I'm currently working on a problem involving complex numbers and exponentiation, and I'm having some trouble finding the correct answer. I'm hoping that someone here can give me some guidance on how to approach this problem.
The problem states that if $z=(1+i\sqrt{3})^n$, where $n$ is a positive integer, what is the value of $|z|$? The answer choices are:
(a) $2^{-n}$
(b) 2
(c) $2^n$
(d) none of the above
Here's what I've tried so far: I rewrote $1+i\sqrt{3}$ in trigonometric form as $2e^{i\pi/3}$, and then raised it to the $n$th power using De Moivre's formula. This gave me $z = 2^ne^{in\pi/3}$. Then, I tried to find the modulus of $z$ by taking the absolute value of both sides:
$|z| = |2^ne^{in\pi/3}| = |2^n|\cdot|e^{in\pi/3}| = 2^n\cdot|e^{in\pi/3}|$
However, I'm not sure where to go from here. Can anyone help me figure out the final answer? I'd really appreciate any advice or suggestions you have.
Thank you in advance!
You can do $$ \left|\left(1+\sqrt3i\right)^n\right|=\left|1+\sqrt3i\right|^n=2^n. $$ So, the correct option is (c).
If you insist in using the exponential function, you can use the fact that $$ \left|e^{in\pi/3}\right|=e^{\operatorname{Re}\left(in\pi/3\right)}=e^0=1. $$