Question

$4+5{\left(\frac{-1+i\sqrt{3}}{2}\right)}^{2008}+3{\left(\frac{-1+i\sqrt{3}}{2}\right)}^{2009}$ is equal to

• A. $-i\sqrt{3}$
• B. $i\sqrt{3}$
• C. $1-i\sqrt{3}$
• D. $-1+i\sqrt{3}$

Answer 1

Answer: (B)

$4+5{\left(\frac{-1+i\sqrt{3}}{2}\right)}^{2008}+3{\left(\frac{-1+i\sqrt{3}}{2}\right)}^{2009}$
$=4+5{(\omega )}^{3\times 609+1}+3{(\omega )}^{3\times 609+2}$ $=4+5\omega +3{\omega }^2$
$[\because {\omega }^3=1]$
$=4+2\omega +3(\omega +{\omega }^2)$
$=4+2\left(\frac{-1+i\sqrt{3}}{2}\right)-3$
$[\because 1+\omega +{\omega }^2=0]$
$=i\sqrt{3}$