Well by your definition,

6 to the superpower means $$6^{5!}$$

so

x to the superpower means $$x^{(x-1)!}$$

y to the superpower means $$y^{(y-1)!}$$

z to the superpower means $$z^{(z-1)!}$$

so

I guess you are asking me to solve

$$x^{(x-1)!}=y^{(y-1)!}+z^{(z-1)!}$$

Don't know - you have me here.