Cube root disaster

Let $u = 9+x, v = 9-x$ , which after cubing both sides of the above equation:

$u + 3u^{2/3}v^{1/3} + 3u^{1/3}v^{2/3} + v = 27;$

or $u + v + 3u^{1/3}v^{1/3}(u^{1/3} + v^{1/3}) = 27;$

or $u + v + 3u^{1/3}v^{1/3} \cdot 3 = 27;$

or $(9+x) + (9-x) + 9[(9+x)(9-x)]^{1/3} = 27;$

or $18 + 9[81-x^2]^{1/3} = 27;$

or $9[81-x^2]^{1/3} = 9;$

or $81 - x^2 = 1;$

or $x^2 = 80$ ;

or $x = \sqrt{80} = \boxed{4\sqrt{5}}.$