Chapter 6.8, Problem 31WE

To determine

To simplify the given quotient.

Answer to Problem 31WE

  $\frac{2}{3-i}=\frac{3}{5}+\frac{1}{5}i$

Explanation of Solution

Given information:

  $\frac{2}{3-i}$

We know that $i^2=-1$ and $i=\sqrt{-1}$

Here we shall also use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$

To solve this we have to multiply numerator and denominator by the conjugate of the denominator.

Hence we get,

  $\begin{array}{c}\frac{2}{3-i}\times \frac{3+i}{3+i}=\frac{2\left(3+i\right)}{{\left(3\right)}^2-{\left(i\right)}^2}\\ =\frac{6+2i}{9-i^2}\\ =\frac{6+2i}{9+1}\\ =\frac{6+2i}{10}\\ =\frac{3}{5}+\frac{1}{5}i\end{array}$