Lets assume that $x<0$ , then what is the result of $|x|^4$

Question

Lets assume that $x<0$ , then what is the result of $|x|^4$ ?

I know that if $x$ is negative then its absolute value will be equal to $-x$ . However , when i take its exponential , should i think as $(-x)^4 = x^4$ or $-x^4 = -x^4$ . Can you expain please which one is correct and why .

I thought that $(-x)^4=x^4$ is true but not sure . Thanks in advance

Answer 1

$$|x|^4=(-x)^4=(-1)^4x^4=x^4$$

Note that if $x < 0$, we have $|x|^4 >0$ but $-x^4 < 0$.

You might like to let $x=-1$ to help you understand better.

Answer 2

You should handle the exponential notations with some care though ! $(-x)^n\neq -x^n$ in general. When $x<0$ we have $|x|=-x\Rightarrow |x|^n=(-x)^n$.

Here since $n=4$ is even, the choice of sign doesn't matter much as far as the result is concerned, but that is surely a wrong conception.