Chapter 6.8, Problem 30WE

To determine

To simplify the given quotient.

Answer to Problem 30WE

  $\frac{15}{2-i}=6+3i$

Explanation of Solution

Given information:

  $\frac{15}{2-i}$

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,

  $\begin{array}{c}\frac{15}{2-i}\times \frac{2+i}{2+i}=\frac{15\left(2+i\right)}{{\left(2\right)}^2-{\left(i\right)}^2}\\ =\frac{30+15i}{4-i^2}\\ =\frac{30+15i}{4+1}\\ =\frac{30+15i}{5}\\ =6+3i\end{array}$