Question

Find the modulus of given complex number.

$z = \dfrac{(3 - 4i)( \sqrt{3} - i)}{( \sqrt{3} + i)^{7} }$
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Answer 1

$\Large{\boxed{\bold{[Topic:\ Complex\ numbers]}}}$

The modulus of a complex number is given by -$\large\text{ \cdots\longrightarrow\sqrt{a^{2}+b^{2}}=\sqrt{(a+bi)(a-bi)}=\sqrt{z\bar{z}}. }$

So, the modulus will be the following.

$\large\text{ \cdots\longrightarrow z\bar{z}=\dfrac{(3-4i)(\sqrt{3}-i)}{(\sqrt{3}+i)^{7}}\times\dfrac{(3+4i)(\sqrt{3}+i)}{(\sqrt{3}-i)^{7}} }$

$\large\text{ \cdots\longrightarrow z\bar{z}=\dfrac{(9+16)\times(3+1)}{(3+1)^{7}} }$

$\large\text{ \cdots\longrightarrow z\bar{z}=\dfrac{5^{2}\times2^{2}}{2^{14}} }$

$\large\text{ \cdots\longrightarrow z\bar{z}=\dfrac{5^{2}}{2^{12}} }$

$\large\text{ \cdots\longrightarrow\boxed{\sqrt{z\bar{z}}=\dfrac{5}{64}} }$

Hence, the modulus of $\largez$ is equal to $\large\dfrac{5}{64}$.

$\Large{\boxed{\bold{[Learn\ more]}}}$

$\Large\rightarrow\text{Properties of complex numbers}$

Let $\largez$ represent a complex number.

$\Large\bullet\text{ Complex conjugate}$

$\large\text{ \cdots\longrightarrow\boxed{z=\bar{z}\iff z\text{ is real.}} }$

$\large\text{ \cdots\longrightarrow\boxed{z=-\bar{z}\iff z\text{ is purely imaginary.}} }$

$\Large\bullet\text{ Real numbers}$

$\large\text{ \cdots\longrightarrow\boxed{z+\bar{z}\text{ is always real.}} }$

$\large\text{ \cdots\longrightarrow\boxed{z\bar{z}\text{ is always real.}} }$

$\Large\bullet\text{ Purely imaginary numbers}$

$\large\text{ \cdots\longrightarrow\boxed{z-\bar{z}\text{ is always purely imaginary.}} }$

Answer 2

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

Given complex number is

$\rm \: z \: = \: \dfrac{(3 - 4i)( \sqrt{3} - i)}{( \sqrt{3} + i)^{7} }$

So,

$\rm \: |z| \: = \: \bigg |\dfrac{(3 - 4i)( \sqrt{3} - i)}{( \sqrt{3} + i)^{7} }\bigg|$

We know,

$\boxed{\sf{  \: \: \: \bigg |\dfrac{z _1}{z _2} \bigg| \: = \: \dfrac{ |z _1| }{ |z _2| } \: \: \: }}$

So, using this result, we get

$\rm \: |z| = \dfrac{ |(3 - 4i)( \sqrt{3} - i)| }{ |\sqrt{3} + i)^{7}| }$

We know, that

$\boxed{\sf{  \: \: \: |z _1 \: z _2| \: = \: |z _1| \: |z _2| \: \: }}$

and

$\boxed{\sf{  \: \: \: | {z}^{n} | \: = \: { |z| }^{n} \: \: \: }}$

So, using these results, we get

$\rm \: |z | = \dfrac{ |(3 - 4i) | \: | ( \sqrt{3} - i)| }{ |\sqrt{3} + i)| ^{7} }$

We know, that,

$\boxed{\sf{  \: |x + iy| \: = \: \sqrt{ {x}^{2} + {y}^{2} } \: \: }}$

So, using this result, we get

$\rm \: |z| = \dfrac{ \sqrt{ {3}^{2} + {( - 4)}^{2} } \: \: \times \: \: \sqrt{ {( \sqrt{3} )}^{2} + {( - 1)}^{2} } }{ {\bigg( \sqrt{ {( \sqrt{3} )}^{2} + {(1)}^{2} } \bigg)}^{7} }$

$\rm \: |z| = \dfrac{ \sqrt{ 9 + 16} \: \: \times \: \: \sqrt{ 3 + 1 } }{ {\bigg( \sqrt{ 3 + 1} \bigg)}^{7} }$

$\rm \: |z| = \dfrac{ \sqrt{25} \: \: \times \: \: \sqrt{4} }{ {\bigg( \sqrt{4} \bigg)}^{7} }$

$\rm \: |z| = \dfrac{ 5 \times 2 }{ {2}^{7} }$

$\rm \: |z| = \dfrac{ 5 }{ {2}^{6} }$

$\rm \: |z| = \dfrac{ 5 }{64 }$

Hence,

$\rm\implies \:\boxed{\sf{  \:\rm \: |z| = \bigg |\dfrac{(3 - 4i)( \sqrt{3} - i)}{( \sqrt{3} + i)^{7} }\bigg| = \dfrac{ 5 }{64 } \: }}$

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Additional Information :-

Argument of complex number in different quadrants

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