# Chapter 1 - Section 1.3 - Complex Numbers; Quadratic Equations in the Complex Number System - 1.3 Assess Your Understanding - Page 111: 32

$\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To simplify the given expression, $\left( \dfrac{\sqrt{3}}{2}-\dfrac{1}{2}i \right)^2 ,$ use the square of a binomial and the equivalence $i^2=-1.$ $\bf{\text{Solution Details:}}$ Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \left( \dfrac{\sqrt{3}}{2}\right)^2-2\left( \dfrac{\sqrt{3}}{2}\right)\left(\dfrac{1}{2}i \right)+\left(\dfrac{1}{2}i \right)^2 \\\\= \dfrac{3}{4}-\dfrac{\sqrt{3}}{2}i+\dfrac{1}{4}i^2 .\end{array} Since $i^2=-1,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{3}{4}-\dfrac{\sqrt{3}}{2}i+\dfrac{1}{4}(-1) \\\\= \dfrac{3}{4}-\dfrac{\sqrt{3}}{2}i-\dfrac{1}{4} \\\\= \dfrac{2}{4}-\dfrac{\sqrt{3}}{2}i \\\\= \dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i .\end{array}

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.