Wednesday, January 13, 2016

number theory - Can sqrtasqrtb be rational if sqrta and sqrtb are irrational?




Let a and b be rational numbers, such that a and b are irrational.




Can ab be rational?




I found examples, where the irrational power of an irrational number is rational, but in those examples at least one of those numbers (base and exponent) has not been a square root of a rational.


Answer



Since ab is expressed as an algebraic number not equal to 0 or 1 raised to an irrational algebraic power, the result will be transcendental (and hence irrational) by the Gelfond–Schneider theorem.


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