Question

Answer the $\sf{{25}^{th}}$ question from the attachment.


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Answer 1

Solution

$\rm \to Given \: that \: \sqrt{5} = 2.236 \: \: and \: \sqrt{10} = 3.162$

$\to \: \rm \: \: \dfrac{15}{ \sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80} }$

Now we can write as

$\to \: \rm \: \dfrac{15}{ \sqrt{10} + 2 \sqrt{5} + 2 \sqrt{10} - \sqrt{5} - 4 \sqrt{5} }$

$\rm \: \to \: \dfrac{15}{3 \sqrt{10} - 3 \sqrt{5} }$

Taking common 3 from denominator

$\rm \: \to \: \frac{ \not15}{ \not3( \sqrt{10} - \sqrt{5} )}$

We get

$\rm \: \to \: \dfrac{5}{ \sqrt{10} - \sqrt{5} }$

Now value are given

$\rm \to \sqrt{10} = 3.162$

$\rm \to \sqrt{5} = 2.236$

We get

$\rm \to \dfrac{5}{3.162 - 2.236}$

$\rm \to \dfrac{5}{0.926}$

Answer:-

$\rm \to 5.399$

Answer 2

$\red{\underline{{ \bf Question }}}$

• $\bf{ Evaluate \: \dfrac{15}{ \sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80} } }$

$\underline {\underline{{\purple{ \sf Solution }}}}$

$\sf{ \implies\dfrac{15}{ \sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80} } }$

$\sf{ \implies\dfrac{15}{ \sqrt{10} + \sqrt{4 \times 5} + \sqrt{4 \times 10} - \sqrt{5} - \sqrt{16 \times 5} } }$

$\sf{ \implies\dfrac{15}{ \sqrt{10} + 2 \sqrt{5} + 2\sqrt{ 10} - \sqrt{5} - 4\sqrt{5} } }$

$\sf{ \implies\dfrac{15}{ \sqrt{10} (1 + 2) + \sqrt{5} (2 - 1 - 4) } }$

$\sf{ \implies\dfrac{15}{ 3\sqrt{10} + \sqrt{5} ( - 3) } }$

$\sf{ \implies\dfrac{15}{ 3\sqrt{10} +3 \sqrt{5} } }$

$\sf{ \implies\dfrac{ \cancel{15}}{ \cancel{3}( \sqrt{10} + \sqrt{5} ) } }$

$\sf{ \implies\dfrac{5}{ \sqrt{10} + \sqrt{5} } }$

$\pink { \rm Now, rationalising \: the \: denominatior }$

$\sf{ \implies\dfrac{5}{ \sqrt{10} + \sqrt{5} } } \times \dfrac{ \sqrt{10} + \sqrt{5} }{ \sqrt{10} + \sqrt{5} }$

$\sf{ \implies\dfrac{5( \sqrt{10} + \sqrt{5} )}{ ( \sqrt{10} {)}^{2} - ( \sqrt{ {5}})^{2} } }$

$\sf{ \implies\dfrac{5( \sqrt{10} + \sqrt{5} )}{ ( 10 - 5 )} }$

$\sf{ \implies\dfrac{ \cancel{5}( \sqrt{10} + \sqrt{5} )}{ \cancel{5}} }$

$\sf{ \implies\ \sqrt{10} + \sqrt{5} }$

$\green{ \tt Therefore, Subsisting \: Values \: for \sqrt{5} \: and \sqrt{10} }$

$\sf{ \implies\ 3.162 + 2.236 }$

$\sf{ \implies\ 5.398 }$

Thus, the final answer is 5.398