Question

Questions :
$\\ \sf \: (1). \: Simplify : \: \frac{3 \sqrt{3} - 2 \sqrt{5}}{3 \sqrt{3} + 2 \sqrt{5}} + \dfrac{ \sqrt{12}}{ \sqrt{5} - 3 }$
$\\ \sf \: (2). \: Simplify : \: \dfrac{2 \sqrt{3}}{ \sqrt{3} - \sqrt{2}} + \dfrac{2 \sqrt{3}}{ \sqrt{3} + \sqrt{2}}$

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Answer 1

Answer:

Step-by-step explanation:

1)

$\dfrac{3\sqrt{3}-2\sqrt{5}}{3\sqrt{3}+2\sqrt{5}}+\dfrac{\sqrt{12}}{\sqrt{5}-3}$

Rationalizing the denominators :-

$\dfrac{3\sqrt{3}-2\sqrt{5}}{3\sqrt{3}+2\sqrt{5}}\times\dfrac{3\sqrt{3}-2\sqrt{5}}{3\sqrt{3}-2\sqrt{5}}+\dfrac{\sqrt{12}}{\sqrt{5}-3}\times\dfrac{\sqrt{5}+3}{\sqrt{5}+3}}$

$\dfrac{(3\sqrt{3}-2\sqrt{5})^{2}}{(3\sqrt{3})^2-(2\sqrt{5})^2}+\dfrac{2\sqrt{3}(\sqrt{5}-3)}{(\sqrt{5})^2-(3)^2}$

$\dfrac{27-12\sqrt{15}+20}{27-20}+\dfrac{2\sqrt{15}+6\sqrt{3}}{5-9}$

$\dfrac{47-12\sqrt{15}}{7}+\dfrac{2(\sqrt{15}+3\sqrt{3})}{-4}$

$\dfrac{47-12\sqrt{15}}{7}-\dfrac{\sqrt{15}+3\sqrt{3}}{2}$

L.C.M of 7,2 = 14

$\dfrac{94-24\sqrt{15}}{14}-\dfrac{7\sqrt{15}+21\sqrt{3}}{14}$

$\dfrac{94-24\sqrt{15}-7\sqrt{15}-21\sqrt{3}}{14}$

$\dfrac{94-31\sqrt{15}-21\sqrt{3}}{14}$

2) $\dfrac{2\sqrt{3}}{\sqrt{3}-\sqrt{2}}+\dfrac{2\sqrt{3}}{\sqrt{3}+\sqrt{2}}$

Taking L.C.M :-

$\dfrac{2\sqrt{3}\times(\sqrt{3}+\sqrt{2})+2\sqrt{3}(\sqrt{3}-\sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}$

$\dfrac{6+2\sqrt{6}+9-3\sqrt{6}}{3-2}$

$\dfrac{15-\sqrt{6}}{1}$

$= 15\sqrt{6}$