Answer
See the answer below.
Work Step by Step
We need to show that $w=z^{\frac{1}{4}}$ or $w^{4}=z$
Using de Moivre's theorem, we get
$w^{4}=[2(\cos15^{\circ}+i\sin15^{\circ})]^{4}$
$=(2)^{4}[\cos(4\cdot15^{\circ})+i\sin(4\cdot15^{\circ})]$
$=16(\cos 60^{\circ}+i\sin60^{\circ})$
which in standard form is
$=16(\frac{1}{2}+i\cdot\frac{\sqrt 3}{2})=8+8i\sqrt 3=z$
