Question

please find the answers please fast​

Answer 1

Appropriate question :

$\dfrac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} + \sqrt{7} } = a - b \sqrt{77}$

Solution :

Rationalising the denominator of $\bf\dfrac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}$:

$\green{ : \implies \dfrac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} + \sqrt{7} } \times \bigg( \dfrac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} - \sqrt{7}} \bigg )}$

$\green{ : \implies \dfrac{ (\sqrt{11 } - \sqrt{7} )( \sqrt{11} - \sqrt{7} )}{( \sqrt{11} + \sqrt{7} )( \sqrt{11} - \sqrt{7}) } }$

Using the identity : $\bf (a-b)(a+b)=a^2-b^2$ and $\bf (a-b)^2=a^2+b^2-2ab$ :

$\orange{: \implies \dfrac{ {( \sqrt{11}) }^{2} + {( \sqrt{7} )}^{2} - 2( \sqrt{11} )( \sqrt{7} )}{ {( \sqrt{11}) }^{2} - {( \sqrt{7}) }^{2} } }$

$\orange{: \implies \dfrac{11 + 7 - 2 \sqrt{77} }{11 - 7} }$

$\pink{ : \implies \dfrac{18 - 2 \sqrt{77} }{4} }$

$\pink{: \implies \dfrac{2(9 - \sqrt{77} )}{2(2)} }$

$\blue{: \implies \dfrac{ \cancel2(9 - \sqrt{77}) }{ \cancel2(2)} }$

$\blue{: \implies \dfrac{9 - \sqrt{77} }{2} }$

When comparing this with RHS :

$\red{\bigstar} \: \: \bf a = \dfrac{9}{2} \\ \red{\bigstar} \: \: \bf b = \dfrac{1}{2}$