Math Help - Irrational Numbers Question

1. 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?

2. 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.

3. Re: Irrational Numbers Question

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

4. 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.