Chapter A.7, Problem 30AYU

To determine

To write:The given expression in the standard form $a+ib$ .

Answer to Problem 30AYU

The given expression in the standard form $a+ib$ is $\frac{1}{2}-\frac{\sqrt{3}}{2}i$ .

Explanation of Solution

Given information:

The expression is ${\left(\frac{\sqrt{3}}{2}-\frac{1}{2}i\right)}^2$ .

Calculation:

Consider the expression.

  $\begin{array}{c}{\left(\frac{\sqrt{3}}{2}-\frac{1}{2}i\right)}^2={\left(\frac{1}{2}i\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2-2\left(\frac{1}{2}i\right)\left(\frac{\sqrt{3}}{2}\right)\\ =\frac{1}{4}\left(-1\right)-\frac{\sqrt{3}}{2}i+\frac{3}{4}\\ =-\frac{1}{4}-\frac{\sqrt{3}}{2}i+\frac{3}{4}\\ =\frac{1}{2}-\frac{\sqrt{3}}{2}i\end{array}$