Let $a$ and $b$ be rational numbers, such that $\sqrt{a}$ and $\sqrt{b}$ are irrational.

Can $\sqrt{a}^\sqrt{b}$ be rational?

I found examples, where the irrational power of an irrational number is rational, but in those examples at least one of those numbers (base and exponent) has not been a square root of a rational.


1 Answer 1


Since ${\sqrt{a}}^{\sqrt{b}}$ is expressed as an algebraic number not equal to $0$ or $1$ raised to an irrational algebraic power, the result will be transcendental (and hence irrational) by the Gelfond–Schneider theorem.


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