Question

lf $1,\omega ,{\omega }^2$ are cube roots of unity then the value of

$(a+2b{)}^2+(a\omega +2b{\omega }^2{)}^2+(a{\omega }^2+2b\omega {)}^2=$

• A. $4ab$
• B. $8ab$
• C. $12ab$
• D. $16ab$

Answer 1

The correct option is C $12ab$

$a^2+4b^2+4ab+a^2{\omega }^2+4ab{\omega }^3+4b^2{\omega }^4+a^2{\omega }^4+4b^2{\omega }^2+4ab{\omega }^3$
$=a^2(1+{\omega }^2+{\omega }^4)+4b^2(1+{\omega }^4+{\omega }^2)+4ab({\omega }^3+1+{\omega }^3)$
$=a^2(1+\omega +{\omega }^2)+4b^2(1+\omega +{\omega }^2)+4ab(1+1+1)$
$=12ab$