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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.

J. M. ain't a mathematician
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David Lewis
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    what do you mean? whats wrong with $x^{m/n}$ with $m/n$ in lowest terms? – yoyo Mar 30 '11 at 00:17
  • 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. – David Lewis Mar 30 '11 at 00:24

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What's "simplest" is highly context dependent - context which you have not supplied. However, one important aspect of simplicity is computational effectivity. In order to have any hope of effectively computing with such radicals it is essential to normalize them so that any algebraic relationships are made explicit. For example, suppose that you are computing with expressions involving $\:5\:$ raised to the powers $\ 1/6,\ 1/10,\:$ and $\:1/15\:.\:$ Then such expressions can all be expressed as polynomials in the single radical $\:5^{1/30}\:,\:$ e.g. $\rm\:5^{1/6} = (5^{1/30})^5\:.\:$ Said in the language of field theory $\rm\ \mathbb Q(5^{1/6},\:5^{1/10},\:5^{1/15}) \subset \mathbb Q(5^{1/30})\:.\:$ Therefore arithmetic operations on such radical expressions reduce to simple efficient polynomial operations, viz. operations in the ring $\rm\ Q[x]\ (mod\ x^{30}-5)\:.\:$

Such normalization is even more crucial when working with multiple radicands. For example $\ \sqrt{6}\ \sqrt{10}\ \sqrt{15}\ =\ 30\:,\:$ so this algebraic dependence needs to be eliminated by choosing two of the three radicands as a basis, say $\:\sqrt{6},\ \sqrt{10}\:,\ $ then eliminating $\:\sqrt{15}\ $ using $\rm\ \sqrt{15}\ =\ \sqrt{6}\ \sqrt{10}/2\:.\:$ Developing effective algorithms for computing with such radicands (e.g. denesting) is a highly nontrivial task, based on the Galois theory of radical extensions (Kummer theory). This is the theory at the basis of such effective algorithms implemented in computer algebra systems.

Bill Dubuque
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  • Well, I did kinda state a context -- it's a teaching question, by which I meant high school algebra 2. In that context, simplest form rules seem kind of arbitrary, motivated as much by driving algebraic mastery as any mathematical basis. For example, the old rule about no radicals in the denominator sees, as far as I can determine, to come from avoiding long decimals in the divisor when doing long division in the pre-calculator era. So while I appreciate hearing about the math behind normal forms, etc, I have a more modest goal. – David Lewis Mar 30 '11 at 12:39
  • One of the advantages to simplifying a radical expression is that two expressions are known to be or not be equal once they are both simplified completely. For high school education, this reason should suffice since multiple choice questions are popular in the US. The choice between writing your expressions with positive or negative exponents does seem to be a matter of taste though. – Joshua Shane Liberman Mar 30 '11 at 14:52
  • @David: It's not clear what "rules" you refer to (besides the laws of exponents). As I said, the driving force here for rule-based computation are certain normal forms that facilitate effective computation in radical extension fields. Just as in your prior question about rationalizing denominators, these normal forms reduce complexity - here by eliminating algebraic dependences among radical monomials. I suspect that any "rules" you may find in high-school textbooks are ad-hoc and probably should be ignored. To properly appreciate these matters requires knowledge of abstract algebra. – Bill Dubuque Mar 30 '11 at 16:28
  • @David: As for your prior question on rationalizing denominators, I hoped that the examples I gave their might serve to convince you that the motivations for such are much deeper than trivial reasons such as "avoiding long decimals in the divisor". I'm disappointed to learn that you were apparently not swayed by these arguments. Please feel free to continue the discussion in comments there if further clarification may prove helpful. – Bill Dubuque Mar 30 '11 at 16:28
  • @Joshua: Of course deciding equality or, equivalently, zero-equivalence, is one component of what I refer to above as effective calculation in radical extensions. This is one of the primary reasons for reducing expressions to tractable normal forms. – Bill Dubuque Mar 30 '11 at 16:37
  • @Bill: I agree with you. I didn't label my comment but it was directed towards David's comment. For completeness, I'd say that the rules that I have seen taught for simplifying a radical expression have four components: 1) No fractions inside the radical. 2) No radicals in the denominator. 3) The base exponent should not be larger than the radical degree. 4) The GCD of the base exponent and the radical degree is 1. – Joshua Shane Liberman Mar 30 '11 at 17:41
  • @Bill -- Don't be disappointed -- I do buy your arguments. I tell students that this is an important skill (improves your algebra), and has important uses in mathematics, some of which you might even see in high school (meaning calculus). As to whether it is really "simpler" in a way they can grasp now, well, obviously not. But do it anyway! And eat your spinach too! – David Lewis Mar 30 '11 at 23:47
  • @Bill -- but please address "simplest form" in the context of rational exponents. OK, I accept the equivalent of radical "simplest" form as ~one~ important canonical form even for rational exponents. But seems to me there is another, more natural form that is in a palpable sense truly "simplest". And that's: factored into primes (including variables) with exponents, one such prime factor per prime, and exponents reduced. That is a true normal form which can facilitate comparison and other results. And, it's truly "simplest" in an elegant sense, a natural generalization of prime factorization. – David Lewis Mar 30 '11 at 23:57
  • @David: Yes, one could completely factor everything as you suggest. But this could lead to combinatorial explosion, turning what might be a polynomial time/space algorithm into an exponential one. In number theory and algebra often it suffices to work with gcd-free bases or similar partially factored objects. Sometimes one can design algorithms so that the factorizations are computed "lazily", only when need be, e.g. assume q is prime, but if something breaks down, e.g. a zero-divisor is discovered mod q, then split q and continue appropriately. – Bill Dubuque Mar 31 '11 at 01:19
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"Lowest terms" for a rational is usually understood to mean $a/b$, with $a$ and $b$ integers, $b\gt 0$, and $\gcd(a,b)=1$. (That is, the "sign" goes with the numerator).

For $p/q$ written in lowest terms, the definition of $a^{p/q}$ is that $a^{p/q} = r$ where $r$ is the unique nonnegative real number such that $r^q = a^p$; that is, $\displaystyle a^{p/q} = \sqrt[q]{a^p}$. This includes the cases with $p\gt q$ and with $-p\gt q$. If you want only fractional exponents of the form $p/q$ with $0\lt p\lt q$, then of course this can be done fairly easily, but it does not seem to me to lead to much simplification for algebraic manipulations (though it may for actual computations).

Rational exponents in lowest terms with even denominator are only, a priori, defined for positive bases when working in the reals.

Arturo Magidin
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