# rule for the power of absolute value expressions

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

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

If the above is true, what is the proof?

• If one feels it requires proof (I don't) the result is clear if $x\ge 0$. If $x\lt 0$, then $x=-|x|$, and therefore $|x^n|=|(-1)^n |x|^n|=(|(-1)^n|)|x|^n=|x|^n$. – André Nicolas Sep 26 '14 at 21:29

## 1 Answer

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 $$|x^n|=\left|\frac{1}{x\cdots x}\right|=\frac{1}{|x\cdots x|}=\frac{1}{|x|\cdots |x|}=\frac{1}{|x|^{-n}}=|x|^n$$

• This doesn't seem correct. Let x = -1 and n = 1. Then we have (sqrt(x^2))^1 = x^(2*1/2) which expands to 1 = -1. – Milo Brandt Sep 26 '14 at 21:52
• @Meelo, please see my edit. – k170 Sep 27 '14 at 0:25
• how is abs(x)......abs(x) = (abs(x))^-n ? - for the case n<0 how do you get from the 4th term to the 5th term – zebra1729 Oct 4 '14 at 12:29
• @zebra1729, if $n\lt 0$, then $n$ is negative and $-n$ is a positive number. So the absolute value of $x$ is multiplied $-n$ times. It's a bit tricky because it's counter intuitive to think in negative values. As an exercise, let $n=-3$ and try to follow every step while keeping the negative sign in front of the $3$. – k170 Oct 4 '14 at 12:54