Power of Complex numbers

How to calculate power of complex numbers? This is a very simple problem, if you know the main idea of this kind of tasks. First of all, it is nesesary to convert any complext number into exponetial form and then apply power to it. Then it is important to understand, that the results mean. So, let's consider simple example:

Example: \( (1+i)^5 = ? \)

This equation can be solved via converting to exponential form:

\[(1+i)^5 = \left(2^\frac{1}{2}e^{i\frac{\pi}{4}}\right)^5 = 2^\frac{5}{2}e^{i\frac{5\pi}{4}} = \frac{25}{2}e^{i(\frac{\pi}{4}+\pi)} = -4-4i\]

For graphical explanation please check this video:

Published: 2023-05-08 13:29:28
Updated: 2023-05-08 14:58:34