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 page you refer to actually says the following.  (The emphasis below is mine.)

The following identities hold for all integer exponents, provided that the base is non-zero

$0.5$ is not an integer, so the property does not apply.
But as you've seen, $(x^2)^{0.5} = x$ is true for $x \ge 0$.  The reason it fails for $x < 0$ is because raising to the exponent $0.5$ is the same as taking the square root (as long as we're working only with real numbers, which I'm assuming is the case here).  That is, $y^{0.5} = \sqrt{y}$.  And when we simply say $\sqrt y$, it is understood that we are talking about the positive square root.  For example, $\sqrt9 = 3$, not $-3$.
In general when we take the square root of a square, we have $\sqrt{x^2} = |x|$.  For example, if $x = -4$ then we have $$x^{0.5} = ((-4)^2)^{0.5} = (16)^{0.5} = 4 \ne x.$$
