Irrationality

Question:
For irrational numbers x,y show that x^y is not necessarily irrational.

How would I go about this question? I thought about proof by contradiction but don't think it will work because x^y could also be irrational. Do I just find a counter-example?

Quote:

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Originally Posted by Gusbob

Do I just find a counter-example?

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This seems to be a convenient way to go about answering the question.

Solving for x:



Now you just have to show that is irrational.

Quote:

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Originally Posted by Gusbob

Question:
For irrational numbers x,y show that x^y is not necessarily irrational.

How would I go about this question? I thought about proof by contradiction but don't think it will work because x^y could also be irrational. Do I just find a counter-example?

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I'm not entirely sure what you're entitle to assume. Easies example

Statement : For irrational numbers x,y show that x^y is not necessarily irrational.

This statement is equivalent to disproving the statement : For irrational numbers x,y show that x^y is necessarily irrational.

Therefore, one counter-example (such as the one given by Drexel) is enough to disprove the latter, and thus to prove the original statement.