1a) Prove that there exist irrational numbers a and b so that a^b is rational. look at the numbers √2^√2 and (√2^√2)^√2

Question

1b) use similar idea to prove that there exists a rational number a and an irrational number b so that a^b is irrational

1c) prove that if 2^√2is irrational then √2^√2is irrational.

Thanks for your help.

Answer 1

(√2^√2)^√2 = √2^(√2*√2) = √2^2 = 2

√2^√2 = (2^1/2)^√2 = 2^(1/√2)

That should get you started