Positive rational raised to positive irrational is always irrational?

Question

I've been puzzling over this question about number theory.

Ignoring complex numbers, and working only with positive numbers:

Can one say that a positive rational number raised to a positive irrational exponent is always irrational?

Thank you, Hein

Servaes · Accepted Answer · 2018-03-04 18:20:50Z

3

No. For example $x:=\log_2(3)$ is irrational but $2^x=3$.

answered Mar 4, 2018 at 18:20

Servaes

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1

$\begingroup$ Thanks so much, much appreciated $\endgroup$
– hein
Commented Mar 4, 2018 at 18:20
3

$\begingroup$ And the reason it's irrational: if we had $\log_2 3 = \frac{p}{q}$ with $p, q$ positive integers, then we would have $2^p = 3^q$, contradicting the fundamental theorem of arithmetic (unique factorization into primes). $\endgroup$
– Daniel Schepler
Commented Mar 4, 2018 at 18:23
$\begingroup$ Thanks @DanielSchepler $\endgroup$
– hein
Commented Mar 4, 2018 at 18:26

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