TITLE: Question about power of square root
QUESTION [0 upvotes]: I am currently prepping for next year's math course. Currently, I am practicing complex numbers and I have come across something I don't understand.
The problem is how to write the following complex number in rectangular form:
$(1+i)^{13}$
I know the argument(z) is $\dfrac{(13 \cdot \pi)}{4}$
but when trying to find modulus, the solution says it is $\sqrt{2}^{13}$. I follow this so far but I don't get what the rule is for the next step, which is $2^6 \cdot \sqrt(2)$.
What is happening here?
Any help is very much appreciated!

REPLY [2 votes]: Hint: $x^{13}=x^{12} x = (x^2)^6 x$

REPLY [0 votes]: The absolute value is$$\left|(1+i)^{13}\right|=|1+i|^{13}=\sqrt2^{13}$$and\begin{align}\sqrt2^{13}&=\sqrt2^{12}\times\sqrt2\\&=\left(2^{1/2}\right)^{12}\times\sqrt2\\&=2^6\sqrt2.\end{align}
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