Question

Simplify
$\frac{3\sqrt{5}}{\sqrt{10} -\sqrt{3}} + \frac{5\sqrt{3} }{\sqrt{6} +\sqrt{5}}$

Answer 1

Solution!!

$\sf \dfrac{3\sqrt{5}}{\sqrt{10}-\sqrt{3}}+\dfrac{5\sqrt{3}}{\sqrt{6}+\sqrt{5}}$

$\sf =\left(\dfrac{3\sqrt{5}}{\sqrt{10}-\sqrt{3}}\times \dfrac{\sqrt{10}+\sqrt{3}}{\sqrt{10}+\sqrt{3}}\right)+\dfrac{5\sqrt{3}}{\sqrt{6}+\sqrt{5}}$

$\sf =\left(\dfrac{(3\sqrt{5})(\sqrt{10}+\sqrt{3})}{(\sqrt{10}-\sqrt{3})(\sqrt{10}+\sqrt{3})}\right)+\dfrac{5\sqrt{3}}{\sqrt{6}+\sqrt{5}}$

$\sf =\left(\dfrac{(3\sqrt{5})(\sqrt{10}+\sqrt{3})}{(\sqrt{10})^{2}-(\sqrt{3})^{2}}\right)+\dfrac{5\sqrt{3}}{\sqrt{6}+\sqrt{5}}$

$\sf =\left(\dfrac{(3\sqrt{5})(\sqrt{10}+\sqrt{3})}{10-3}\right)+\dfrac{5\sqrt{3}}{\sqrt{6}+\sqrt{5}}$

$\sf =\left(\dfrac{(3\sqrt{5})(\sqrt{10}+\sqrt{3})}{7}\right)+\dfrac{5\sqrt{3}}{\sqrt{6}+\sqrt{5}}$

$\sf =\left(\dfrac{(3\sqrt{5})(\sqrt{10}+\sqrt{3})}{7}\right)+\left(\dfrac{5\sqrt{3}}{\sqrt{6}+\sqrt{5}}\times \dfrac{\sqrt{6}-\sqrt{5}}{\sqrt{6}-\sqrt{5}}\right)$

$\sf =\left(\dfrac{(3\sqrt{5})(\sqrt{10}+\sqrt{3})}{7}\right)+\left(\dfrac{(5\sqrt{3})(\sqrt{6}-\sqrt{5})}{(\sqrt{6}+\sqrt{5})(\sqrt{6}-\sqrt{5})}\right)$

$\sf =\left(\dfrac{(3\sqrt{5})(\sqrt{10}+\sqrt{3})}{7}\right)+\left(\dfrac{(5\sqrt{3})(\sqrt{6}-\sqrt{5})}{(\sqrt{6})^{2}-(\sqrt{5})^{2}}\right)$

$\sf =\left(\dfrac{(3\sqrt{5})(\sqrt{10}+\sqrt{3})}{7}\right)+\left(\dfrac{(5\sqrt{3})(\sqrt{6}-\sqrt{5})}{6-5}\right)$

$\sf =\left(\dfrac{(3\sqrt{5})(\sqrt{10}+\sqrt{3})}{7}\right)+(5\sqrt{3})(\sqrt{6}-\sqrt{5})$

$\sf =\left(\dfrac{3\sqrt{50}+3\sqrt{15}}{7}\right)+5\sqrt{18}-5\sqrt{15}$

$\sf =\dfrac{15\sqrt{2}+3\sqrt{15}}{7}+15\sqrt{2}-5\sqrt{15}$

$\sf =\dfrac{15\sqrt{2}+3\sqrt{15}+105\sqrt{2}-35\sqrt{15}}{7}$

$\sf =\dfrac{120\sqrt{2}-32\sqrt{15}}{7}$