Question

If $x=a+b$, $y=a{\omega }^2+b\omega$ and $z=a\omega +b{\omega }^2$, where $\omega$ is a complex cube root of unity, then the product of $xyz$ is equal to ____

• A. $a^3+b^3$
• B. $a+b$
• C. $a^2+b^2$
• D. $a^4+b^4$

Answer 1

The correct option is A $a^3+b^3$

$x\times y\times z$
$(a+b)(a{w}^2+bw)(aw+b{w}^2)$
$=(a+b)(a^2{w}^3+ab{w}^4+ab{w}^2+b^2{w}^3)$
$=(a+b)(a^2+ab{w}^{3}w+ab{w}^2+b^2)$
$=(a+b)(a^2+abw+ab{w}^2+b^2)$
$=(a+b)(a^2+ab(w+w^2)+b^2)$
$=(a+b)(a^2-ab+b^2)$
$=(a+b)(a^2-ab+b^2)$
$=a^3+b^3$