fplot(@(x) sqrt(2-x^2)) ;

  hold on

   fplot(@(x) -sqrt(2-x^2)) ;

  fplot(@(x) abs((3-x^3)^(1/3))*sign((3-x^3)));

  axis equal

  grid on

  

plots this graph of the two functions, on either side of the equation.

The square function is real only between -sqrt(2) and sqrt(2), and in this range is always below the cubic function, so there are no real solutions to the equation.

The closest the two curves come is cuberoot(3) - sqrt(2), at the top of the circle and at its right side.