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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?

Thomas Andrews
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John Daley
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3 Answers3

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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. :)

Thomas Andrews
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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:

$$ \sqrt[n]{x^n}=\begin{cases} x & \text{if $n$ is odd} \\ |x| & \text{if $n$ is even} \\ \end{cases}$$

Adriano
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  • but $ x^{\frac{3}{3}}=x=x^{\frac{23}{23}}=\left | x \right |$ !! – mnsh Aug 08 '13 at 04:33
  • Again, you have to be careful with parentheses. Ordinarily $x^{\frac{a}{b}}$ would be interpreted as $x^{(a/b)}$, i.e. first you take $a/b$, then $x$ to that power. So if $a = b$ (and, of course, $b \ne 0$), $a/b = 1$ and $x^{\frac{a}{b}} = x$. In particular, $x^{\frac{2 \cdot 3}{3 \cdot 3}} = x$, not $|x|$. $(x^{2 \cdot 3})^{1/(2\cdot 3)}$ would be $|x|$, but that's not the same thing. – Robert Israel Aug 08 '13 at 04:47
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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.

zcc
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