Can the sum of two cube roots of positive non-perfect cubes be an integer?

Give reasons for your answer.

assume ³√a + ³√b where a, b integers but not cubes is an integer
cube the expression
a + 3³√a²*³√b + 3³√a³√b² + b
this must also be an integer
factor
a + 3³√a³√b(³√a+³√b) + b
The bold term must also be an integer.

Implication:
If the sum of two cube roots of positive non-perfect cubes is an integer then their product is also an integer.

The answer to the question posed is probably no.
It may be easier to show that if the product of the cube roots is not an integer then their sum also cannot be.

Edited on August 14, 2015, 11:48 pm

Posted by Jer on 2015-08-14 12:50:16