I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$.
However, I do not understand why if I substitute $-5$ in for $x$ that I can't do the following using rules of exponents:
$\left|-5\right| = \sqrt{(-5)^2} = \left[(-5)^2\right]^\frac 12 = (-5)^\frac 22 = (-5)^\frac 11 = -5$.
Clearly we know that the end result isn't right, but I can't find any logical problems with my reasoning. Can someone shed some light onto the situation for me?
Thanks!
Best Answer
The rule you're using: $$(a^b)^c = a^{bc} $$ applies for all $b, c \in \mathbb R$ if and only if $a \geq 0$. If $b, c \in \mathbb Z$, then it also holds for $a \lt 0$.
So $$((-5)^2)^{\large\frac 12} = (25)^{1/2} = 5$$