# Irrational Numbers Question

• September 25th 2011, 02:16 PM
thamathkid1729
Irrational Numbers Question
Is the set of irrational numbers closed under the operation of exponentiation? That is, for all irrational numbers a and b, is a^b irrational? Why?
• September 25th 2011, 05:26 PM
Deveno
Re: Irrational Numbers Question
i do not know if $\sqrt{2}^{\sqrt{2}}$ is rational, but if it is not, then certainly $(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = (\sqrt{2})^2 = 2$ is.
• September 26th 2011, 12:36 AM
mr fantastic
Re: Irrational Numbers Question
Quote:

Originally Posted by thamathkid1729
Is the set of irrational numbers closed under the operation of exponentiation? That is, for all irrational numbers a and b, is a^b irrational? Why?

http://www.math.sc.edu/~filaseta/gra...h785Notes8.pdf
• September 26th 2011, 12:20 PM
ebaines
Re: Irrational Numbers Question
Consider $e^{\ ln A} = A$. if $A$ is a rational number than $\ln A$ is irrational. So here you have an example of an irrational number raised to an irrational number yielding a rational number result.