Posted by Soroban on October 08, 2002 at 01:45:15:
In Reply to: irrational numbers posted by loser who needs help on October 07, 2002 at 23:06:05:
: can an irrational number to an irrational exponent ever be rational? disprove or prove by example...
Hello!
If you're calling yourself a loser because of this problem,
don't beat yourself up over it. It's quite an advanced problem.
There is a fiendish clever proof for: "An irrational number
raised to an irrational power CAN be rational."
I will use R2 to mean "square root of 2".
Let x = (R2)^(R2), an irrational number raised to an irrational power.
There are only two possibilities for x:
(1) x is rational, or (2) x is irrational.
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If (1) x is rational, Q.E.D.
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If (2) x is irrational, raise the equation to the power R2.
We will have: x^(R2) = {(R2)^R2)^R2 = (R2)^2 = 2, a rational number.
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You may not realize it, but our proof is complete.
Look at the left member: x^(R2).
We agreed in (2) that x is irrational.
Hence, x^(R2) is an "irrational to an irrational power".
And we have just shown that it equal 2, a rational number.
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