Question

Is there a quick way of calculating powers of complex numbers please?

Answer 1

eg. \[(8 + i)^{5}\]

Answer 2

There is, it's called using DeMoivre's theorem. It involves converting a complex number into trigonometric/polar form:
\[z=x+iy~~\iff~~z=re^{i\theta}=r(\cos\theta+i\sin\theta)\]
The theorem itself states that
\[(re^{i\theta})^n=r^ne^{in\theta}\]

Answer 3

...which leads to the conclusion that
\[[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)\]

Answer 4

Thank you!