\left[\because j^2=-1\right]\\ =-9\end{array}$

Hence, the given statement is false.