Are there positive integers $x, y$ and $z$ such that $2^{x} · 3^{4} · 14^{y} = 126^{z}$

Question

Can anyone give me a tip on how to approach this. Possibly a theorem of some sort that allows me to work with powers using modular arithmetic. Thanks for the help.

Answer 1

$14=2\cdot7\implies14^y=2^y7^y$

$$\implies2^x3^414^y=2^{x+y}3^47^y$$

Again, $126=2\cdot7\cdot3^2\implies126^z=(2\cdot7\cdot3^2)^z=2^z7^z3^{2z}$

As $3,2,7$ are pairwise prime, equate the exponents of each of them