Irrational Raised to Irrational

[mathdad]

Precalculus by David Cohen 3rd Edition
Chapter 1, Section 1.1.

Question 66, page 6.

Can an irrational number raised to an irrational power yield an answer that is rational?

Let A = (sqrt{2})^(sqrt{2}).

Now, either A is rational or irrational. If A is rational, we are done. Why? If A is irrational, we are done. Why?

 

[Opalg]

Precalculus by David Cohen 3rd Edition
Chapter 1, Section 1.1.

Question 66, page 6.

Can an irrational number raised to an irrational power yield an answer that is rational?

Let A = (sqrt{2})^(sqrt{2}).

Now, either A is rational or irrational. If A is rational, we are done. Why? If A is irrational, we are done. Why?

I would start this from the other end! Take a rational number, say $4$. Can you express $4$ in the form $4 = a^b$, where $a$ and $b$ are both irrational?

 

[mathdad]

A friend replied by saying the following:

"A useful result here is Gelfond's Theorem:

If a is an algebraic number, and b is an

algebraic irrational number, then a^b is

transcendental."

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Let me see if I can answer your question.

Say a = (√2)^(√2)

Say b = 2√2

a^b = ((√2)^(√2))^(2√2) = (√2)^(√2*2√2) = (√2)^4 = 4

The answer is yes.