Examples

Complex product, quotient, power using Euler

Define:

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

Product $z w$ :

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}}

Quotient $z / w$ :

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}}

Power $z^{8}$ :

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

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:

3 + 3 i ≫≫ 32 (\frac{1}{2} + \frac{1}{2} i) ≫≫ 32 e^{i \frac{π}{4}}

Compute the power:

(3 + 3 i)^{4} ≫≫ (32 e^{i \frac{π}{4}})^{4} ≫≫ 324 e^{iπ} ≫≫ - 324