Will irrational exponents always yield irrational results?

Question

Earlier today in my calculus class my professor asked for an estimate value of $4^\sqrt{2}$. Although that is easy if you just say that $\sqrt{2} \simeq 1.41$ and proceed with $4^{1.41} = 4^{\frac{141}{100}} = \sqrt[100]{4^{141}}$ and so on I couldn't stop wondering if, since $\sqrt{2} \in \mathbb{R}\setminus \mathbb{Q}$, does $4^{\sqrt{2}} \in \mathbb{R}\setminus \mathbb{Q}$ as well?

In a more generic sense, is it true that:

$$\forall x\in\mathbb{R},y\in\mathbb{R}\setminus \mathbb{Q} \hspace{5pt} \colon \hspace{5pt} x^y\in\mathbb{R} \setminus \mathbb{Q}$$

How may I prove (or disprove) it?

If you believe $\log 3 / \log 2$ is irrational, then $2^{\log 3 / \log 2 = 3$ is rational. — AJY, Sep 19 '16 at 19:28
At least one of ${\sqrt 2}^{\sqrt 2}$ and $\left({\sqrt 2}^{\sqrt 2}\right)^{\sqrt 2}$ is a counterexample. — Dave L. Renfro, Sep 19 '16 at 19:29
In an informal sense, note that $f(x) = x^b$ for most values of $b$ (and under a typical definition) varies continuously between $f(0) = 0$ and $f(1) = 1$, and thus must pass through a countably infinite number of values $x$ for which $f(x)$ is a rational value. However, this would generally not constitute a proof. — Brian Tung, Sep 19 '16 at 20:32

lhf · Accepted Answer · 2016-09-20T13:48:23.510

5

The comments give you counterexamples, including the classic one mentioned by Dave L. Renfro.

A general theorem in that direction is the Gelfond–Schneider theorem:

If $a$ and $b$ are algebraic numbers with $a \ne 0,1$ and $b$ irrational, then $a^b$ is a transcendental number.

Note that every transcendental number is irrational.

edited Sep 20 '16 at 13:48

answered Sep 19 '16 at 19:30

lhf

216,483

score 2 · Answer 2 · answered Sep 19 '16 at 19:44

The answer is no. For example take $$a^{\log_a(x)}=x$$ Take now $a$ rational such that $\log_a(x)$ don't be rational for $x$ rational.

However the irrational $\log_a(x)$ must be trascendental because if it is algebraic $\ne 0,1$ then x must be trascendental (Gelfond- Schneider theorem. Applying this same theorem we know that $\log_a(x)$ is trascendental for $x$ rational and distinct of $a^r$ with $r$ rational)

Will irrational exponents always yield irrational results?

2 Answers2