Chapter 9, Problem 25RE

To determine

To find: The expression is ${\left(3+4i\right)}^4$ .

Answer to Problem 25RE

The expression is ${\left(3+4i\right)}^4=-527-336i$ .

Explanation of Solution

Given information:

The given expression is ${\left(3+4i\right)}^4$ .

Calculation:

Calculate the expression,

The magnitude of ${\left(3+4i\right)}^4$ ,

  $\begin{array}{l}=\sqrt{3^2+4^2}\\ =\sqrt{25}\\ =5\end{array}$

Now,

  $\begin{array}{c}\left(3+4i\right)=5\left(\frac{3}{5}+\frac{4}{5}i\right)\\ =5\left(\cos 53.13°-i\sin 53.13°\right)\end{array}$

By the De Mover’s theorem,

  $\begin{array}{c}{\left(3+4i\right)}^4={\left[5\left(\cos 53.13°-i\sin 53.13°\right)\right]}^6\\ =625\left[\cos 212.52°-i\sin 212.52°\right]\end{array}$

Substitute values of $\cos 212.52°$ and $\sin 212.52°$

Thus,

  $\begin{array}{c}{\left(3+4i\right)}^4=625\left[\cos 212.52°-i\sin 212.52°\right]\\ =625\left(-0.8432-i0.5376\right)\\ =-527-336i\end{array}$

Therefore, the expression is ${\left(3+4i\right)}^4=-527-336i$ .