What are the steps to solve this problem?
x^(2/3) - x^(1/3) - 6 = 0
This is actually a quadratic equation in $\displaystyle x^{1/3}$.
Recalling the laws of indices we can say that $\displaystyle x^{2/3} = \left(x^{1/3}\right)^2$
To illustrate the point let $\displaystyle u = x^{1/3}$ - thus we have $\displaystyle u^2-u-6=0$
Solve the quadratic using your favourite method (this one does factorise) to get two values of u.
Once you have the values of u you put $\displaystyle x^{1/3} = root_1$ and $\displaystyle x^{1/3} = root_2$ where root_1 and root_2 are your values of u. It should then be straightforward to find x
In that case you have u= 2 and u=3
Since we defined u as being equal to x^{1/3} then x^{1/3} must equal 2 and 3.
$\displaystyle u = x^{1/3} = 2$ and $\displaystyle u = x^{1/3} = 3.$
You can then solve for x, you will get two solutions but this simply means two values of x satisfy this equation