Question regarding the square root of a squared number. I've learnt that the square root of a number squared is equal to the absolute value of that number, but I haven't really understood why. I have looked through other questions on MSE but didn't really find a good answer.
As an example: for me there are two ways to arrive at a solution for $\sqrt{(-5)^2}$
First: $\sqrt{(-5)^2}= $ $\sqrt{25}=5$
Second: $\sqrt{(-5)^2}= $ $\ (-5)^{\frac{2}{2}}$$\ =(-5)^1=-5$
but according to the $\sqrt{x^2} = | x | $  rule 5 is the only solution.
What is the flaw in my logic in getting -5 as a solution. I would really appreciate a comprehensive explanation that clears this up. Thanks in advance.
 A: We define $\sqrt k$ to be the positive square root of $k$. This means that taking square roots is a valid function on the positive reals - if we allowed it to have multiple values, then it wouldn't be a function. 
This is different to the solutions of the equation $x^2=k$. Here we are looking for all real numbers whose square is $k$, so we must also consider the negative square root. 
In you second case, the flaw in the argument is that you are implicitly taking the square root of a negative number - you are saying that $$\sqrt{(-5)^2}=\sqrt{-5}^2$$which gets messy, as taking the square root of a negative number, even if we allow complex numbers, requires a choice of root, and the rules of indices don't apply in the complex case in the same way they do in the real case. 
A: You can think this in this way.
Let $$f: \mathbb{R}\rightarrow\mathbb{R}^+$$ 
$$x\mapsto x^2$$
where $x\in\mathbb{R}$
and $$g:\mathbb{R}^+\rightarrow\mathbb{R}^+$$
$$y\mapsto \sqrt{y}$$
where  $y\in\mathbb{R}^+$
So to solve this problem, we must follow this order of arithmetic just like your first illustration and your second should be $\sqrt{(-5)^2}=[(-5)^2]^{\frac{1}{2}}$ 
A: How do you know if $25$ is $5^2$ or $(-5)^2$ ? Numbers have no "memory".
