Find the shortest distance from the origin to the surface of $xyz^2=2$, i.e. minimize $f(x,y,z)=x^2+y^2+z^2$ subject to $xyz^2-2=0$ hence we obtain $L(x,y,z,\lambda)=x^2+y^2+z^2+\lambda(xyz^2-2)$,taking partial derivatives now... $$L_x=2x+\lambda yz^2=0 \\ L_y=2y+\lambda xz^2=0 \\ z^2(x^2-y^2)=0\Rightarrow x^2=y^2 \\ L_z=2z+2\lambda xy=0\\ y(2x^2-z^2)=0\Rightarrow z^2=2x^2 $$
I left out some of the manipulations but it was just solving for $\lambda$ and then getting everything in terms of $x$. Now I'm confused about how to solve the last equation in which I have $L_\lambda=xyz^2-2=0$, my solution manual randomly arrives at $x^2y^24z^2=4$, unfortunately I can't see how they're arriving here. The furthest I can get is multiplying both sides by $xy$ to obtain $2xy=x^2y^2z^2=x^2(x^2)(2x^2)=2x^6\Rightarrow x^6-xy=x(x^5-y)=0$