Science:Math Exam Resources/Courses/MATH152/April 2022/Question A04/Solution 1

Recall that we can write complex numbers as vectors in $\mathbb {R} ^{2}$. Knowing this, the vector $\left[{\begin{array}{c}-1\\1\end{array}}\right]$ represents the complex number $-1+i$, and has norm ${\sqrt {2}}$ and polar angle $3\pi /4$. So, we have

$$-1+i={\sqrt {2}}e^{i{\frac {3\pi }{4}}}.$$

Thus, we may compute

$$(-1+i)^{30}=({\sqrt {2}}e^{i{\frac {3\pi }{4}}})^{30}=2^{15}e^{i{\frac {90\pi }{4}}}=2^{15}e^{i\left({\frac {88\pi }{4}}+{\frac {2\pi }{4}}\right)}=2^{15}e^{i\pi /2}=2^{15}i.$$