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Rational Result? Ridiculous! (Posted on 2004-12-20) Difficulty: 3 of 5
Can an irrational number, raised to an irrational power, give a rational result?

See The Solution Submitted by Old Original Oskar!    
Rating: 2.7500 (4 votes)

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Solution an example--solution | Comment 5 of 19 |

Euler's number, e, was the first to have been proved to be transcendental (thus, all the more so, irrational).

e^(ln(2)) = 2 by definition.

But is ln(2) irrational?

The equation above could be written:

2^(1/ln(2)) = e

If ln(2) were rational, then so would be 1/ln(2).  But then e would be an integral root of an integral power of 2, and thus not transcendental.  So ln(2) is irrational also, and e^(ln(2)) represents an irrational number raised to an irrational power that gives a rational result.

  Posted by Charlie on 2004-12-20 17:11:30
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