Complex number and exponents

Hello everyone,

I need to perform the following operation: (1+2i)^214. Can I just follow the same guideline as if I didn't have a real part, ie (1+2i)^((53x4)+2) = (1+2i)^2 = -3+4i, or does this little trick only apply when the exponent is attached only to the imaginary part?

If I cannot perform the operation this way, how should I proceed about it?

Many thanks!

Egoyan

Re: Complex number and exponents

You have to start by expressing the number in exponential or polar form.

Re: Complex number and exponents

Quote:

Originally Posted by

**Egoyan** I need to perform the following operation: (1+2i)^214.

If I cannot perform the operation this way, how should I proceed about it?

If you are not working on an actual application, then this is a pointless problem.

Let $\displaystyle \theta=\arctan\left(2\right)$ then $\displaystyle 1+2i=\sqrt{5}\exp(i\theta)$.

Thus $\displaystyle (1+2i)^{214}=5^{107}\exp(214i\theta)$

Of course that can be reduced: $\displaystyle \mod(214\cdot\theta,2\pi)\sim 4.451969$

Re: Complex number and exponents

It's just a question there is in a math homework I have.

Right, I should have thought about that. I expressed the number in polar form and it was indeed pretty easy from there. Thanks guys.