Let's plug in a = 9 and b = 14
[tex]a^3 + b^3 = (a+b)(a^2 - ab + b^2)\\\\9^3 + 14^3 = (9+14)(9^2 - 9*14 + 14^2)\\\\9^3 + 14^3 = (9+14)(81 -126 + 196)\\\\9^3 + 14^3 = (23)(151)\\\\9^3 + 14^3 = 3473\\\\[/tex]
It is possible to calculate the terms on the right hand side without a calculator. You can either memorize things, do mental math, or use pencil/paper to do the scratch work.
As a way to verify things,
[tex]9^3 = 9*9*9 = 729\\\\14^3 = 14*14*14 = 2744\\\\9^3+14^3 = 729+2744 = 3473\\\\\text{Therefore, } 9^3 + 14^3 = 3473 \text{ has been proven correct}\\\\[/tex]
