Background
So I was watching this youtube video. It poses a problem solve: $$ 2^x = x^4$$
My Method
$$2^x = x^2 x^2$$
Or:
$$ 2^{x/2} 2^{x/2} = x^2 x^2$$
By prime factorization theorem and symmetry:
$$x/2 = 2$$
Extending this method to other problems
We only restrict ourselves to integer solutions:
$$a^b c^d = x^4 y^6$$
where $d > b$, then:
$$ a^b c^d = (xy)^4 y^2$$
Or:
$$ a^{b-2} c^{d-2} (ac)^2 = (xy)^4 y^2$$
Using prime factorization theorem it should be possible to rearrange the primes such that one solution amongst the set of solutions:
$$ac = y^2$$ $$a^{b-2} c^{d-2} = (xy)^4$$
Simplyfying:
$$ac = y^2 \implies y = a^{1/2}c^{1/2} $$ $$a^{b-2} c^{d-2} = (xy)^4 \implies x = a^{b/4-1} c^{d/4-1} $$
Question
What are these set of problems in number theory known as? Are there any interesting applications where they show up?
