Why does $\sqrt{x^2}$ seem to equal $x$ and not $|x|$ when you multiply the exponents? I understand that $\sqrt{x^2} = |x|$ because the principal square root is positive.
But since $\sqrt x = x^{\frac{1}{2}}$ shouldn't $\sqrt{x^2} = (x^2)^{\frac{1}{2}} = x^{\frac{2}{2}} = x$ because of the exponents multiplying together?
Also, doesn't $(\sqrt{x})^2$ preserve the sign of $x$? But shouldn't $(\sqrt{x})^2 = (\sqrt{x})(\sqrt{x}) = \sqrt{x^2}$?
How do I reconcile all this? What rules am I not aware of?
Edit: Since someone voted to close my question, I should probably explain the difference between my question and Proving square root of a square is the same as absolute value, regardless of how much I think the difference should be obvious to anyone who reads the questions. Cole Johnson was asking if there's any way to prove that $\sqrt{x^2} = |x|$. I am not asking that; I already accept the equation as fact. I'm asking how to resolve some apparent contradictions that arise when considering square roots of squares, and how I should approach these types of problems. (Cameron, please read.)

Consider the following:$$1^{\frac1n}$$Look up roots of unity.
The nth root of a number produces n different answers, creating a regular n-sided shape with a "radius" of 1 on the complex plane.
But what does this mean?  Well, you could consider this...
The nth root of a positive number in general.
This removes all of the other possible answers, leaving us with only one answer.  Among the answers removed was the original answer, but because real-life math usually can ignore this, we're all good.
Now signs, as you may imagine, have been messed up.  We, as a general society, have decided that it be positive.  This has made math much easier for younger pupils still learning math.  And as we take more roots, we pretty much lose signs.  We end up with $i$'s, plusses, and minuses.  It is not that you have messed up anywhere, it was just a simple decision made to make the square root function positive.
In many maths, you will find that we will say the n-th root of a number does indeed produce n amount of answers, but for lower level math, we stick with the positive answer, or the result of no positive answer.
I mean, think about it, what does $\sqrt1=-1,1$ do for us in real life?  Not much, as it turns out.  When we need both the positive and negative answers, it is known that that is what we are finding.  But it doesn't have much application to real life.
So the rule is that square roots are positive, and just keep a spot in the back of your mind remembering that this is not the only answer.
