- #1

#### nycmathguy

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

- Relevant Equations
- A = (sqrt{2})^(sqrt{2})

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

the answer is “yes.” (However, if you study the following solution very carefully, you’ll see that even though we’ve answered the question in the affirmative, we’ve not

pinpointed the specific case in which an irrational number raised to an irrational power is rational.)

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

Now, either A is rational or A is irra-

tional. If A is rational, we are done. Why?

(b) If A is irrational, we are done. Why?

Hint Given: Consider A^(sqrt{2})

the answer is “yes.” (However, if you study the following solution very carefully, you’ll see that even though we’ve answered the question in the affirmative, we’ve not

pinpointed the specific case in which an irrational number raised to an irrational power is rational.)

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

Now, either A is rational or A is irra-

tional. If A is rational, we are done. Why?

(b) If A is irrational, we are done. Why?

Hint Given: Consider A^(sqrt{2})