Chapter 6.8, Problem 7E

To determine

With the help of $\frac{1}{z}=\frac{\overline{z}}{{\left|z\right|}^2}$ evaluate the value of $\frac{1}{z}$ for $z=-4+3i$ .

Answer to Problem 7E

The value of $\frac{1}{z}$ is equal to $-\frac{4}{25}-\frac{3}{25}i$ .

Explanation of Solution

Given information: It is given that,

  $z=-4+3i\mathrm{....}\left(1\right)$

Formula used: The formula used to evaluate $\frac{1}{z}$ is written as,

  $\frac{1}{z}=\frac{\overline{z}}{{\left|z\right|}^2}\mathrm{....}\left(2\right)$

Calculation: Since $z=-4+3i$ therefore $\overline{z}=-4-3i$ . Now evaluate ${\left|z\right|}^2$ using (1). Therefore from (1),

  $\begin{array}{c}{\left|z\right|}^2={\left(-4\right)}^2+{\left(3\right)}^2\\ =16+9\\ =25\end{array}$

Now from (2) evaluate $\frac{1}{z}$ ,

  $\begin{array}{c}\frac{1}{z}=\frac{\overline{z}}{{\left|z\right|}^2}\\ =\frac{-4-3i}{\left(25\right)}\\ =-\frac{4}{25}-\frac{3}{25}i\end{array}$