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 Now, either A is rational or A is irrational. If A is rational, we are done. Why?
(b) If A is irrational, we are done. Why?
Hint: Consider
Remark: For more about this problem and related questions, see the article “Irrational Numbers,” by J. P. Jones and S. Toporowski in American Mathematical Monthly, vol. 80 (1973), pp. 423–424.
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