TITLE: How does the exponent of a function effect the result?
QUESTION [25 upvotes]: The $x^{2/2}$ can be represented by these ways: 
$$\begin{align}
x^{2\over2}=\sqrt{x^2} = |x|\\
\end{align}
$$
And
$$\begin{align}
x^{2\over2}=x^{1} = x\\
\end{align}
$$
Which one is correct?
And what is the domain of $x^{2 \over 2}$?

REPLY [3 votes]: The discussion on this question has been an education to me, as it is clear that a fair amount of "sloppiness" has crept into the typical presentation of fractional exponents in introductory courses or general practice.  I, for one, was shown a long time ago that $ \ \sqrt{x^2} \ = \ |x| \ . $  [In fact, in the early days of computer languages before "library subroutines" such as ABS(X) were available, a standard way of obtaining absolute value was to write SQRT(X*X).]
I realized a while after writing my comments (far too late at night) that I was not even clear about how such exponents are read.  The way I have presented this notation, which is somewhat cumbersome, is:
As a function for real numbers, if we are to have the interpretation $ \ x^{p/q} \ = \ \sqrt[q]{x^p} \ = \ (\sqrt[q]{x})^p \ $ , then, with integers $ \ p \ \ne \ 0 \ , \ q \ > \ 0 \ ,  $ and the ratio $ \ \frac{p}{q} \ $ not to be reduced,
with $ \ p \ > \ 0 \ , $
for $ \ q \ $ odd:  domain and range are $ \ \mathbb{R} \ , $
for $ \ q \ $ even: domain is $ \ x \ \ge \ 0 \ , $ range is $ \ y \ \ge \ 0 \ ; $
with $ \ p \ < \ 0 \ , $
for $ \ q \ $ odd:  domain and range are $ \ \mathbb{R}  - \{0\} \ , $
for $ \ q \ $ even: domain is $ \ x \ > \ 0 \ , $ range is $ \ y \ > \ 0 \ . $
So saying, it is now clear that it is not consistent for me to have said that $ \ x^{2/2} \ = \ \sqrt{x^2} \ = \ |x| \ . $ $  ^* \ $ (So I will stop going around saying that...)  It should rather be said that the absolute value issue simply does not arise because writing $ \ x^{2/2} \ $ requires that $ \ x \ $ be non-negative.  So the complete statement would be more like $ \ x^{2/2} \ = \ x \ , $ for $ \ x \ \ge \ 0 \ $ only (and not defined for $ \ x \ < \ 0 \ $).  One should not "reduce" the fraction, as that changes the intent of the notation.
$ ^* \ $ What I should be saying to students is: the "square-root" operation does give $ \sqrt{x^2} \ = \ |x| \  $  , but this is not the same as $ \ x^{2/2} \ . $
I note in reading this thread that mathematicians apparently avoid the ambiguity entirely by restricting the domain to $ \ x \ > \ 0 \ . $  It is clear that writing fractional exponents should be done advisedly.
EDIT -- In light of the excellent points raised, I see that there is a certain "friction" between the overly-broad use of fractional exponents and the need for compatibility with the exponential function $ \ a^x \ , \ a > 0 \ . $  The definition I proposed above ends up having too many conditions to be particularly useful, so I see that I should reject it and limit the domain to $ \ x > 0 \ . $  Thank you to everyone for a helpful discusssion!