MHB Irrational Raised to Irrational

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An irrational number raised to an irrational power can indeed yield a rational result, as demonstrated with the example A = (sqrt{2})^(sqrt{2}). If A is rational, the question is resolved. If A is irrational, raising it to another irrational power, such as 2√2, results in a rational number, specifically 4. This conclusion is supported by Gelfond's Theorem, which states that an algebraic number raised to an algebraic irrational number is transcendental. Thus, the discussion confirms that the initial question can yield a rational outcome.
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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?
 
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RTCNTC said:
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?
 
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."

- - - Updated - - -

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.
 
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