Chapter 6.8, Problem 32WE

To determine

To simplify the given quotient.

Answer to Problem 32WE

  $\frac{10}{1+7i}=\frac{1}{5}-\frac{7}{5}i$

Explanation of Solution

Given information:

  $\frac{10}{1+7i}$

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,

  $\begin{array}{c}\frac{10}{1+7i}\times \frac{1-7i}{1-7i}=\frac{10\left(1-7i\right)}{{\left(1\right)}^2-{\left(7i\right)}^2}\\ =\frac{10-70i}{1-49i^2}\\ =\frac{10-70i}{1+49}\\ =\frac{10-70i}{50}\\ =\frac{1}{5}-\frac{7}{5}i\end{array}$