The value of $(1 + i)^{3} + (1 - i)^{6}$ is

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2 (- 1 + 5 i)

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1 - 5 i

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none of these

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Solution

The correct option is B $2 (- 1 + 5 i)$
$1 + i = \sqrt{2} [\frac{1 + i}{\sqrt{2}}]$
$= \sqrt{2} . e^{i \frac{π}{4}}$ ...(i)
And
$1 - i = \sqrt{2} [\frac{1 - i}{\sqrt{2}}]$
$= \sqrt{2} . e^{i \frac{- π}{4}}$ ...(ii)
Hence
$(1 + i)^{3} + (1 - i)^{6}$
$= (\sqrt{2})^{3} e^{i \frac{3 π}{4}} + 2^{3} . e^{i \frac{- 3 π}{2}}$
$= 2 [\sqrt{2} (\frac{- 1 + i}{\sqrt{2}}) + 4 (i)]$
$= 2 [- 1 + i + 4 i]$
$= 2 [- 1 + 5 i]$

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