Do there exist irrational numbers a and b such that a raised to the power b (aᵇ) is a rational number?
Yes
Consider the number A = √2√2.
If A is rational — great, problem solved: both √2 and √2√2 are irrational and rational respectively.
If A is irrational, then
A√2 = √2^√2^√2 = √2² = 2,
which is rational.
In both cases, we have irrational numbers a and b such that aᵇ is rational.
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