I need to find the critical points and relative max/min for this equation:
f(x,y)=x^2+y^2+(1/x^2*y^2)

I found the partials with respect to x and y. I equal them to zero in order to find the critical points, and got the equation:
y^4*x^2=y^2*x^4

My question is how can I solve this equation for x and y?

2x - 2/(y^2*x^3)=0 (i.e. partial derivative wrt x); 2y - 2/(y^3*x^2) = 0 (p.d. wrt to y). I.e. you have two equations: x^2 = 1/y^4; and x^4 = 1/y^2. This system has four solutions: (-1,-1); (-1,1); (1,-1); (1,1). Here are your critical points. Include also (0,0) in this list too as the derivatives do not exist in this point. To solve your equation (which I suspect has nothing to do with this particular problem due to non-legitimate manipulations with equations)): because x^2 and y^2 are positive and not zeros, your equation is identical to x^2=y^2. This equation has quite a different set of solutions: y=x; y=-x, which is wrong for this problem.