If x=8 and y=64 then x+y=72 and x⅓+y⅓=2+4=6.
The other solution is x=64 and y=8.
The clue was to find two cubes whose sum is 72. Perfect cubes up to 72 are:
1, 8, 27, 64 and only 8 (23) and 64 (43). Graphically x+y=72 is a straight line while the other equation is a curve. A straight line can only intersect a curve of this type in two places, which are therefore the only solutions (8,64) and (64,8).