Irrationality of square roots of non-squares

Statement

Let $n$ be a natural number. Then $\sqrt{n}$ is irrational if and only if $n$ is not a perfect square.

Remarks

If $n = m^2$, then $\sqrt{n} = m$ is rational. The content is the other direction. A proof: if $\sqrt{n} = a/b$, then $a^2 = n b^2$. By unique factorization, each prime has an even exponent in $a^2$ and in $b^2$, so also in $n$. Then $n$ is a square. For example, $\sqrt{2}$, $\sqrt{3}$ and $\sqrt{12}$ are irrational. The same argument shows that the $k$-th root of $n$ is irrational unless $n$ is a $k$-th power.

Explanations

L1 human

The square root of a whole number is either a whole number or an irrational number. It is never a fraction such as $\frac{3}{2}$.