Prove that there exist irrational numbers r and s, such that r^s is rational. Do not use the Gelfond and Schneider Theorem, however, you may g

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Prove that there exist irrational numbers r and s, such that r^s is rational. Do not use the Gelfond and Schneider Theorem, however, you may give irrational numbers without proving that they are irrational.

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Step 1

Let x=22  and y=2 

we know y is irrational but it is not clear whether x is rational or irrational. On one hand if x is rational then we have an irrational number to an irrational power that is rational:

xy=222=22=2

On the other hand if 22 is rational, then let x=2  and y=2

xy=22

which is rational (According to our assumption).

This is example of Non-constructive proof.

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