Question

(3+i√3) (3-i√5)÷(√3+√2i) (√3-√2i​

Answer 1

$\bf \huge{ \underline{ \underline{ \red{Correct \: \: Question:-}}}}$

$\sf \large \: \frac{(3 + i \sqrt{5} )(3 - i \sqrt{5}) }{( \sqrt{3} + \sqrt{2} i) - ( \sqrt{3} - i \sqrt{2} )}$

$\bf \huge{ \underline{ \underline{ \red{Answer:-}}}}$

$: \longmapsto\sf \large \: \frac{(3 + i \sqrt{5} )(3 - i \sqrt{5}) }{( \sqrt{3} + \sqrt{2} i) - ( \sqrt{3} - i \sqrt{2} )}$

$\sf \large \: we \: \: know \: \: that \: :$

$\boxed{ \huge{ \mathfrak{ \red{(a + b)(a - b) = {a}^{2} - {b}^{2} }}}}$

$: \longmapsto\sf \large \: \frac{ {(3)}^{2} - {(i \sqrt{5} })^{2} }{ \sqrt{3} + \sqrt{2} i - \sqrt{3} + i \sqrt{2} }$

$: \longmapsto\sf \large \: \frac{9 - {i}^{2} \times 5 }{ \sqrt{3} - \sqrt{3} + \sqrt{2} i + i \sqrt{2} }$

$\sf \underline { let \: \: us \: \: assume \: \: that \: \: {i}^{2} = - 1 \: }$

$: \longmapsto\sf \large \: \frac{9 - ( - 1) \times 5}{0 + \sqrt{2} i + i \sqrt{2} }$

$: \longmapsto\sf \large \: \frac{9 + 5}{ \sqrt{2} i + \sqrt{2}i }$

$: \longmapsto\sf \large \: \frac{14}{2 \sqrt{2} i}$

Rationalising the denominator:

$: \longmapsto\sf \large \: \frac{14}{2 \sqrt{2}i } \times \frac{ \sqrt{2}i }{ \sqrt{2}i }$

$: \longmapsto\sf \large \: \frac{28 \sqrt{2} i}{8 \times {i}^{2} }$

$: \longmapsto\sf \large \: \frac{28 \sqrt{2}i }{8 \times ( - 1)} \: \: \: \: ( \therefore \: \: {i}^{2} = - 1)$

$: \longmapsto\sf \large \: \frac{28 \sqrt{2} i}{ - 8}$

$: \longmapsto\sf \large \: \frac{ - 7 \sqrt{2} i}{2}$

$: \longmapsto\sf \large \: \frac{ - 7 \sqrt{2} }{2} i$

$: \longmapsto\sf \large \: 0 + \left ( \dfrac{ - 7 \sqrt{2} }{2} \right )i$