Question

Evaluate
$\frac{15}{ \sqrt{10} + \sqrt{20 } + \sqrt{40} - \sqrt{5} - \sqrt{80} }$

given
$\sqrt{5 } = 2.2 \\ and \\ \sqrt{10} = 3.2$

Answer 1

$\frac{15}{ \sqrt{10} + \sqrt{20 } + \sqrt{40} - \sqrt{5} - \sqrt{80} }\\ = \frac{15}{ \sqrt{10} + 2 \sqrt{5} + 2\sqrt{10} - \sqrt{5} - 4 \sqrt{5} } \\ = \frac{15}{3 \sqrt{10} - 3 \sqrt{5} } \\ = \frac{(3)(5)}{3( \sqrt{10} - \sqrt{5} )} \\ = \frac{5}{ \sqrt{10} - \sqrt{5} }$
Rationalize the denominator
$\frac{5}{ \sqrt{10} - \sqrt{5} } \times \frac{ \sqrt{10} + \sqrt{5} }{ \sqrt{10} + \sqrt{5} } \\ = \frac{5 (\sqrt{10} + \sqrt{5}) }{10 - 5} \\ = \frac{ 5(\sqrt{10} + \sqrt{5} )}{5} \\ = \sqrt{10} + \sqrt{5}$
Given, \sqrt{10} =3.2 and \sqrt{5}=2.2

Substitute for them and you'll get.
3.2 +2.2 = 5.4

Answer 2

Given :-

$\sqrt{5} = 2.2 \\ \\ \sqrt{10} = 3.2$

To Find :-

$\frac{15}{ \sqrt{10} + \sqrt{20 } + \sqrt{40} - \sqrt{5} - \sqrt{80} } \: =$

__________________________

$\</strong><strong>r</strong><strong>e</strong><strong>d</strong><strong>{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

=>

$\frac{15}{ \sqrt{10} + \sqrt{20 } + \sqrt{40} - \sqrt{5} - \sqrt{80} } \:$

$= \frac{15}{ \sqrt{10} + \sqrt{2 \times 2 \times 5} + \sqrt{2 \times 2 \times 2 \times 5} - \sqrt{5} - \sqrt{2 \times 2 \times 2 \times 2 \times 5} } \\ \\ = \frac{15}{ \sqrt{10} + 2 \sqrt{5} + 2 \sqrt{10} - \sqrt{5} - 4 \sqrt{5} } \\ \\ = \frac{15}{3 \sqrt{10} - 3 \sqrt{5} }$

$= \frac{15}{3( \sqrt{10} - \sqrt{5} )} \\ \\ = \frac{5}{ (\sqrt{10} - \sqrt{5} )}$

Now Put the value of √10 and √5 →

$= \frac{5}{3.2 - 2.2} \\ \\ = \frac{5}{1} \\ \\ \boxed{ = 5}$

Or By Rationalising

$\frac{5}{ (\sqrt{10} - \sqrt{5} )}$

$\frac{5}{ (\sqrt{10} - \sqrt{5} )} \times \frac{\sqrt{10} + \sqrt{5} )}{ (\sqrt{10} + \sqrt{5} )}$

$\frac{5\times(\sqrt{10} + \sqrt{5} )}{ 10-5 }$

$\frac{5\times(\sqrt{10} + \sqrt{5} )}{ 5 }$

$(\sqrt{10} + \sqrt{5} )$

Put the values of √10 and √5 -

= 3.2 + 2.2

= 5.4

$\rule{200}{1}$

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