Synthetic division by root b gives:
b | 1 0 0 -b^3
1 b b^2 b^3
1 b b^2 | 0 = a^2+ab+b^2
So a^3-b^3=(a-b)(a^2+ab+b^2)=513
But ab=54 so (a-b)(a^2+54+b^2)=513
(a-b)^2=a^2-2ab+b^2=a^2+b^2-108 so a^2+b^2=(a-b)^2+108.
Let x=a-b, then a^2+b^2=x^2+108 and x(x^2+108+54)=513,
x(x^2+162)=513, x^3+162x-513=0=(x-3)(x^2+3x+171) (use synthetic division to get this)
There is only one real root, x=3 so a-b=3.
The factors of 54 include 9*6. These numbers differ by 3 (a-b=3=9-6) and 9^3-6^3=729-216=513, making a=9 and b=6.