Question

If $\alpha ,\beta$ are complex cube roots of unity, then the value of ${\alpha }^2{\beta }^2+{\alpha }^{10}{\beta }^2+{\alpha }^2{\beta }^{10}$ equals

• A. 1
• B. -1
• C. 3
• D. 0

Answer 1

Answer: (D)

0

Therefore
$\alpha =w$
$\beta =w^2$
Hence
${\alpha }^2{\beta }^2+{\alpha }^2{\beta }^2({\alpha }^8+{\beta }^8)$
$=(w.w^2{)}^2+w^2.w^4(w^8+w^{16})$
$=1+(w^8+w^{16})$
$=1+w^8(1+w^6.w^2)$
$=1+w^6.w^2(1+w^2)$
$=1+w^2(-w)$
$=1-1$
$=0$
Hence, option 'D' is correct.