First slide

Algebra of complex numbers

Question

If $i = \sqrt{- 1}, then 4 + 3 {(- \frac{1}{2} + i \frac{\sqrt{3}}{2})}^{127} + 5 {(- \frac{1}{2} + i \frac{\sqrt{3}}{2})}^{124}$ is equal to

Moderate

A

$4 \sqrt{3} i$
B

$2 \sqrt{3} i$
C

$1 \sqrt{3} - i$
D

$1 \sqrt{3} + i$

Solution

Let $ω = - \frac{1}{2} + i \frac{\sqrt{3}}{2} then ω^{3} = 1$
$z = 4 + 3 ω^{127} + 5 ω^{124}$
$= 4 + 3 {(ω^{3})}^{42} ω + 5 {(ω^{3})}^{41} ω$
$= 4 + 3 ω + 5 ω = 4 + 8 ω$
$= 4 - 4 + 4 \sqrt{3} i = 4 \sqrt{3} i$

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