Question

A rational number raised to a rational power can either be rational or irrational. For example, 4^( 1/ 2) is rational because 4 ^1 /2 = 2, and 2^ 1 /4 is irrational because 2^ 1/ 4 = √2/ 4 .

In this problem, you will investigate the possibilities for an irrational number raised to an irrational powerCan the value of an irrational number raised to an irrational power ever be rational?

Answer 1

Step-bThis is an interesting problem. To solve it I should find two irrational numbers r and s such that rs is rational.

I am not sure I am able to do that. However, I am confident that the following argument does solve the problem.

As we know, √2 is irrational. In particular, √2 is real and also positive. Then √2√2 is also real. Which means that it is either rational or irrational.

If it's rational, the problem is solved with r = √2 and s = √2.

Assume √2√2 is irrational. Let r = √2√2 and s = √2. Then rs = (√2√2)√2 = √2√22 = √22 = 2. Which is clearly rational.

Either way, we have a pair of irrational numbers r and s such that rs is rational. Or do we? If we do, which is that?

(There's an interesting related problem.)

Chris Reineke came up with an additional example. This one is more direct and constructive. Both log(4) and √10 are irrational. However

√10 log(4) = 10log(2) = 2.y-step explanation: