I'm not sure how to deal with the cube roots.
\[\large\begin{eqnarray}\begin{split}x+y=&3383\\\sqrt[3]x+\sqrt[3]y=&17\\xy=&?\end{split}\end{eqnarray}\]
+128408 x^(1/3) + y^(1/3) = 17 square both sides
(x^1/3 + y^1/3)^2 = 17^2
(x^2/3 + 2(xy)^1/3 + y^2/3) = 289
x^2/3 + y^2/3 = 289 - 2(xy)^ (1/3)
Factor x + y as a sum of cubes
x + y = (x^1/3 + y^1/3) ( x^(2/3) - (xy)^1/3 + y^(2/3) )
3383 =(17) ( x^(2/3) - (xy)^(1/3) + y^(2/3) )
3383 /17 = x^(2/3) + y^(2/3) - (xy)^(1/3)
199 = 289 - 2(xy)^(1/3 ) - (xy)^(1/3)
189 - 289 = - 3(xy)^(1/3)
-90 = -3(xy)^(1/3)
-90 / -3 = (xy)^1/3
30 = (xy)^(1/3 cube both sides
27000 =xy
