I'm confused as to why my calculator is saying that $$x^\frac{n}{n}=x$$ when $n$ is an even integer. I was under the impression that $$x^\frac{n}{n}=\sqrt[n]x^n=|x|$$ when $n$ is an even integer. Thanks for any clarification.
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$\begingroup$The basic problem is that the "laws of exponents" such as $(x^a)^{1/b} = x^{a/b} = (x^{1/b})^a$ work for positive $x$ but not always for negative $x$. Things get even more interesting when complex numbers are involved.
I would have to insist that when you write $x^{n/n}$ it has the meaning $x^{(n/n)}$, and since $n/n = 1$ (at least for $n \ne 0$) this is equal to $x^1 = x$. Most computer algebra packages take the same view, indeed in a text-based interface you generally can't enter this without the parentheses:
x^(n/n)If you want $(x^{n})^{1/n}$ or $(x^{1/n})^n$, you put in parentheses accordingly.
$\endgroup$ 2 $\begingroup$The calculator is correct. $\frac {n}{n} = 1$ for all $n \neq 0$. Any argument to the contrary must address that.
$\endgroup$ $\begingroup$$x^{(\frac nn)} =x^1 = x$
$(x^{n})^{\frac 1n} = |x|$
Note the difference... brackets affect a lot
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