Generalized way of solving this types of equations $x^3 +y^4 =z^5$

Question

$$x^3 +y^4 =z^5$$
How can I solve this equation.I only know trial and error method, but it's not a generalized way.
Please tell me a generic way to solve this type of equation.

Answer 1

The equation $x^3 + y^4 = z^5$ has infinitely many solutions in positive integers. An infinite family of solutions is generated by $$x = a(a^3 + b^4)^{8k},\qquad y=b(a^3 + b^4)^{6k},\qquad z=(a^3 + b^4)^{5k}.$$ There are probably other solutions; I doubt that an exhaustive list of solutions is known.

Beal's conjecture would imply that the equation has no relatively prime integer solutions. But this conjecture remains unproved, and there is a $1 million prize for a proof or counterexample.

Answer 2

$$2^{24} +2^{24}=2^{25}$$
$$(2^{8})^{3} + (2^{6})^{4}=(2^{5})^{5}$$
So,$$ x=256\ ,\ y=64 \ ,\ z=32$$