Just a couple of small technical point here. If x and n are real numbers, do we have to write $x ^ {n/n} = |x|$? Or can we just reduce it to $x^{n/n} = x$?
One reason I ask is because then we would arrive at $x = x^1 = x ^{n/n} = |x|$. Does this mean, then, that $x$ is not equal to $x^{n/n}$? Or that $x ^ 1$ is not equal to $x^{n/n}$?
Similarly, if I write $(\sqrt x)^2 = x^{2/2} = \sqrt{x^2}$, am I correct? Or does the order matter here?
On
You seem to be confusing $x^{n/n}$ with $(x^n)^{1/n}$. These are not quite equivalent (nor are any of them equivalent to $(x^{1/n})^n$. As others have pointed out, $x^{n/n}=x^1=x$.
More interestingly, consider $(x^n)^{1/n}=\sqrt[n]{x^n}$. Note that if $x\in \Bbb R$ and $n$ is a positive integer, then:
$$ \sqrt[n]{x^n}=\begin{cases} x & \text{if $n$ is odd} \\ |x| & \text{if $n$ is even} \\ \end{cases}$$
On
if we interpret $x^{n/n}$ as $(x^{n})^{1/n}$ or $(x^{1/n})^{n}$, they are different:
The first one contains a multivalued function $$ f(z) = z^{1/q} $$ so that $(x^{n})^{1/n} = \{xe^{2\pi i/n}, i = 0,1,\ldots,n-1\}$
The second one can be simplified to just a single value $x$, since suppose $x = re^{i\theta}$, then $$ x^{1/n} = \{r e^{(2\pi i + \theta)/n}, i = 0,1,\ldots,n-1\} $$ Then the mapping $z \mapsto z^n$ merges then together to a single $x$ again.
You are incorrect.
Once you are dealing with non-integer exponents, you either only define $x^y$ for $x$ positive, or you no longer have the property that $(x^y)^z = x^{yz}$ in general.
The expression $x^y$ gets complicated when $y$ is rational, it gets stranger when $y$ is irrational, and it gets insane when $y$ is a complex number. :)
Finally, the expression $x^{n/n}= x^1$ for any $x$, since $n/n=1$, so we have the order of operations - $n/n$ isn't some representation that is "like" $1$, it is $1$.
If you define $x^1=|x|$, you are losing a lot of rich mathematics. :)