Prove that the diophantine equation $1/x^4 + 1/y^4 = 1/z^2$ has no solutions in nonzero integers, x, y, z

Question

Prove that the Diophantine equation $1/x^4 + 1/y^4 = 1/z^2$ has no solutions in nonzero integers, x, y, z.

I've tried to solve this using the Theorem: "The Diophantine equation $x^4 + y^4 = z^2$ has no solutions in nonzero integers x, y, z.", and transformed the equation into $y^4z^2 + x^4z^2 = x^4y^4$. What are the next steps?

Answer 1

The next step is: $z^2(x^4+y^4) = x^4y^4 = (x^2y^2)^2\implies x^4+y^4 = k^2$, contradiction to the result you quoted.