Complex expression with powers in exponential form

Problem

Compute in algebraic form $(1-i)^{5}+\frac{1}{(1+\sqrt3\,i)^{3}}$ using the exponential form to evaluate the powers.

Solution

Write $1-i$ in exponential form: $|1-i|=\sqrt2$ and $\arg(1-i)=-\dfrac\pi4$, so $1-i=\sqrt2\,e^{-i\pi/4}$. Raising to the fifth power, $(1-i)^5=(\sqrt2)^5e^{-i5\pi/4}=4\sqrt2\,e^{-i5\pi/4}=4\sqrt2\left(-\frac{\sqrt2}2+\frac{\sqrt2}2 i\right)=-4+4i.$

Similarly $1+\sqrt3\,i=2e^{i\pi/3}$, so $(1+\sqrt3\,i)^3=8e^{i\pi}=-8$ and therefore $\frac1{(1+\sqrt3\,i)^3}=-\frac18.$

Adding the two results: $(1-i)^5+\frac1{(1+\sqrt3\,i)^3}=-4+4i-\frac18=-\frac{33}8+4i.$

$\boxed{-\frac{33}{8}+4i}$