Is that too big?

Taking $10! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 = 1 \cdot 2^1 \cdot 3^1 \cdot 2^2 \cdot 5^1 \cdot 2^{1}3^{1} \cdot 7^1 \cdot 2^3 \cdot 3^2 \cdot 2^{1}5^{1}= 2^{8} 3^{4} 5^{2} 7^{1}$ , then $3^{x} 5^{y} \Rightarrow x = 4, y = 2$ and $x + y = \boxed{6}.$

Relevant Wiki: Legendre's formula

$x$ , which is the exponent for 3, can be obtained by:

$\bigg\lfloor \dfrac{10}{3^1} \bigg\rfloor + \bigg\lfloor\dfrac{10}{3^2} \bigg\rfloor+ \red{\underbrace{\ldots}_{<1}} = 4$

Similarly, $y$ can also be found out:

$\bigg\lfloor \dfrac{10}{5^1} \bigg\rfloor + \red{\underbrace{\ldots}_{<1}} = 2$

Thus, the sum is:

$x + y = \boxed{6}$