Rational number to the power of irrational number = irrational number. True?

I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question.

Now, his inital solution was like this: let's take a rational number $r$ and an irrational number $i$. Let's assume

$$a = r^i$$ $$b = \frac{1}{i}$$

So we have

$$a^b = (r^i)^\frac{1}{i} = r$$

which is rational per initial supposition. $b$ is obviously irrational if $i$ is. My friend says that it is also obvious that if $r$ is rational and $i$ is irrational, then $r^i$ is irrational. I quickly objected saying that $r = 1$ is an easy counterexample. To which my friend said, OK, for any positive rational number $r$, other than 1 and for any irrational number $i$ $r^i$ is irrational. Is this true? If so, is it easily proved? If not, can someone come up with a counterexample?

Let's stick to real numbers only (i.e. let's forget about complex numbers for now).

• But of course, a cardinality argument shows that for $0 < r \neq 1$, $r^i$ is irrational for most (all but countably many) irrational $i$. So the example isn't irrepairably wrong. – Daniel Fischer Jul 22 '13 at 13:04
• Can anyone explain why math people say "prove that there exist irrational numbers a and b..." instead of the more intuitive "prove that irrational numbers a and b exist...". The second approach makes more sense to me. – goodguys_activate Jul 22 '13 at 16:38
• @makerofthings7: comments to other questions are not the way to ask new questions. Use the "ask question" link. – Chris Eagle Jul 22 '13 at 17:17
• @makerofthings7 Might be because $\exists$ is a prefix operator, and then the habit carries over into one's language. – Jack M Jul 22 '13 at 17:18
• @makerofthings7 It's probably a combination of many things: (a) you speak the way people around you speak, in mathematics or anywhere else - this is just a kind of jargon, (b) there are lots of non-native English speakers reading and writing English papers, so clarity and consistency is very important, (c) "prove that x exists such that..." sounds very sloppy to me, because "such" modifies "x" ("prove that such an x exists that..."??), and in any case it might mislead you into thinking you were given a formula or algorithm for x earlier on, and are being asked to check it makes sense. – Billy Jul 22 '13 at 18:55

Consider $2^{\log_2 3}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$

• Ah, of course! So obvious now that you've said it! – Armen Tsirunyan Jul 22 '13 at 12:46
• I suppose the obvious followup question is how to prove that $\log_2 3$ is not rational. – Ilmari Karonen Jul 22 '13 at 17:00
• @IlmariKaronen : if $2^{p/q}=3$, then $2^p=3^q$. – user10676 Jul 22 '13 at 17:24
• Gotta be one of the shortest best answers. – LarsH Jul 22 '13 at 18:50
• And by the Fundamental theorem of arithmetic... – Malvolio Feb 15 '14 at 0:50

A supplement to the answer by Chris above:

Let $r$ be a positive rational number and $i$ a positive irrational number. If $r^{i}$ is rational, then $r^i=\frac{a}{b}$ for $a,b\in \mathbb{Z}$ such that $b\neq 0$. In particular, $i=\log_{r}\left(\frac{a}{b}\right)$. Therefore, Chris Eagle's answer is, in fact, prototypical. (Note also, that if $r=1$, then we get a contradiction as you observed.)

• Do you know an elementary proof of this, please? – Alex M. Sep 23 '16 at 21:26