$$\eqalign{
& \root 3 \of {15x - 1} = 4\root 3 \of x - \root 3 \of {13x + 1} \cr
& \Leftrightarrow \root 3 \of {15x - 1} + \root 3 \of {13x + 1} = 4\root 3 \of x \cr
& \Leftrightarrow 15x - 1 + 13x + 1 + 3\root 3 \of {15x - 1} .\root 3 \of {13x + 1} \left( {\root 3 \of {15x - 1} + \root 3 \of {13x + 1} } \right) = 64x \cr
& \Leftrightarrow 3\root 3 \of {15x - 1} .\root 3 \of {13x + 1} \left( {\root 3 \of {15x - 1} + \root 3 \of {13x + 1} } \right) = 36x \cr
& \Leftrightarrow \root 3 \of {15x - 1} .\root 3 \of {13x + 1} \left( {\root 3 \of {15x - 1} + \root 3 \of {13x + 1} } \right) = 12x\,\,\left( * \right) \cr
& Dat\,\,\left\{ \matrix{
\root 3 \of {15x - 1} = u \hfill \cr
\root 3 \of {13x + 1} = v \hfill \cr} \right. \Rightarrow \left\{ \matrix{
{u^3} = 15x - 1 \hfill \cr
{v^3} = 13x + 1 \hfill \cr} \right. \Rightarrow {u^3} + {v^3} = 28x \cr
& \Rightarrow \left( {u + v} \right)\left( {{u^2} - uv + {v^2}} \right) = 28x \cr
& \left( * \right) \Rightarrow uv\left( {u + v} \right) = 12x \cr
& TH1:\,\,u + v = 0 \Rightarrow 12x = 0 \Leftrightarrow x = 0 \cr
& u = 0 \Leftrightarrow \cr
& Thu\,\,lai:\,\,\root 3 \of { - 1} = 4\root 3 \of 0 - \root 3 \of 1 \,\,\left( {Dung} \right) \Rightarrow x = 0\,\,la\,\,nghiem\,\,cua\,\,pt \cr
& TH2:\,\,u + v \ne 0 \cr
& \Rightarrow {{uv\left( {u + v} \right)} \over {\left( {u + v} \right)\left( {{u^2} - uv + {v^2}} \right)}} = {{12x} \over {28x}} = {3 \over 7} \cr
& \Leftrightarrow {{uv} \over {{u^2} - uv + {v^2}}} = {3 \over 7} \cr
& \Leftrightarrow 7uv = 3{u^2} - 3uv + 3{v^2} \cr
& \Leftrightarrow 3{u^2} - 10uv + 3{v^2} = 0 \cr
& \Leftrightarrow \left( {u - 3v} \right)\left( {3u - v} \right) = 0 \cr
& \Leftrightarrow \left[ \matrix{
u = 3v \hfill \cr
u = {1 \over 3}v \hfill \cr} \right. \Leftrightarrow \left[ \matrix{
\root 3 \of {15x - 1} = 3\root 3 \of {13x + 1} \hfill \cr
\root 3 \of {15x - 1} = {1 \over 3}\root 3 \of {13x + 1} \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
15x - 1 = 27\left( {13x + 1} \right) \hfill \cr
27\left( {15x - 1} \right) = 13x + 1 \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
336x = - 28 \hfill \cr
392x = 28 \hfill \cr} \right. \Leftrightarrow \left[ \matrix{
x = - {1 \over {12}} \hfill \cr
x = {1 \over {14}} \hfill \cr} \right. \cr
& \Rightarrow S = \left\{ {0; - {1 \over {12}};{1 \over {14}}} \right\} \cr} $$
