Rational exponent question

Question

Do all rational exponents have whole number base integer derived from the fraction?

$a^b=c$ where $a$ and $c$ are whole numbers and $b$ is fraction

example: $b= 5/4$

How do I get $a$ that gives first whole $c$, please?

Thank you in advance for answer I should have known.

well for $a= 1$, $c$ is a whole number, Do you mean for $a \gt 1$? — The Integrator, Jun 04 '18 at 09:33
I´ve got exponent 5/4 and want to know what a gives result of integer c — , Jun 04 '18 at 09:34
@Yves to the Integrator, but to you as well. First integer c for exponent 5/4 — , Jun 04 '18 at 09:35
I do assume (don´t know whether correctly) all rational exponent have whole number solution (c) in some base (a) — , Jun 04 '18 at 09:37
If i understand you correctly, you want some value of $a$ for which $c$ is a whole number when you are given $b$? — The Integrator, Jun 04 '18 at 09:37
first if there are more solutions, more $a$´s that give integer c — , Jun 04 '18 at 09:37
@Yves yes, the smallest (to say first instead is my language custom) — , Jun 04 '18 at 09:40

score 1 · Accepted Answer · answered Jun 04 '18 at 09:42

1

For $a^{p/q}$ to be an integer, assuming $p/q$ irreducible, $a^{1/q}$ must itself be an integer.

The smallest $a^{1/q}$ being $2$, the answer is

$$a=2^q.$$

For $b=\dfrac54$, $a=16$ and $c=32$.

answered Jun 04 '18 at 09:42

1 Answers1