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 $\pi /2$ :tongue:

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 $$(ab^2)^3$$?