# Chapter 7 - Section 7.7 - Complex Numbers - 7.7 Exercises: 75

$-2$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To divide the given expression, $\dfrac{3+i}{i} ,$ multiply both the numerator and the denominator by $i$. Use $i^2=-1.$ $\bf{\text{Solution Details:}}$ Multiplying both the numerator and the denominator by $i,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{3+i}{i}\cdot\dfrac{i}{i} \\\\= \dfrac{(3+i)i}{i^2} .\end{array} Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{(3)i+(i)i}{i^2} \\\\= \dfrac{3+i^2}{i^2} .\end{array} Since $i^2=-1,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{3+(-1)}{-1} \\\\= \dfrac{2}{-1} \\\\= -2 .\end{array}

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