Proof that the square root of any nonperfect square positive integer is irrational

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Let us assume that $\sqrt{n}$ is rational where $n$ is a nonperfect square positive integer. Then it can be expressed as $\frac{p}{q}$. Thus $\frac{p^2}{q^2}=n$. That means that $(q^2)n=p^2$. But no perfect square times a nonperfect square positive integer is a perfect square. Therefore $\sqrt{n}$ is irrational.