First slide

Algebra of complex numbers

Question

If $x = 9^{1 / 3} 9^{1 / 9} 9^{1 / 27} \dots \infty, y = 4^{1 / 3} 4^{- 1 / 9} 4^{1 / 27} \dots \infty,$ and $z = \sum_{r = 1}^{\infty} (1 + i)^{- r}$ then arg(x + yz) is equal to

Moderate

A

$0$
B

$π - \tan^{- 1} (\frac{\sqrt{2}}{3})$
C

$- \tan^{- 1} (\frac{\sqrt{2}}{3})$
D

$- \tan^{- 1} (\frac{2}{\sqrt{3}})$

Solution

$x = 9^{\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots} = 9^{\frac{\frac{1}{3}}{1 - \frac{1}{3}}} = 9^{\frac{1}{2}} = 3$
$\begin{array}{l} y = 4^{\frac{1}{3} - \frac{1}{9} + \frac{1}{27} + \dots} = 4^{\frac{\frac{1}{3}}{1 + \frac{1}{3}}} = 4^{\frac{1}{4}} = \sqrt{2} \\ z = \sum_{r = 1}^{\infty} (1 + i)^{- r} = \frac{1}{1 + i} + \frac{1}{(1 + i)^{2}} + \frac{1}{(1 + i)^{3}} + \dots \\ = \frac{\frac{1}{1 + i}}{1 - \frac{1}{1 + i}} = \frac{1}{i} = - i \end{array}$
Let $a = x + yz = 3 - \sqrt{2} i$ (fourth quadrant). Then
$\arg α = - \tan^{- 1} (\frac{\sqrt{2}}{3})$

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