Rational?

$\large \sqrt{\frac{m+1}{m}} ~\text{or} ~ \sqrt{\frac{m}{m+1}}$ The point to note is $\gcd(m,m+1)=1$ . Therefore, they would always be coprime. Even if any one number is a square, other won't be and hence it wouldn't be square free and hence would not follow the definition of a rational number. Example- $\sqrt{\dfrac 45} = \dfrac{2}{\sqrt{5}} \ne \text{rational number}$

Another argument could be:

Squares of two distinct positive integers always differ greater than or equal than $3$ .

So,according to the question,the two integers are consecutive. So we are guaranteed that atleast the square root of one integer will be an irrational number.

So the given expression will always be an irrational number.