Intereting Posts

Adjoint functors as “conceptual inverses”
How to understand why $x^0 = 1$, where $x$ is any real number?
If $\lim_{h\to 0} \frac{f(x_0 + h) – f(x_0 – h)}{2h} = f'(x_0)$ exists, is f differentiable at $x_0$?
Is there a slowest divergent function?
Prove that $f(1999)=1999$
Evaluating limits at positive and negative infinity
When does equality hold in the Minkowski's inequality $\|f+g\|_p\leq\|f\|_p+\|g\|_p$?
What is the proper notation for integer polynomials: $\Bbb Y=\{p\in\Bbb Q\mid p:\Bbb Z\to \Bbb Z\}$?
Generalizing Bellard's “exotic” formula for $\pi$ to $m=11$
Determining if a quadratic polynomial is always positive
Sum equals integral
Uniform continuity of $x^3+ \sin x$
Determining the parameters for a spiral tangent to an arc and intersecting a specified point in 2D.
Adriaan van Roomen's 45th degree equation in 1593
Prove $0<a_k\in \mathbb R$ and $\prod\limits_{k=1}^n a_k =1$, then $\prod\limits_{k=1}^n (1+a_k) \ge 2^n$

Can an irrational number raised to an irrational power be rational?

If it can be rational, how can one prove it?

- Prove $\sqrt{2} + \sqrt{5}$ is irrational
- What is a real-world metaphor for irrational numbers?
- General Continued Fractions and Irrationality
- If $(a_n)$ is increasing and $\lim_{n\to\infty}\frac{a_{n+1}}{a_1\dotsb a_n}=+\infty$ then $\sum\limits_{n=1}^\infty\frac1{a_n}$ is irrational
- $45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?
- Irrationality proofs not by contradiction

- How to prove that for all natural numbers, $4^n > n^3$?
- a square root of an irrational number
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- Sum of absolute values and the absolute value of the sum of these values?
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- Critiques on proof showing $\sqrt{12}$ is irrational.
- Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$
- How to prove CYK algorithm has $O(n^3)$ running time

There is a classic example here. Consider $A=\sqrt{2}^\sqrt{2}$. Then A is either rational or irrational. If it is irrational, we have $A^\sqrt{2}=\sqrt{2}^2=2$.

Yes, it can, $$ e^{\log 2} = 2 $$

**Summary of edits**: If $\alpha$ and $\beta$ are *algebraic* and irrational, then

$\alpha^\beta$ is not only irrational but transcendental.

Looking at your other question, it seems worth discussing what happens with square roots, cube roots, algebraic numbers in general. I heartily recommend Irrational Numbers by Ivan Niven, NIVEN.

So, the more precise question is about numbers such as $$ {\sqrt 2}^{\sqrt 2}.$$ For quite a long time the nature of such a number was not known. Also, it is worth pointing out that such expressions have infinitely many values, given by all the possible values of the expression

$$ \alpha^\beta = \exp \, ( \beta \log \alpha ) $$

in $\mathbb C.$ The point is that any specific value of $\log \alpha$ can be altered by $2 \pi i,$ thus altering $\beta \log \alpha$ by $2 \beta \pi i,$ finally altering the chosen interpretation of $\alpha^\beta.$ Of course, if $\alpha$ is real and positive, people use the principal branch of the logarithm, where $\log \alpha$ is also real, so just the one meaning of $\alpha^\beta$ is intended.

Finally, we get to the Gelfond-Schneider theorem, from Niven page 134: If $\alpha$ and $\beta$ are algebraic numbers with $\alpha \neq 0, \; \alpha \neq 1$ and $\beta$ is not a real rational number, then any value of $\alpha^\beta$ is transcendental.

In particular, any value of $$ {\sqrt 2}^{\sqrt 2}$$ is transcendental, including the “principal” and positive real value that a calculator will give you for $\alpha^\beta$ when both $\alpha, \; \beta$ are positive real numbers, defined as usual by $ e^{\beta \log \alpha}$.

There is a detail here that is not often seen. One logarithm of $-1$ is $i \pi,$ this is Euler’s famous formula

$$ e^{i \pi} + 1 = 0.$$ And $\alpha = -1$ ** is** permitted in Gelfond-Schneider. Suppose we have a positive real, and algebraic but irrational $x,$ so we may take $\beta = x.$ Then G-S says that

$$ \alpha^\beta = (-1)^x = \exp \,(x \log (-1)) = \exp (i \pi x) = e^{i \pi x} = \cos \pi x + i \sin \pi x $$ is transcendental. Who knew?

If $r$ is any positive rational other than $1$, then for all but countably many positive reals $x$ both $x$ and $y = \log_x r = \ln(r)/\ln(x)$ are irrational (in fact transcendental), and $x^y = r$.

Consider, for example, $2^{1/\pi}=x$, where $x$ should probably be irrational but $x^\pi=2$. More generally, 2 and $\pi$ can be replaced by other rational and irrational numbers, respectively.

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