Note that 8 is 2 cubed, 27 is 3 cubed, 12 is 2 times 6 and 18 is 3 x 6. Also I would cross multiply and note what I just said in the prior sentence to try to work out a solution.
I agree with this up to the line and . This clearly implies the following line, but is not equivalent to it. In fact, appears to be a tautology, valid for all x. I certainly don't understand how it implies that .
Starting from and I would take the positive square root on both sides of those equations. The first one gives . For the second equation we need to be more careful. If x>0 then and we get . Thus the two equations together give , from which and therefore x=1.
But if x<0 then the positive square root of is , and a similar procedure to the previous paragraph gives , so that x=–1.
Thus there are two solutions, x = 1 and x = –1.