Chapter A.7, Problem 29AYU

To determine

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

Answer to Problem 29AYU

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 ${\left(\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)}^2$ .

Calculation:

Consider the expression.

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