Question Type 7: Using DeMoivre's theorem to apply powers onto complex numbers in Polar form Exercises

Compute $\left(2\left(\cos\frac{\pi}{6}+ i\sin\frac{\pi}{6}\right)\right)^2$ in the form $a+bi$.

Compute $\left(5\left(\cos\frac{\pi}{9}+ \mathrm{i}\sin\frac{\pi}{9}\right)\right)^3$ and express your answer in the form $a+b\mathrm{i}$, giving exact values for $a$ and $b$.

Let $z=2\left(\cos \frac{\pi}{5} + i\sin \frac{\pi}{5}\right)$. Find $z^7$ in polar form.

Calculate $\left(\sqrt{2}\left(\cos\left(\frac{\pi}{8}\right)+ i\sin\left(\frac{\pi}{8}\right)\right)\right)^6$ and express the result in the form $a+bi$.

Compute $\left(5\left(\cos\left(\frac{3\pi}{10}\right)+i\sin\left(\frac{3\pi}{10}\right)\right)\right)^9$ and give the result in exact $a+bi$ form.

Express $z=-1 + i\sqrt{3}$ in polar form and then compute $z^4$, giving your answer in $a+bi$ form.

Compute $\left(4\left(\cos\frac{\pi}{4}+ i \sin\frac{\pi}{4}\right)\right)^4$ in the form $a+bi$.

Compute $(7(\cos(5\pi/12)+ i\sin(5\pi/12)))^2$ and express the answer in $a+bi$ form.

Express $z=1 - i$ in polar form and find $z^{12}$, giving the result in $a+bi$ form.

Express $\displaystyle\frac{1+i}{\sqrt{2}}$ in polar form and then compute its 8th power, giving your answer in $a+bi$ form.

Compute $(6(\cos(7\pi/18)+ i\sin(7\pi/18)))^3$ and express your result in $a+bi$ form.

Find $\left(3\left(\cos\left(\frac{2\pi}{3}\right)+ i\sin\left(\frac{2\pi}{3}\right)\right)\right)^5$, giving the result in the form $a+bi$.