Complex powers

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Complex powers

So, what happens when we multiply with the same complex number over and over again? We have that

\displaystyle z=a+bi=p\left( {\cos \alpha +i\sin \alpha } \right)

That means that

\displaystyle {{z}^{n}}={{p}^{n}}{{\left( {\cos \alpha +i\sin \alpha } \right)}^{n}}

But when multiplying numbers we add their arguments. That means that

\displaystyle {{z}^{n}}={{p}^{n}}{{\left( {\cos \alpha +i\sin \alpha } \right)}^{n}}={{p}^{n}}\left( {\cos \left( {n\alpha } \right)+i\sin \left( {n\alpha } \right)} \right)

This will give us a practical way of getting higher powers of a complex number without having to multiply over and over again.

You can test this on this page:

You can grab and move the green point marked z.

Up a level : Complex numbers
Previous page : Complex multiplication
Next page : Complex rootsLast modified: Nov 11, 2025 @ 12:16