Irrational equation, How to solve it? The equation 
$$\sqrt[3]{x^2-1} + x = \sqrt{x^3-2}$$ has a solution $x = 3.$ How to solve this eqution?

You have to rewrite this as
$$
  \sqrt[3]{x^2-1}= \sqrt{x^3-2}-x
$$
and take the 3rd power of both members obtaining
$$
   x^2-1=(x^3-2)\sqrt{x^3-2}-3x(x^3-2)+3x^2\sqrt{x^3-2}-x^3.
$$
Now, you can isolate the square root obtaining
$$
  3x^4+x^3+x^2-6x-1=(x^3+3x^2-2)\sqrt{x^3-2}.
$$
I think from here you can go on by yourself.
