Let a and b be rational numbers, such that √a and √b are irrational.
Can √a√b 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 √a√b 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.
No comments:
Post a Comment