# Find power of complex number online with step by step solution

Our online calculator lets you to find power of complex number with step by step solution. To use the calculator you should choose representation of complex number (algebraic, trigonometric or exponential) and enter corresponding data. We also represent some basic knowledge of complex numbers.

Suppose we have complex number

z=x+i∙y

To find its n-th power (n∈Z), one need to calculate expression:

z^{n}=(x+i∙y)^{n}

To accomplish this task, one can use short multiplication formulas. For example,

z^{2}=(x+i∙y)^{2}=
x^{2}+2∙x∙i∙y+(i∙y)^{2}=
x^{2}+2∙x∙y∙i-y^{2}=
x^{2}+y^{2}+2∙x∙y∙i

However, for high values of n it would be simplier to use de Moivre's formula:

z^{n}=r^{n}∙
(cos(n∙φ)+i∙
sin(n∙φ))

As expected, to use this formula, complex number should be given in the trigonometric form.

Finally, the simplest way to find n-th power of complex number is to use exponential representation:

z=r∙
e^{i∙φ}

Then:

z^{n}=r^{n}∙
e^{i∙n∙φ}