Question

(4 root 2 /15 root -3 root 2) + (3 root 5 / root 10 -root 3) + (5 root 5 / 6 + root 5) rationalisation of denominator​

Answer 1

Answer:

⇒ $\frac{1911 + 465\sqrt{2} - 210\sqrt{5} +868\sqrt{6} + 93\sqrt{15}}{217}$

Step-by-step explanation:

Given :

To rationalise the denominator of :

$\frac{4\sqrt{2}}{\sqrt{3} - \sqrt{2}} + \frac{3\sqrt{5}}{\sqrt{10} - \sqrt{3} } + \frac{5\sqrt{5} }{6+\sqrt{5}}$

Solution :

$\frac{4\sqrt{2}}{\sqrt{3} - \sqrt{2}} + \frac{3\sqrt{5}}{\sqrt{10} - \sqrt{3} } + \frac{5\sqrt{5} }{6+\sqrt{5}}$

⇒ $(\frac{4\sqrt{2}}{\sqrt{3} - \sqrt{2}}) + (\frac{3\sqrt{5}}{\sqrt{10} - \sqrt{3} }) + (\frac{5\sqrt{5} }{6+\sqrt{5}})$

__

$\frac{4\sqrt{2}}{\sqrt{3} - \sqrt{2}}$

⇒ $\frac{4\sqrt{2}}{\sqrt{3} - \sqrt{2}} \times \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}}$

⇒ $\frac{4\sqrt{2}(\sqrt{3} + \sqrt{2})}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{4\sqrt{2}(\sqrt{3} + \sqrt{2})}{3 - 2} = \frac{4\sqrt{2}(\sqrt{3} + \sqrt{2})}{1} =4\sqrt{2}(\sqrt{3} + \sqrt{2})$

__

$\frac{3\sqrt{5}}{\sqrt{10} - \sqrt{3} }$

⇒ $\frac{3\sqrt{5}}{\sqrt{10} - \sqrt{3} } \times \frac{\sqrt{10} + \sqrt{3} }{\sqrt{10} + \sqrt{3} }$

⇒ $\frac{3\sqrt{5}(\sqrt{10} + \sqrt{3})}{(\sqrt{10})^2 - (\sqrt{3})^2} = \frac{3\sqrt{5}(\sqrt{10} + \sqrt{3})}{10 - 3} = \frac{3\sqrt{5}(\sqrt{10} + \sqrt{3})}{7}$

__

$\frac{5\sqrt{5} }{6+\sqrt{5}}$

⇒ $\frac{5\sqrt{5} }{6+\sqrt{5}} \times \frac{6-\sqrt{5}}{6-\sqrt{5}}$

⇒ $\frac{5\sqrt{5}(6-\sqrt{5})}{6^2-(\sqrt{5})^2} = \frac{5\sqrt{5}(6-\sqrt{5})}{36-5} = \frac{5\sqrt{5}(6-\sqrt{5})}{31}$

__

$\frac{4\sqrt{2}}{\sqrt{3} - \sqrt{2}} + \frac{3\sqrt{5}}{\sqrt{10} - \sqrt{3} } + \frac{5\sqrt{5} }{6+\sqrt{5}} = 4\sqrt{2}(\sqrt{3} + \sqrt{2}) + \frac{3\sqrt{5}(\sqrt{10} + \sqrt{3})}{7} - \frac{5\sqrt{5}(6-\sqrt{5})}{31}$

⇒ $4\sqrt{2}(\sqrt{3} + \sqrt{2}) + \frac{3\sqrt{5}(\sqrt{10} + \sqrt{3})}{7} - \frac{5\sqrt{5}(6-\sqrt{5})}{31}$

⇒ $\frac{4\sqrt{2}(\sqrt{3} + \sqrt{2}) \times 31 \times 7}{31 \times 7} + \frac{3\sqrt{5}(\sqrt{10} + \sqrt{3})\times 31}{7 \times 31} - \frac{5\sqrt{5}(6-\sqrt{5}) \times 7}{31 \times 7}$

⇒ $\frac{868\sqrt{2}(\sqrt{3} + \sqrt{2})}{217} + \frac{93\sqrt{5}(\sqrt{10} + \sqrt{3})}{217} - \frac{35\sqrt{5}(6-\sqrt{5})}{217}$

⇒ $\frac{868\sqrt{2}(\sqrt{3} + \sqrt{2}) + 93\sqrt{5}(\sqrt{10} + \sqrt{3}) - 35\sqrt{5}(6-\sqrt{5})}{217}$

⇒ $\frac{868\sqrt{6} + 1736 + 93\sqrt{15} + 465\sqrt{2} - 210\sqrt{5}+175}{217}$

⇒ $\frac{1911 + 465\sqrt{2} - 210\sqrt{5} +868\sqrt{6} + 93\sqrt{15}}{217}$