rule for the power of absolute value expressions

Question

Is $|x^n|=|x|^n$

for any rational $n$ and for any real number $x$?

If the above is true, what is the proof?

Answer 1

Definition of Absolute Value \[ |x|=\left\{ \begin{array}{cc} x & : x\ge 0 \\ -x & : x<0 \\ \end{array} \right. \] Multiplicative Identity of Absolute Value \[ |xy|=|x|\cdot|y| \] So if $n\ge 0$, we have $$ |x^n|=|x\cdots x|=|x|\cdots |x|=|x|^{1+\dots +1}=|x|^n $$ However, if $n\lt 0$ and $x\neq 0$, we have $$ |x^n|=\left|\frac{1}{x\cdots x}\right|=\frac{1}{|x\cdots x|}=\frac{1}{|x|\cdots |x|}=\frac{1}{|x|^{-n}}=|x|^n $$