Using Euler's Formula to Evaluate Complex Numbers in Rectangular Form

[ezperkins]

"Use Euler's formula to evaluate the following and write your answer in rectangular form."
A. (2i)⁵
B. (1+i)^-.5

I referred to my precal book and various websites and am still clueless. I started to work out A. but I'm not sure of anything. Here's what I did:

(2i)⁵ = 32i

On the imaginary/real plane, that forms a 90 degree angle.

\theta = \frac{\pi}{4}

e^{i \theta } = cos \theta + isin \theta

cos \frac{\pi}{4} = 0 & isin\frac{\pi}{4} = i

e^ {\frac{i\pi}{4}} = i

e^ {\frac{\pi}{4}} = ? . . .

Whenever I don't know what I'm doing, I just mimic, and I feel like I'm mimicking incorrectly.
I would really like to know how to do this but can't figure it out on my own. Thanks in advance :)

 

[Mentallic]

Well, firstly, a 90^o angle isn't \pi /4, it's \pi /2

If e^{\frac{i\pi}{2}}=i

then 2e^{\frac{i\pi}{2}}=2i

and \left(2e^{\frac{i\pi}{2}}\right)^5=(2i)^5

Can you take it from here?

 

[ezperkins]

haha thanks, I have a habit of making dumb mistakes like that.

I'm working it a few different ways but keep winding up with:

\cos \theta + i \sin \theta = i

I've flown through all of the other problems on this stupid homework, but I've been working on this problem for about two hours and I still don't know what to do.

 

[Mentallic]

You're looking at the problem in entirely the wrong way.

You need to simplify:

\left(2e^{\frac{i\pi}{2}}\right)^5

Do it like you would any other real number. What is (ab^2)^3?