Chapter 6.5, Problem 94E

To determine

To find: The plot of the second, third and forth power of a function in a graph.

Answer to Problem 94E

The points move in a clockwise manner in a specific direction.

Explanation of Solution

Given information:

The given expression is:

  $z=\left(\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)$

Consider the given expression as:

  $z=\left(\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)$

Find the square of the above expression:

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

Find the cube of the above expression:

  $\begin{array}{c}{z}^3=z^2.z\\ =\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)\left(\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)\\ =-\frac{1}{4}-\frac{\sqrt{3}}{4}i+\frac{\sqrt{3}}{4}i+\frac{3}{4}{i}^2\\ =-\frac{1}{4}-\frac{\sqrt{3}}{4}i+\frac{\sqrt{3}}{4}i+\frac{3}{4}\left(-1\right)\\ =-1\end{array}$

Find the forth power of the above expression:

  $\begin{array}{c}{z}^4=z^3.z\\ =-1\left(\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)\\ =-\frac{1}{2}-\frac{\sqrt{3}}{2}i\end{array}$

Plot the above results in a graph paper:

  

Hence, it is seen from the above graph that the points move in a clockwise manner in a specific direction