Why is $-x = (x^2) ^ {\frac{1}{2}}$ not logically equivalent to $(-x) ^ 2 = ((x^2) ^ {\frac{1}{2}}) ^ 2$?

Question

Why is $-x = (x^2) ^ {\frac{1}{2}}$ not logically equivalent to $(-x) ^ 2 = ((x^2) ^ {\frac{1}{2}}) ^ 2$ for all values of x?

First equation:

$-x = (x^2) ^ {\frac{1}{2}}$

Second equation:

$(-x) ^ 2 = ((x^2) ^ {\frac{1}{2}}) ^ 2$

For example, if I plug x = -1 into both equations, only one of them evaluates to true. I would expect both of them to do so because I performed identical operations on both sides. Why are the two equations not logically equivalent?

Answer 1

Why are the following equations not equivalent?

$$x = a$$

$$x^2 = a^2$$

Can you figure out why one implies the other but not the other way around?

Answer 2

Your first equation does not hold. It would be instead

$-x= -(x^2)^{\frac{1}{2}}$