Friday, May 31, 2019

radicals - What rational numbers have rational square roots?

All rational numbers have the fraction form ab, where a and b are integers(b0).



My question is: for what a and b does the fraction have rational square root? The simple answer would be when both are perfect squares, but if two perfect squares are multiplied by a common integer n, the result may not be two perfect squares. Like:49818



And intuitively, without factoring, a=8 and b=18 must qualify by some standard to have a rational square root.




Once this is solved, can this be extended to any degree of roots? Like for what a and b does the fraction have rational nth root?

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