Write the expression in rectangular form, x+yi, and in exponential form reiθ. (13−i)4 The rectangular...

${\displaystyle {(\sqrt{13}-i)}^4}$
${\displaystyle {(\sqrt{13}-i)}^2{(\sqrt{13}-i)}^2}$ 
Now ${\displaystyle {(A-B)}^2=A^2+B^2-2AB}$ 
${\displaystyle (13+i^2-2\sqrt{13}i)(13+i^2-2\sqrt{13}i)}$ 
${\displaystyle i^2=-1}$ 
${\displaystyle (12-2\sqrt{13}i)(12-2\sqrt{13}i)}$ 
${\displaystyle (144+4\times 13i^{-2}-4\sqrt{13}i)}$ 
${\displaystyle (144-52-4\sqrt{13}i)}$ ${\displaystyle =92-4\sqrt{13}i\to }$ rectangular form 
${\displaystyle r=\sqrt{{\left(92\right)}^2+{\left(4\sqrt{13}\right)}^2}}$ 
${\displaystyle r=\sqrt{8464+208}\Rightarrow r\sqrt{8672}}$ 
${\displaystyle r=4\sqrt{542}}$ 
${\displaystyle \theta ={\tan }^4(-\frac{4\sqrt{13}}{92})}$ 
${\displaystyle \theta ={\tan }^{-1}(-\frac{\sqrt{13}}{23})}$ ${\displaystyle =r{e}^{i\theta }\Rightarrow 4\sqrt{542}{e}^{i{\tan }^{-1}(-\frac{\sqrt{13}}{23})}}$

Say ${\displaystyle {(\sqrt{13}-i)}^4}$ 
${\displaystyle \Rightarrow {\left({(\sqrt{13}-1)}^2\right)}^2}$ 
${\displaystyle \Rightarrow {({\left(\sqrt{13}\right)}^2+{\left(i\right)}^2-2\cdot \sqrt{13}.i)}^2}$ 
${\displaystyle \Rightarrow {(13+(-1)-2\sqrt{13}i)}^2}$ 
${\displaystyle \Rightarrow {(12-2\sqrt{13}i)}^2}$ 
${\displaystyle \Rightarrow {\left(12\right)}^2+{\left(2\sqrt{13}i\right)}^2-2\cdot 12\cdot 2\sqrt{13}i}$ 
${\displaystyle \Rightarrow 144+52i^2-48\sqrt{13}i}$ 
${\displaystyle \Rightarrow 144-52-48\sqrt{13}i}$ 
${\displaystyle \Rightarrow 92+(-48\sqrt{13}i)}$ 
This is in the form of ${\displaystyle x+iy}$, ${\displaystyle x=92,y=(-48\sqrt{13})}$ 
if ${\displaystyle r{e}^{i\theta }}$ 
${\displaystyle v=\sqrt{{\left(92\right)}^2+{(-48\sqrt{13})}^2}}$ 
${\displaystyle =\sqrt{x^2+y^n}}$ 
${\displaystyle \Rightarrow r=\sqrt{38,416}}$ 
${\displaystyle \Rightarrow r=196}$. 
${\displaystyle \Rightarrow \theta =({360}^{\circ }-{61.99}^{\circ })}$ 
${\displaystyle \Rightarrow \theta ={298.01}^{\circ }}$ 
${\displaystyle \Rightarrow \theta \approx {298}^{\circ }=\frac{77}{45}}$ ${\displaystyle =r{e}^{i\theta }=196e^{i\cdot {298}^{\circ }}}$${\displaystyle =196\cdot e^{i\frac{77}{45}}}$ (answer)