IB Mathematics Applications & Interpretation 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 $\frac{1+i}{\sqrt{2}}$ 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 $a+bi$.