Rational $n^\text{th}$ Root

Looking at the exponent of each prime factor, for the term to be a rational number, the value of $r$ must be the divisor of $\text{gcd}(36,48) = 12$ . Since the divisors of $12$ are $1,2,3,4,6$ and $12$ , there are $\boxed{6}$ positive integers $r$ that make $\sqrt[r]{2^{36}3^{48}}$ rational.