Re: irrational numbers

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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.

~~~~~~~~~

If (1) x is rational, Q.E.D.

~~~~~~~~~~

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.

~~~~~~~~~~

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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