Examples

Complex product, quotient, power using Euler

Define:

z = 2 e^{i \frac{π}{2}} w = 5 e^{i \frac{π}{3}}

Product $z w$ :

\begin{matrix} z w ≫ ≫ (2 e^{i \frac{π}{2}}) \cdot (5 e^{i \frac{π}{3}}) \\ ≫ ≫ (2 \cdot 5) (e^{i \frac{π}{2}}) (e^{i \frac{π}{3}}) ≫ ≫ 10 e^{i \frac{π}{2} + i \frac{π}{3}} ≫ ≫ 10 e^{i \frac{5 π}{6}} \end{matrix}

Quotient $z / w$ :

\begin{matrix} z / w ≫ ≫ (2 e^{i \frac{π}{2}}) / (5 e^{i \frac{π}{3}}) \\ ≫ ≫ \frac{2 e^{i \frac{π}{2}}}{5 e^{i \frac{π}{3}}} ≫ ≫ \frac{2}{5} e^{i \frac{π}{2}} e^{- i \frac{π}{3}} ≫ ≫ \frac{2}{5} e^{i \frac{π}{6}} \end{matrix}

Power $z^{8}$ :

\begin{matrix} z^{8} ≫ ≫ {(2 e^{i \frac{π}{2}})}^{8} \\ ≫ ≫ 2^{8} {(e^{i \frac{π}{2}})}^{8} ≫ ≫ 512 e^{i \cdot 4 π} \end{matrix}

Notice:

e^{i \cdot 4 π} ≫ ≫ {(e^{2 π i})}^{2} ≫ ≫ 1^{2} ≫ ≫ 1

Simplify:

512 e^{i \cdot 4 π} ≫ ≫ 512

Compute ${(3 + 3 i)}^{4}$ .

Solution

First convert to exponential form:

\begin{matrix} 3 + 3 i ≫ ≫ 3 \sqrt{2} (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i) \\ ≫ ≫ 3 \sqrt{2} e^{i \frac{π}{4}} \end{matrix}

Compute the power:

\begin{matrix} {(3 + 3 i)}^{4} ≫ ≫ {(3 \sqrt{2} e^{i \frac{π}{4}})}^{4} \\ ≫ ≫ 324 e^{i π} ≫ ≫ - 324 \end{matrix}