Question

If $i=\sqrt{-1}$, then $4+5{(\frac{-1}{2}+i\frac{\sqrt{3}}{2})}^{334}-3{(\frac{1}{2}+i\frac{\sqrt{3}}{2})}^{365}$ is equal to

• A. $1-i\sqrt{3}$
• B. $-1+i\sqrt{3}$
• C. $4\sqrt{3}i$
• D. $-i\sqrt{3}$

Answer 1

The correct option is C $4\sqrt{3}i$

$\because \omega =\frac{-1}{2}+i\frac{\sqrt{3}}{2},{\omega }^2=-\frac{1}{2}-i\frac{\sqrt{3}}{2},{\omega }^3=1$

Now $4+5{(\frac{-1}{2}+i\frac{\sqrt{3}}{2})}^{334}-3{(\frac{1}{2}+i\frac{\sqrt{3}}{2})}^{365}$
$=4+5(\omega {)}^{334}-3(-{\omega }^2{)}^{365}$
$=4+5({\omega }^3{)}^{111}\times \omega +3({\omega }^3{)}^{243}\times \omega$
$=4+5\omega +3\omega$
$=4+8\omega$
$=4-4+4\sqrt{3}i=4i\sqrt{3}$