Proving an irrational to an irrational is rational

In summary, the conversation discusses the possibility of an irrational number raised to another irrational number resulting in a rational number. The conversation includes a suggested approach to proving this, as well as a clarification that it is not necessary to prove it if someone else has already done so.
  • #1
hew
5
0

Homework Statement


prove that it is possible that an irrational number raised to another irrational, can be rational.
you are given root2 to root2 to root2


Homework Equations





The Attempt at a Solution


i have shown that root2 to root2 to root2 is rational, but would appreciate a hint on showing root2 to root2 is irrational
 
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  • #2
Suppose [itex]\sqrt{2}^\sqrt{2}[/itex] is rational. Then you are done!

If it is not rational, then
[tex]\left(\sqrt{2}^\sqrt{2}\right)^\sqrt{2}[/tex]
is again an "irrational to an irrational power".

Now, what is
[tex]\left(\sqrt{2}^\sqrt{2}\right)^\sqrt{2}[/tex]?

Do you see how, even though we don't know whether [itex]\sqrt{2}^\sqrt{2}[/itex] is rational or irrational, either way we have an irrational number to an irrational power that is rational?
 
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  • #3
wow, i thought you somehow had to prove it. Thanks
 
  • #4
Either that or get someone to prove it for you!
 

1. How can you prove that an irrational number multiplied by an irrational number is rational?

To prove that an irrational number multiplied by an irrational number is rational, we can use the proof by contradiction method. This means assuming that the product of two irrational numbers is rational and then showing that this leads to a contradiction.

2. Can you provide an example of two irrational numbers whose product is rational?

Yes, an example of two irrational numbers whose product is rational is √2 and √2. Both of these numbers are irrational, but when multiplied together, they equal 2, which is a rational number.

3. Is it possible for an irrational number multiplied by an irrational number to equal an irrational number?

No, it is not possible for an irrational number multiplied by an irrational number to equal an irrational number. This is because multiplying any two irrational numbers will always result in a rational number.

4. How does this concept relate to the fundamental theorem of arithmetic?

This concept relates to the fundamental theorem of arithmetic because it shows that the product of two irrational numbers can always be expressed as a rational number, which follows the idea that all numbers can be broken down into prime factors.

5. Can this proof be extended to other operations, such as division or exponentiation?

Yes, this proof can be extended to other operations such as division or exponentiation. The same concept of using a proof by contradiction can be applied to show that the quotient or power of two irrational numbers will always result in a rational number.

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