Question

Question!!
express the given complex number in the form of (a + ib)

$\sf { \bigg( - 2 - \dfrac{1}{3} i \bigg) }^{3}$
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Answer 1

Step-by-step explanation:

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Answer 2

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

Given complex number is

$\rm :\longmapsto\:\sf { \bigg( - 2 - \dfrac{1}{3} i \bigg) }^{3}$

can be rewritten as

$\rm \:  =  \: {\bigg[ - \sf { \bigg(2 + \dfrac{i}{3} \bigg) }\bigg]}^{3}$

$\rm \:  =  \: - \: {\bigg[\sf { \bigg(2 + \dfrac{i}{3} \bigg) }\bigg]}^{3}$

We know,

$\boxed{ \tt{ \: {(x + y)}^{3} = {x}^{3} + {3x}^{2}y + {3xy}^{2} + {y}^{3} \: }}$

So, using this identity, we get

$\rm \:  =  - \:\bigg[ {(2)}^{3} + 3 {(2)}^{2} \times \dfrac{i}{3} + 3 (2) {\bigg[\dfrac{i}{3} \bigg]}^{2} + {\bigg[\dfrac{i}{3} \bigg]}^{3}\bigg]$

$\rm \:  =  - \:\bigg[8 + 4i + \dfrac{2}{3} {i}^{2} + \dfrac{ {i}^{3} }{27}\bigg]$

We know,

$\rm :\longmapsto\:\boxed{ \tt{ \: {i}^{2} = - 1}} \: \: \: and \: \: \: \boxed{ \tt{ \: {i}^{3} = - \: i \: }}$

So, on substituting, these values, we get

$\rm \:  =  - \:\bigg[8 + 4i - \dfrac{2}{3} - \dfrac{i }{27}\bigg]$

$\rm \:  =  - \:\bigg[\bigg(8 - \dfrac{2}{3}\bigg) + i\bigg(4 - \dfrac{1}{27}\bigg)\bigg]$

$\rm \:  =  - \:\bigg[\bigg(\dfrac{24 - 2}{3}\bigg) + i\bigg(\dfrac{108 - 1}{27}\bigg)\bigg]$

$\rm \:  =  - \:\bigg[\bigg(\dfrac{22}{3}\bigg) + i\bigg(\dfrac{107}{27}\bigg)\bigg]$

$\rm \: \:   =  - \: \dfrac{22}{3} - \: i\dfrac{107}{27}$

Hence,

$\red{\bf\implies \:\boxed{ \tt{ \: \sf { \bigg( - 2 - \dfrac{1}{3} i \bigg) }^{3}  =  - \: \dfrac{22}{3} - \: i\dfrac{107}{27} \: }}}$

More to know :-

$\boxed{ \tt{ \: |z| = |\overline{z}| \: }}$

$\boxed{ \tt{ \: z. \: \overline{z} \: = { |z| }^{2} \: }}$

$\boxed{ \tt{ \: \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \: }}$

$\boxed{ \tt{ \: \overline{z_1 - z_2} = \overline{z_1} - \overline{z_2} \: }}$

$\boxed{ \tt{ \: \overline{z_1 \times z_2} = \overline{z_1} \times \overline{z_2} \: }}$

$\boxed{ \tt{ \: \overline{z_1 \div z_2} = \overline{z_1} \div \overline{z_2} \: }}$