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A teaching question. What is or might be simplest form for rational exponents? My search turns up little for the concept other than a translation of the simplest form for n-th root radicals. But that leads to ugly things such as...

$\large{5^\frac{7}{3}\rightarrow 5^2\centerdot 5^\frac{1}{3}}$

$\large{x^{-\frac{2}{3}}\rightarrow x^{-1}\centerdot x^\frac{1}{3}}$

...when the whole point of rational exponents is beauty and elegance.

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    what do you mean? whats wrong with $x^{m/n}$ with $m/n$ in lowest terms?2011-03-30
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    I mean, is there a concept that makes sense. Lowest terms for m/n works, as long as you stick to non-negative bases. Seems you should also factor into primes and combine factors with the same base. But beyond that, I can't see the point of mimicking the radical rules. Actually, that is enough to serve as a nice normal form for monomials with monomial factors.2011-03-30

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