Question

solve this in the form of a+ib:

$\frac{(3 + \iota \sqrt{5}) (3 - \iota \sqrt{5} )}{ (\sqrt{3} + \sqrt{2} \iota) - ( \sqrt{3} - \iota \sqrt{2} )}$
​

Answer 1

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

$\green{\therefore{\text{Complex\:number=}}\frac{ - 7 \iota}{ \sqrt{2} }}$

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

$\green{ \underline \bold{Given : }}\\ : \implies\frac{(3 + \iota \sqrt{5}) (3 - \iota \sqrt{5} )}{ (\sqrt{3} + \sqrt{2} \iota) - ( \sqrt{3} - \iota \sqrt{2} )} \\ \\ \red{ \underline \bold{To \: Find : }} \\ : \implies \frac{(3 + \iota \sqrt{5}) (3 - \iota \sqrt{5} )}{ (\sqrt{3} + \sqrt{2} \iota) - ( \sqrt{3} - \iota \sqrt{2} )} in \: form \: of \: a + \iota b = ?$

• According to given question :

$: \implies \frac{(3 + \iota \sqrt{5}) (3 - \iota \sqrt{5} )}{ (\sqrt{3} + \sqrt{2} \iota) - ( \sqrt{3} - \iota \sqrt{2} )} \\ \\: \implies \frac{ ({3})^{2} - (\iota \sqrt{5} )^{2} }{ \sqrt{3} + \sqrt{2} \iota - \sqrt{3} + \sqrt{2} \iota} \\ \\\bold{\circ \: \: {a}^{2} - {b}^{2} = (a + b)(a - b)} \\ \\ : \implies \frac{9 - { \iota}^{2} \times 5}{2 \sqrt{2} \iota } \\ \\ : \implies \frac{9 - ( - 1) \times 5 }{2 \sqrt{2} \iota} \\ \\ \bold{\circ \: \: { \iota}^{2} = - 1} \\ \\ : \implies \frac{9 + 5}{2 \sqrt{2} \iota } \times \frac{ - \sqrt{2} \iota }{ - \sqrt{2} \iota} \\ \\ \bold{ \circ \: \: Rationalising } \\ \\ : \implies \frac{ - 14 \sqrt{2} \iota }{ - 2 \times 2 { \iota}^{2} } \\ \\ : \implies \frac{ - 14 \sqrt{2} \iota }{ - 4 \times ( - 1)} \\ \\ : \implies \frac{ - 14 \sqrt{2} \iota }{4} \\ \\ : \implies \frac{ - 7 \iota}{ \sqrt{2} } \\ \\ \green{\therefore \text{Complex \: number }= 0 + \frac{ - 7 \iota}{ \sqrt{2} } } \\ \\ \green{\therefore \text{Re(z) = 0}} \\ \\ \green{ \therefore \text{Im(z) =} \frac{ - 7 \iota}{ \sqrt{2} } }$

Answer 2

QuesTion :-

$\star \: \frac{(3 + \iota \sqrt{5}) (3 - \iota \sqrt{5} )}{ (\sqrt{3} + \sqrt{2} \iota) - ( \sqrt{3} - \iota \sqrt{2} )} \:$

Answer :-

$\to \boxed{ a + \iota \: b = 0 + \frac{( - 7)}{ \sqrt{2} } \iota }\\ \\ \sf \: here \\ \to \boxed{ a = 0 \: \: \: and \: \: \: b = - \frac{7}{ \sqrt{2} } } \:$

Used Identity :-

→ a² - b ² = ( a - b )( a + b )

SoluTion :-

We have ,

$\mapsto \frac{(3 + \iota \sqrt{5}) (3 - \iota \sqrt{5} )}{ (\sqrt{3} + \sqrt{2} \iota) - ( \sqrt{3} - \iota \sqrt{2} )} \: \\ \\ \mapsto \frac{ {3}^{2} - {( \iota \sqrt{5}) }^{2} }{ (\sqrt{3} + \sqrt{2} \iota) - ( \sqrt{3} - \iota \sqrt{2} )} \\ \\ \mapsto \frac{ 9 - 5 { \iota}^{2} }{ (\sqrt{3} + \sqrt{2} \iota) - ( \sqrt{3} - \iota \sqrt{2} )} \\ \\ \because \: { \iota}^{2} = - 1 \\ \\ \mapsto \frac{ 9 + 5 }{ \sqrt{3} + \sqrt{2} \iota - \sqrt{3} + \iota \sqrt{2} } \: \\ \\ \mapsto \: \frac{14}{2 \iota \sqrt{2} } \\ \\ \mapsto \: \frac{7}{ \iota \sqrt{2} } \times \frac{ \iota \sqrt{2} }{ \iota \sqrt{2} } \\ \\ \mapsto \: \frac{7 \iota \sqrt{2} }{ {( \iota \sqrt{2} )}^{2} } \\ \\ \mapsto \: \frac{7 \iota \sqrt{2} }{2 { \iota}^{2} } \\ \\ \because \: { \iota}^{2} = - 1 \\ \\ \mapsto \: \frac{7 \iota \sqrt{2} }{ - 2} \times \frac{ \sqrt{2} }{ \sqrt{2} } \\ \\ \mapsto \: \frac{7 \iota \times 2}{ - 2 \sqrt{2} } \\ \\ \mapsto \: \frac{ - 7 \iota}{ \sqrt{2} } \: \sf \: (imaginary \: number) \\ \\ \rm \: hence \\ \\ \to \boxed{ a + \iota \: b = 0 + \frac{( - 7)}{ \sqrt{2} } \iota }\\ \\ \sf \: here \\ \to \boxed{ a = 0 \: and \: b = - \frac{7}{ \sqrt{2} } }$