Rewrite the equation (sqrt(5))^x - ([3]2)^y = 17 using fractional exponents and determine the form of x and y for integer powers.

The given equation is $(\sqrt{5})^x - (\sqrt[3]{2})^y = 17$.
We can express the roots as fractional exponents: $\sqrt{5} = 5^{1/2}$ and $\sqrt[3]{2} = 2^{1/3}$.
Substituting these into the equation, we get:

Using the exponent rule $(a^b)^c = a^{bc}$:

Since $x, y \in \mathbb{N}$ (natural numbers), for $5^{x/2}$ and $2^{y/3}$ to result in integer values (which is necessary for their difference to be the integer