Question

First simplify each of the following numbers to the$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form or to therole="math" localid="1664869600880" $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

${\mathbf{(}\mathbf{0}\mathbf{.64}\mathbf{+}\mathbf{0}\mathbf{.77}\mathbf{i}\mathbf{)}}^{\mathbf{4}}$.

Short answer

The complex number becomes .$z=-0.93-0.35i$

The graph is mentioned below.

Step-by-step solution

Given Information  

Thecomplex number is${(0.64+0.77i)}^4$

Definition of the Complex number

A complex number is a number that is made up of both real and imaginary numbers.

Find the value.

The complex number is ${(0.64+0.77i)}^4$

Find the values for $(0.64+0.77i)$

Find x and y.

localid="1664870310188" $\begin{array}{c}z=(0.64+0.77i)\\ x=0.64\\ y=0.77\end{array}$

Find the value of r

localid="1664870316970" $\begin{array}{c}r=\sqrt{x^2+y^2}\\ r=\sqrt{x{\left(0.64\right)}^2+{\left(0.77\right)}^2}\\ =1\end{array}$

Find the value of .$\theta$

localid="1664870321932" $\begin{array}{c}\theta ={\cos }^{-1}\left(\frac{x}{r}\right)\\ \theta ={\cos }^{-1}\left(\frac{0.64}{1}\right)\\ =0.876\text{rad}\end{array}$

localid="1664870326981" $0.64+0.77i=e^{i\left(0.876\right)}$

For the complex number .localid="1664870330678" ${(0.64+0.77i)}^4$

localid="1664870335743" $\begin{array}{c}z\text{t}={\left(0.64+0.77i\right)}^4\\ ={\left(e^{i\left(0.876\right)}\right)}^4\\ =\left(e^{i(4\times 0.876)}\right)\\ \left(e^{i(4\times 0.876)}\right)=e^{i\left(3.5\right)}\end{array}$

Find x and y.

localid="1664870340314" $\begin{array}{c}x=\cos \left(3.5\text{rad}\right)\\ =-0.93\\ y=\sin \left(3.5\text{rad}\right)\\ =-0.35\end{array}$

The complex number becomes localid="1664870345449" $z=-0.93-0.35i$

The graph is mentioned below.

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