# Is there a rational number #x# such that #sqrt(x)# is irrational, but #sqrt(x)^sqrt(x)# is rational?

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The motivation for this question is to consider: #sqrt(2)^sqrt(2)# and #(sqrt(2)^sqrt(2))^sqrt(2)# .

Note that #sqrt(2)# is irrational, and #(sqrt(2)^sqrt(2))^sqrt(2) = sqrt(2)^2 = 2# .

So either #sqrt(2)^sqrt(2)# is rational and we find that this rational number raised to an irrational power #sqrt(2)# is also rational, or #sqrt(2)^sqrt(2)# is irrational and we find that this irrational number raised to an irrational power is rational.

Going back to the question asked, I think there is no such rational number #x# , but I also suspect that it is difficult to prove.

The motivation for this question is to consider:

Note that

So either

Going back to the question asked, I think there is no such rational number

##### 1 Answer

#### Answer:

No. If

#### Explanation:

The Gelfond-Schneider theorem says that if

A proof of the Gelfond-Schneider theorem can be found at http://people.math.sc.edu/filaseta/gradcourses/Math785/Math785Notes8.pdf

In our example,

Hence with