Power

Let $x = 5^{a}7^{b}$ and $y = 5^{c}7^{d} (a,b,c,d \in \mathbb{N}).$ This leads to $N = 5x^7 = 7y^5 \Rightarrow 5(5^{a}7^{b})^7 = 7(5^{c}7^{d})^5 \Rightarrow 5^{7a+1}7^{7b} = 5^{5c}7^{5d+1}$ .

Setting the respective exponents for the bases of 5 and 7 equal to each other yields $7a +1 = 5c, 7b = 5d+1$ , one finds that equality occurs when:

$a = 2 + 5(k-1), c = 3 + 7(k-1)$ and $b = 3 + 5(k-1), d = 4 + 7(k-1)$

for $k \in \mathbb{N}.$ hence, there are infinitely many values for $N$ for $x,y \in \mathbb{N}$ under these conditions.