An Irrational Power That Is Rational

Problem Statement

Prove that an irrational number raised to the power of an irrational number can be rational, by considering $\sqrt{2}^{\sqrt{2}}$ . You may assume $\sqrt{2}$ is irrational.

Hints

Consider $\alpha = \sqrt{2}^{\sqrt{2}}$ . By the Law of Excluded Middle, $\alpha$ is either rational or irrational. Examine both cases:

If $\alpha$ is rational, you're done immediately.
If $\alpha$ is irrational, compute $\alpha^{\sqrt{2}}$ and simplify.

This is a non-constructive existence proof---you prove something exists without determining which case actually holds!

Solutions

Proof by Cases:

Let $\alpha = \sqrt{2}^{\sqrt{2}}$ . We consider two exhaustive cases:

Case 1: If $\alpha = \sqrt{2}^{\sqrt{2}}$ is rational, then we have found irrational base $\sqrt{2}$ and irrational exponent $\sqrt{2}$ such that $\sqrt{2}^{\sqrt{2}}$ is rational. Done.

Case 2: If $\alpha = \sqrt{2}^{\sqrt{2}}$ is irrational, consider:

\begin{aligned} \alpha^{\sqrt{2}} &= \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}} \\ &= \sqrt{2}^{\sqrt{2} \cdot \sqrt{2}} \quad \text{(exponent law)} \\ &= \sqrt{2}^{2} \\ &= 2 \end{aligned}

Since $2$ is rational and both $\alpha$ (irrational by case assumption) and $\sqrt{2}$ (given as irrational) are irrational, we have found an example.

Conclusion: In either case, there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. $\blacksquare$

Note: This proof doesn't tell us whether $\sqrt{2}^{\sqrt{2}}$ is actually rational or irrational---and we don't need to know! This is the beauty of non-constructive existence proofs.

Takeaways

Reconstruct the full proof from the hint and the solution outline, and justify every transformation explicitly.
Check edge cases and verify where each assumption is used in the argument.

Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills:

HSC Proofs: https://vumaths.com/booklets/hsc-proofs/
HSC Trigonometry: https://vumaths.com/booklets/hsc-trigonometry/
HSC Differential Equations: https://vumaths.com/booklets/hsc-differential-equations/

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