taking the absolute value of complex numbers to an arbitrary power

Question

I need $|\frac{i^{n}}{n}|$ and I have seen the problem simplified to $\frac{|i^{n}|}{n}$ and I am confused by this as isn't $\frac{1}{n}$ the coefficient of i so we could just square it and take the square root to find the absolute value. Further I do then not understand how $|i^{n}| = 1$

Answer 1

I am not quite sure I get your question. What do you mean by "isn't $\frac{1}{n}$ the reciprocal of i? In general, it holds that: $\vert z_1z_2 \vert = \vert z_1\vert\vert z_2\vert$ and $\vert z_1^{n} \vert = \vert z_1 \vert^{n}$. If $z_1 = \frac{1}{n}, z_2=i^{n}$ and use of property 1 followed by property 2 gives: $ \frac{|i|^{n}}{|n|}$, and I take it that n is non-negative so you might remove the modulus from the denominator.