Question

Simplify abd express the result in the form of a+ib.
${(\frac{4 \iota^{3} - \iota}{2\iota+ 1} )}^2$
​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\therefore{\text{Complex\:number}=3+4\iota}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{ \underline \bold{Given : }} \\ : \implies {(\frac{4 \iota^{3} - \iota}{2\iota+ 1} )}^2 \\ \\ \red{ \underline \bold{To \: Find : }} \\ : \implies {(\frac{4 \iota^{3} - \iota}{2\iota+ 1} )}^2 \: in \: form \: of \: a + \iota b = ?$

• According to given question :

$: \implies{(\frac{4 \iota^{3} - \iota}{2\iota+ 1} )}^2 \\ \\ \bold{\circ \: \: { \iota}^{3} = - \iota} \\ \\ : \implies (\frac{4 \times - \iota - \iota}{2 \iota + 1} )^{2} \\ \\ : \implies (\frac{ - 5 \iota}{2 \iota + 1 })^{2} \\ \\ \circ \: \: (a - b)^{2} = {a}^{2} + {b}^{2} - ab \\ \\ : \implies \frac{25 { \iota}^{2} }{ {2}^{2} \times {i}^{2} + {1}^{2} +2 \times 2 \iota \times 1 } \\ \\ \bold{ \circ \: \: { \iota}^{2} = - 1} \\ \\ :\implies \frac{25 \times ( - 1)}{4\times (- 1) + 1 + 4 \iota} \\ \\ : \implies \frac{ - 25}{ - 3 + 4 \iota} \\ \\ \bold{\circ \: \: Rationalising} \\ \\ : \implies \frac{ 25}{3- 4 \iota} \times \frac{3+ 4 \iota}{3 + 4 \iota} \\ \\ : \implies \frac{75 + 100 \iota}{ {(3})^{2} - ( {4 \iota})^{2} } \\ \\ : \implies \frac{75 + 100 \iota}{9 - 16 \times ( - 1)} \\ \\ : \implies \frac{75}{25} + \frac{100 \iota}{25} \\ \\ \green{: \implies 3 + 4 \iota} \\ \\ \green{\therefore \text{ Complex \: number} = 3 + 4\iota} \\ \\ \green{\therefore \text{Re(z) = 3}} \\ \\ \green{\therefore \text{Im(z) }= 4 \iota}$

Answer 2

$\huge{\underline{\mathfrak{\pink{Solution}}}}$

$( { \frac{4 {i}^{3} - i }{2i + 1}) }^{2}$

$\mathbb \pink{as \: we \: know \: that \: {i}^{3} = - i}$

$\mathbb \blue{putting \: value \: of \: {i}^{3} \: in \: the \: question}$

${( \frac{ - 4i - i}{2i + 1}) }^{2}$

$( { \frac{ - 5i}{2i + 1}) }^{2}$

(a-b)² = a² + b² - 2ab

$\frac{25 {i}^{2} }{4 {i}^{2} + 1 - 4i }$

Value of i² = - 1

$\frac{-25i}{ - 3 - 4i}$

Multiplying and dividing by -3+4i

$\frac{-25i}{ - 3 - 4i} \times \frac{ - 3 + 4i}{ - 3 + 4i}$

$\frac{ - 75i + 100 {i}^{2} }{25}$

$\frac{ 75i + 100}{25}$

$\frac{ 100}{25} - \frac{75i}{25}$

→ 4 + 3i

Re(z) = 4

im(z) = 3