Question

express (-1-i)^-30 in the form of a + ib

Answer 1

$\green{\large\underline{\sf{Solution-}}}$

Given complex number is

$\rm :\longmapsto\: {( - 1 - i)}^{ - 30}$

$\rm \:  =  \: \dfrac{1}{ {( - 1 - i)}^{30} }$

$\rm \:  =  \: \dfrac{1}{ {( - [1 + i])}^{30} }$

$\rm \:  =  \: \dfrac{1}{ {[1 + i]}^{30} }$

$\rm \:  =  \: \dfrac{1}{ {[ {(1 + i)}^{2} ]}^{15} }$

$\rm \:  =  \: \dfrac{1}{ {(1 + {i}^{2} + 2i)}^{15} }$

$\rm \:  =  \: \dfrac{1}{ {(1 - 1 + 2i)}^{15} } \: \: \: \: \: \: \: \{ \because \: {i}^{2} = - 1 \}$

$\rm \:  =  \: \dfrac{1}{ {(2i)}^{15} } \: \: \: \: \: \: \:$

$\rm \:  =  \: \dfrac{1}{ {2}^{15} {(i)}^{15} } \: \: \: \: \: \: \:$

$\rm \:  =  \: \dfrac{1}{ {2}^{15} {(i)}^{14} \times i} \: \: \: \: \: \: \:$

$\rm \:  =  \: \dfrac{1}{ {2}^{15} {( {i}^{2} )}^{7} \times i} \: \: \: \: \: \: \:$

$\rm \:  =  \: \dfrac{1}{ {2}^{15} {( - 1 )}^{7} \times i} \: \: \: \: \: \: \:$

$\rm \:  =  \: - \: \dfrac{1}{ {2}^{15} i} \: \: \: \: \: \: \:$

$\rm \:  =  \: - \: \dfrac{1}{ {2}^{15} i} \times \dfrac{i}{i} \: \: \: \: \: \: \:$

$\rm \:  =  \: - \: \dfrac{i}{ {2}^{15} {i}^{2} }$

$\rm \:  =  \: - \: \dfrac{i}{ {2}^{15}( - 1)}$

$\rm \:  =  \: \: \dfrac{i}{ {2}^{15}}$

Hence,

$\rm \implies\:\boxed{\tt{ {( - 1 - i)}^{ - 30} =0 + \frac{i}{ {2}^{15} } \: }}$

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Additional Information :-

Argument of complex number

$\begin{gathered}\boxed{\begin{array}{c|c} \bf Complex \: number & \bf arg(z) \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf x + iy & \sf  {tan}^{ - 1}\bigg |\dfrac{y}{x} \bigg|   \\ \\ \sf  - x + iy & \sf \pi - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf  - x - iy & \sf  - \pi + {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf x - iy & \sf  - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \end{array}} \\ \end{gathered}$