Question

Root 11 - root 7 / root 11 + root 7 = a - b root 77
Pls answer fast

Answer 1

$\huge\sf\mathbb\color{black} \underline{\colorbox{white}{☯SoLuTiOn☯}}$

Step-by-step explanation:

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

$\frac{ (\sqrt{11} + \sqrt{7} {)}^{2} }{( { \sqrt{11} )}^{2} - {( \sqrt{7}) }^{2} } = a - b \sqrt{77}$

$\frac{11 + 7 + 2 \sqrt{77} }{11 - 7} = a - b \sqrt{77}$

$\frac{18 + 2 \sqrt{77} }{4} = a - b \sqrt{77}$

$\frac{9 + \sqrt{77} }{2} = a - b \sqrt{77}$

$\frac{9}{2} + \frac{ \sqrt{77} }{2} = a - b \sqrt{77}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: compare \: \: both \: \: sides$

$a \: = \frac{9}{2}$

$b \: = \frac{ - 1}{2}$

Answer 2

ANSWER:

Given:

$\:\:\:\:\bullet\:\:\:\:\dfrac{\sqrt{11}-\sqrt7}{\sqrt{11}+\sqrt7}=a-b\sqrt{77}$

To Find:

• Value of a and b.

Solution:

$\text{We are given that,}\\\\:\longrightarrow\dfrac{\sqrt{11}-\sqrt7}{\sqrt{11}+\sqrt7}=a-b\sqrt{77}\\\\\text{Simplifying LHS,}\\\\:\implies\dfrac{\sqrt{11}-\sqrt7}{\sqrt{11}+\sqrt7}\\\\\text{On rationalizing the denominator,}\\\\:\implies\dfrac{\sqrt{11}-\sqrt7}{\sqrt{11}+\sqrt7}\times\dfrac{\sqrt{11}-\sqrt7}{\sqrt{11}-\sqrt7}\\\\:\implies\dfrac{(\sqrt{11}-\sqrt7)\times(\sqrt{11}-\sqrt7)}{(\sqrt{11}+\sqrt7)\times(\sqrt{11}-\sqrt7)}\\\\\text{We know that,}\\\\:\hookrightarrow(x+y)(x-y)=x^2-y^2\\\\\text{So,}\\\\:\implies\dfrac{(\sqrt{11}-\sqrt7)^2}{(\sqrt{11})^2-(\sqrt7)^2}$

$\text{We know that,}\\\\:\hookrightarrow(x-y)^2=x^2-2xy+y^2\\\\\text{So,}\\\\:\implies\dfrac{(\sqrt{11})^2-2(\sqrt{11})(\sqrt7)+(\sqrt7)^2}{11-7}\\\\:\implies\dfrac{11-2\sqrt{77}+7}{4}\\\\:\implies\dfrac{18-2\sqrt{77}}{4}\\\\:\implies\dfrac{2\!\!\!/(9-\sqrt{77})}{4\!\!\!/_{\:2}}\\\\:\implies\dfrac{9-\sqrt{77}}{2}\\\\\text{So,}\\\\:\implies\dfrac{9-\sqrt{77}}{2}=a-b\sqrt{77}\\\\:\implies\dfrac{9}{2}-\dfrac{\sqrt{77}}{2}=a-b\sqrt{77}\\\\\text{Hence, on comparing the like terms,}\\\\\bf{:\implies a=\dfrac{9}{2}\:\:and\:\:b=\dfrac{1}{2}}$

Formulae Used:

• (x+y)(x-y)=x^2 - y^2
• (x-y)^2=x^2 - 2xy + y^2