<ol class="abc-list"><li><div class="list-item-paragraph">Express the complex number \(1 + i\) in the form \(\sqrt{a}\mathrm{e}^{i\frac{\pi}{b}}\), where&nbsp;\(a, b \in \mathbb{Z}^+\).</div></li><li><div class="list-item-paragraph">Using the result from (a), show that&nbsp;\(\left(\frac{1 + i}{\sqrt{2}}\right)^n\), where&nbsp;\(n \in \mathbb{Z}\) has only eight distinct values.</div></li><li><div class="list-item-paragraph">Hence solve the equation \(z^8 - 1 = 0\).</div></li></ol>

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 \(z^8 - 1 = 0\).

Special Results involving Exponential Forms [Out of Syllabus]

Complex Numbers

International Baccalaureate Higher Level Mathematics

\(n\)th roots of a complex number

\(n\)th roots of a complex number