Chapter 6.8, Problem 8E

To determine

With the help of $\frac{1}{z}=\frac{\overline{z}}{{\left|z\right|}^2}$ evaluate the value of $\frac{1}{z}$ for $z=12-5i$ .

Answer to Problem 8E

The value of $\frac{1}{z}$ is equal to $\frac{12}{169}+\frac{5}{169}i$ .

Explanation of Solution

Given information: It is given that,

  $z=12-5i\mathrm{....}\left(1\right)$

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 $z=12-5i$ therefore $\overline{z}=12+5i$ . Now evaluate

  ${\left|z\right|}^2$ using (1). Therefore from (1),

  $\begin{array}{c}{\left|z\right|}^2={\left(12\right)}^2+{\left(-5\right)}^2\\ =144+25\\ =169\end{array}$

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

  $\begin{array}{c}\frac{1}{z}=\frac{\overline{z}}{{\left|z\right|}^2}\\ =\frac{12+5i}{169}\\ =\frac{12}{169}+\frac{5}{169}i\end{array}$