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. $4\sqrt{(}3i)$
• D. $-i\sqrt{3}$

Answer 1

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

$\omega =\frac{-1}{2}+i\frac{\sqrt{3}}{2}$
${\omega }^2=\frac{-1}{2}-i\frac{\sqrt{3}}{2}$
Thus, $4+5{[-\frac{1}{2}+i\frac{\sqrt{3}}{2}]}^{334}-3{[\frac{1}{2}+i\frac{\sqrt{3}}{2}]}^{365}$
$=4+5(\omega {)}^{334}-3(-{\omega }^2{)}^{365}$
$=4+5\left(\omega \right)-3(-1{)}^{365}(\omega {)}^{730}$
$=4+5\omega +3\omega$
$=4+8\omega$
$=4+8(\frac{-1}{2}+i\frac{\sqrt{3}}{2})$
$=4-4+i4\sqrt{3}$
$=i4\sqrt{3}$