Answered: Prove that there exist irrational…

Name: Prove that there exist irrational numbers r and s, such that r^s is ra
Uploaded: 2024-03-20T06:38:44.000Z
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Description: 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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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.

Expert Solution

Step 1

Let $x = {\sqrt{2}}^{\sqrt{2}}$ and $y = \sqrt{2}$

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:

$x^{y} = {({\sqrt{2}}^{\sqrt{2}})}^{\sqrt{2}} = {(\sqrt{2})}^{2} = 2$

On the other hand if ${\sqrt{2}}^{\sqrt{2}}$ is rational $,$ then let $x = \sqrt{2} a n d y = \sqrt{2}$

$x^{y} = {\sqrt{2}}^{\sqrt{2}}$

which is rational $($ According to our assumption $)$ .

This is example of Non-constructive proof.

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