Question

Aratati ca numarul ( 1+i radical din 3) la puterea a 3-a este intreg

Answer 1

$\displaystyle (a+b)^3=(a+b)(a+b)(a+b)=(a^2+b^2+2ab)(a+b)=a^3+a^2b+ab^2+b^2+2a^2b+2ab^2=a^3+3a^2b+3ab^2+b^3 \\ (1+i\sqrt3)^3=1+3i\sqrt3+9i^2+3i^3\sqrt3=1+3i\sqrt3-9-3i\sqrt3=1+(-9)=-8\in Z$