Prove that if 'a' is a positive integer and the nth root of 'a' is rational, then the nth root of 'a' must be an integer.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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It is given that a is a positive integer and the nth root of a is rational. A rational number can be expressed in the form p/q where p and q are integers.

a^(1/n) = p/q

=> a = (p/q)^n

If p is not an integral multiple of q, p^n is not an integral multiple of q^n, in which case p^n/q^n cannot be an integer. But we have (p/q)^n = a which is an integer.

Therefore p/q has to be an integer.

This proves that the nth root of a is an integer.

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