# Question about power of square root

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!

• Welcome to MSE, please, learn how to use mathjax to type your questions – jjagmath Dec 28 '20 at 10:38

## 2 Answers

Hint: $$x^{13}=x^{12} x = (x^2)^6 x$$

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}

• @Forester I've edited my answer. Thank you. – José Carlos Santos Dec 28 '20 at 10:38