Question

In Problems 41–52, write each expression in the standard form $a+bi$.

$52.{(1-\sqrt{5}i)}^8$

Short answer

${(1-\sqrt{5}i)}^8=-1263.6-286.42i$.

Step-by-step solution

Step 1. Rewrite a+bi in the form of r(cos∅+isin∅).

The magnitude of $(1-i\sqrt{5})$is

$$\begin{gathered} \sqrt{1^2+{(-\sqrt{5})}^2}=\sqrt{1+5} \\ =\sqrt{6} \end{gathered}$$

which gives

$$\begin{gathered} (1-i\sqrt{5})=\sqrt{6}(\frac{1}{\sqrt{6}}-\frac{\sqrt{5}}{\sqrt{6}}i) \\ =\sqrt{6}(\cos 294.1°-i\sin 294.1°) \end{gathered}$$

Step 2. Using De Moivre's Theorem.

If a complex number is $z=r(\cos \varnothing +i\sin \varnothing )$then $z^n=r^n\left[\cos \right(n\varnothing )+i\sin (n\varnothing \left)\right]$.

We get

role="math" localid="1646807571178" $$\begin{gathered} {(1-i\sqrt{5})}^8={\left[\sqrt{6}\right(\cos 294.1°-i\sin 294.1°\left)\right]}^8 \\ =\left\{\right(\sqrt{6}{)}^8\left[\cos \right(8\times 294.1)-i\sin (8\times 294.1\left)\right]\} \\ =1296(\cos 2352.8°-i\sin 2352.8°) \\ =1296(-0.975-i0.221) \\ =-1263.6-286.42i \end{gathered}$$