First slide

Algebra of complex numbers

Question

$Least positive argument of the 4^{th} root of the complex number 2 - i \sqrt{12} is:$

Moderate

A

$\frac{π}{6}$
B

$\frac{5 π}{12}$
C

$\frac{7 π}{12}$
D

$\frac{11 π}{12}$

Solution

$\begin{array}{l} z^{4} = 2 (1 - \sqrt{3} i) = 4 (\frac{1}{2} - \frac{\sqrt{3}}{2} i) \\ = 4 [\cos (- \frac{π}{3}) + i \sin (- \frac{π}{3})] \\ z = \sqrt{2} [\cos \frac{2 m π - (\frac{π}{3})}{4} + i \sin \frac{2 m π - (\frac{π}{3})}{4}] \\ F o r m = 1, z = \sqrt{2} [\cos (\frac{5 π}{12}) + i \sin (\frac{5 π}{12})] \\ ∴ T h e l e a s t p o s i t i v e a r g u m e n t o f z i s \frac{5 π}{12} \end{array}$

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