Question

Rationalise the denomineter
$1 \ \div \sqrt[]{5} + \sqrt[]{3} + \sqrt[]{8}$
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Answer 1

Answer:

The answer is $\dfrac{5\sqrt 3+3\sqrt 5-2\sqrt{30}}{30}$

Step-by-step explanation:

$\dfrac{1}{\sqrt 5+\sqrt 3+\sqrt8}=\dfrac{1}{(\sqrt 5+\sqrt 3)+\sqrt8}\times \dfrac{(\sqrt 5+\sqrt 3)-\sqrt8}{(\sqrt 5+\sqrt 3)-\sqrt8}\\\\=\dfrac{\sqrt 5+\sqrt 3-\sqrt 8}{(\sqrt 5+\sqrt 3)^2-(\sqrt 8)^2}=\dfrac{\sqrt 5+\sqrt 3-\sqrt 8}{5+3+2\sqrt{15}-8}} \\\\=\dfrac{\sqrt 5+\sqrt 3-\sqrt 8}{2\sqrt{15}}} \times \dfrac{\sqrt {15}}{\sqrt {15}}\\\\=\dfrac{5\sqrt 3+3\sqrt 5-2\sqrt{30}}{30}$

Note : Rationalizing factor of $\sqrt 5+\sqrt 3+\sqrt 8$ can be taken as $\sqrt 5+\sqrt 3-\sqrt 8$