# Complex number - calculation

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1. Oct 6, 2017

### TheColector

• Member advised to use the homework template for posts in the homework sections of PF.
Hi
I was hoping some of you would give me a clue on how to solve this complex number task.
z = (1 +(√3 /2) + i/2)^24 → x=(1 +(√3 /2), y= 1/2
I think there must be some nice looking way to solve it.
My way was to calculate |z| which was equal to [√(3+2√3)]/2 → cosθ = x/|z|, sinθ= y/|z|
After using De Moivre's formula I got very awful result which was:
z = |z|^24 * (cos(24*θ) + i sin(24*θ))
Can you think of any better looking way to solve this ?

2. Oct 6, 2017

### haruspex

Have you tried simply squaring three times and cubing once?

3. Oct 6, 2017

### Ray Vickson

Work hard at determining $\theta$ as accurately as you can, because the result may surprise you. (Don't just give one or two decimal places; use as many places as is practical, or---even better----try for an "analytic", closed-form expression.)

4. Oct 6, 2017

### Staff: Mentor

It's not clear to me what you're trying to do.
It appears that you want to write z in the form of x + iy. If so, x and y would not be as you show them above.
In future posts, don't delete the homework template. Its use is required.

5. Oct 7, 2017

### haruspex

You can gain a lot of insight into the geometry by either squaring or cubing the expression then comparing the real and imaginary parts.

6. Oct 7, 2017

### TheColector

Sorry about deleting it. I won't do so in the future. What I meant with this → was to show the represenstative form of x and iy as a part of "z"

7. Oct 7, 2017

### kuruman

Better yet, consider the quantity that you raise to the 24th power, $z_1=\frac{\sqrt{3}}{2}+\frac{1}{2}i$. Can you find angle $\varphi$ such that $z_1=e^{i \varphi}$ ? Then raise to the 24th power.

8. Oct 8, 2017

### haruspex

Isn't that effectively what was tried in post #1? And it is 1+½√3+½i.
Looks like TheColector had trouble finding the angle. This is a lot easier after a single squaring or cubing.

9. Oct 8, 2017

### kuruman

Sorry, I misread the parentheses. This makes the problem more interesting.
$$1+\frac{\sqrt{3}}{2}+ \frac{1}{2}i=e^{2i\pi}+e^{i\pi/6}$$
Can you write this as the product of two terms?

(Edited to give less away)

Last edited: Oct 8, 2017