Question about whether $(x^2)^{0.5} = x$. The Wikipedia page on exponentiation suggests that the following identity holds provided the base $b$ is non-zero:
$$(b^m)^n = b^{mn}$$
Consider the following function:
$$y = (x^2)^{0.5}$$
According to the identity above the following should hold:
$$y = (x^2)^{0.5} = x^1$$
However consider the plots of the two functions:
$$y = x$$
$$y = (x^2)^{0.5}$$
The functions are equal for $x \geq 0$ however for $x < 0$ there is a discrepancy. Could you comment on the use of exponentiation rules? Are there other circumstances that similar discrepancies can be found when applying exponentiation rules?

The identity you stated only holds for $x \geq 0$ (as noted in your wikipeda link) so, for general $x \in \mathbb{R}$ we have $(x^2)^{1/2} = |x| \neq x$ for $x <0$, hence the discrepancy 

A clarification; you have two options:


*

*Either the identity holds for non-zero bases provided the exponents are integers 

*The identity holds for real exponents provided the base is non-negative. 
