Hey guys. I understand that the set of irrational numbers aren't closed under addition, subtraction, multiplication, and division. However, I was wondering if it is closed under exponentiation. When I Google it, I see unanimously that it's not closed because you can simply square an irrational number such as sqrt(2) to get a rational number. However, I was under the impression that both the base and exponent had to be irrational numbers to test if it was closed. This is how I remember showing whether or not other number sets are closed under a particular operation.

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So, do both the base and exponent need to be irrational numbers to test for closedness?

If not, I understand how irrationals aren't closed, since raising numbers to positive integer powers is just repeated multiplication, which is already known not to be closed.

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If they do have to be, are irrationals closed under exponentiation?