Express the complex number $1 + i$ in the form $\sqrt{a}\mathrm{e}^{i\frac{\pi}{b}}$, where&nbsp

Question

Express the complex number $1 + i$ in the form $\sqrt{a}\mathrm{e}^{i\frac{\pi}{b}}$, where $a, b \in \mathbb{Z}^+$. Using the result from (a), show that $\left(\frac{1 + i}{\sqrt{2}}\right)^n$, where $n \in \mathbb{Z}$ has only eight distinct values. Hence solve the equation $z^8 - 1 = 0$.

Accepted Answer

(a) $a=2,\ b=4$. (b) Exactly eight values. (c) $z=\mathrm e^{ik\pi/4}$, $k=0,1,\ldots,7$.

\(n\)th roots of a complex number

\(n\)th roots of a complex number

Solution:

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