The left-hand side of the above equation can only be an integer, but the right-hand side is an integer only if q=1.

Application:

The cube root of 2 cannot be rational since it is a solution to the algebraic equation x^{3}−2=0 and it cannot be an integer.

To show that an expression, such as √3−√2, is not rational it is only a matter of finding an algebraic equation with integer
coefficients for which it is a root. For the case of x=√3−√2, it immdiately follows that x² is 3−2√6+2 and hence

This is an algebraic equation with integer coefficient for which √3−√2 is a root.
There is no integer equal to √3−√2. Therefore √3−√2 cannot be rational.

The failure of some expression to be a real rational number should not be construed to mean that it
is a real irrational number. For example, the solution to

x² + 1 = 0

cannot be a real rational number. It is of course a complex number, of an integral nature.