The standard method solving equations like this is as follows.

Introduce new variable 

y = x^(1/3) =        (1).

Then your equation takes the form

 -  -  = .   (2)

It is a quadratic equation for y. Solve it using the quadratic formula:

 =  =  = .

Thus the equation (2) has two solutions:  = 3;    = .

Now, to find x, you need solve two equations ( recall (1) ):

1)  = 3. To solve it, raise both side to the degree 3: x =  = 27.

2)  = . To solve it, raise both side to the degree 3: x =  = .

Answer. There are two solutions of the original equation: x = 27 and x = .