Nth Root of Integer is Integer or Irrational

 

 

    

Theorem

Let n be a natural number.
Let x be an integer.

If the n th root of x is not an integer, it must be irrational.

Proof

Suppose that x 1/n is not an integer.
Suppose for the sake of contradiction that the n th root of x is rational.
Then by Existence of Canonical Form of Rational Number, there exist an integer a and a natural number b which are coprime such that:

				x 1/n	=		a b
		⟹		x	=		a n b n

Since x is an integer, a n and b n must share a common factor if b≠1 .

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If a n and b n are coprime, and if b≠1 , a n /b n would not be an integer because a n /b n in simplest terms would not have 1 as a denominator.
However, since a and b are coprime, a n and b n are coprime because no new prime factors are introduced.
Thus, b must equal 1.

				x	=		a n
		⟹		x 1/n	=		a

Thus, x 1/n must be an integer, which is a contradiction.
Therefore the n th root of an integer must be irrational if it is not an integer.
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