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

Compute $(2(\cos(\pi/6)+ i\sin(\pi/6)))^2$ in the form $a+bi$.

Compute $(5(\cos(\pi/9)+ i\sin(\pi/9)))^3$ and express your answer in $a+bi$ form.

Compute $(4(\cos(\pi/4)+ i\sin(\pi/4)))^4$ in the form $a+bi$.

Compute $(3(\cos(2\pi/3)+ i\sin(2\pi/3)))^5$ and give the result in $a+bi$ form.

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

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

Compute $(\sqrt2(\cos(\pi/8)+ i\sin(\pi/8)))^6$ and express the result in $a+bi$ form.

Express $\displaystyle\frac{1+i}{\sqrt2}$ in polar form and then compute its 8th power, giving your answer in $a+bi$ form.

Compute $(5(\cos(3\pi/10)+i\sin(3\pi/10)))^9$ and give the result in $a+bi$ form.

Express $z=-1 + i\sqrt3$ in polar form and then compute $z^4$, giving your answer in $a+bi$ form.

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