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I need help with algebra of exponentiation.

$$\begin{align*} \sqrt{x^2} &= (x^2)^{1/2} &\qquad &\text{(since }\sqrt[x]{y}=y^{1/x}\text{)}\\ &= x^{2(1/2)} &&\text{(since }(x^y)^z = x^{yz}\text{)}\\ &= x^{(1/2)2} &&\text{(since }xy=yx\text{)}\\ &= (x^{1/2})^2 &&\text{(since }(x^y)^z = x^{yz}\text{)}\\ &= \left(\sqrt{x}\right)^2 &&\text{(since }\sqrt[x]{y} = y^{1/x}\text{)} \end{align*}$$


$\sqrt{x^2}$ is a real number.

$(\sqrt{x})^2$ is a real number, if $x\geq 0$.

$\sqrt{x^2} = (\sqrt{x})^2$.

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    A lengthy calculation! Can you try to frame an explicit question? It would help if you used more words.2011-07-14
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    This is an utter mess. The first line makes absolutely no sense. The equal sign means "these two things are equal", it **does not** mean "and here's something completely different but somewhat related in my head to what came before". $\sqrt{x^2}$ is **not** equal to $\sqrt[x]{y}$. $(x^2)^{1/2}$ is **not** equal to $(x^y)^z$. $x^{2\times(1/2)}$ is **not** equal to $xy=yx$. In short, this is a "stream of consciousness" list of incorrect equalities. **Rule 1:** An equal sign goes only between two things that are equal.2011-07-14
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    @Arturo: This is really unfair. The original text made sense. (at least to me)2011-07-14
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    @Theo: The original text has the exact same incorrect equalities: the first line says "root 2(x^2)", the second line asserts this is equal to "root x y " and to "y^{1/x}". Curly brackets do not indicate comments, they indicate sets; so this makes it even more nonsensical. It's still a "stream of consciousness" list of incorrect equalities.2011-07-14
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    @Arturo: If you look closely enough, Fox is changing the numbers to free variables, writing out the formulas, and then changing the variables back to numbers. Definitely quirky and amateurish the way it's written, but the motivating idea is sound algebra in the end. I think we all agree OP needs to learn how to do it the standard way so that he or she is more efficient and better able to communicate his or her thoughts. EDIT: Theo has apparently edited the question to make apparent Fox's modus here.2011-07-14
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    @Arturo: Here's what I read2011-07-14
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    Edited to put what were meant to be justifications *as* justifications, instead of interspersed with chain of equalities.2011-07-14
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    @Theo: Looks like we engaged in simultaneous edits...2011-07-14
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    @anon: No; $x$ plays multiple roles, and not everything is numbers in between the "justifications". Methinks you are giving too much credit there. In any case, it's not "quirky", it's lousy practice and bad use of notation.2011-07-14
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    @Arturo: your edit is much better than mine so that's fine. However, I think someone was a bit quick to approve the first edit... This "edit gives rep" thing really leads to many people without the skills editing, and we need to pay more attention to retain the meaning of the questions and answers.2011-07-14
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    @Theo: Agreed; that first edit was not very good. Even setting aside other issues, it was very poor LaTeXing.2011-07-15

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