How Can You Simplify the Calculation of a Complex Number Raised to a Power?

[TheColector]

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 ?

 

[haruspex]

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 ?

Have you tried simply squaring three times and cubing once?

 

[Ray Vickson]

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 ?

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.)

 

[Mark44]

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

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.

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 ?

In future posts, don't delete the homework template. Its use is required.

 

[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.

 

[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"

 

[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.

 

[haruspex]

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.

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.

 

[kuruman]

And it is 1+½√3+½i.

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)