Algebra or Numbertheory?

Making a substitution $a=x+y,b=x-y$ , with $x>y$ , we can write the given equation as $8x^2y=1024$ , or $x^2y=2^7$ . We quickly see that the only solution with $x>y$ is $x=8, y=2$ so $a=\boxed{10}$ and $b=6$ .

The given equation can be simplified as :

$a^3 - b^3$ = $(a-b) (a^2 + ab + b^2)$ ----->1

$a^2b - ab^2 = ab(a - b)$ ------>2

(1 + 2) ==> $(a - b)(a^2 + 2ab + b^2)$ = $(a - b)(a + b)^2$ = $1024 = (2^{10})$ .

From above equation, we can conclude that one of the number is a perfect square & (a + b) should carry a greater value  than (a - b).
The only possible answer is (a + b) = 16 which means (a - b) = 4.

By solving we get ,

$\boxed{ a = 10 }$