Question

Please explain how to solve this question given in the attachment

Answer 1

$\underline{\underline{\large\bf{Given\: Expression:-}}}$

$\red{➤}\:$$\sf \dfrac{ 7 + \sqrt{5} }{7 - \sqrt{5} } + \dfrac{ 7 - \sqrt{5} }{7 + \sqrt{5} } = a + \dfrac{7}{11} b \sqrt{5}$

$\underline{\underline{\large\bf{Solution:-}}}$

Solving RHS by rationalisation-

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{ 7 + \sqrt{5} }{7 - \sqrt{5} } + \dfrac{ 7 - \sqrt{5} }{7 + \sqrt{5} } \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{ 7 + \sqrt{5}(7 + \sqrt{5}) }{7 - \sqrt{5} (7 + \sqrt{5})} - \dfrac{ 7 - \sqrt{5}(7 - \sqrt{5}) }{7 + \sqrt{5}(7 - \sqrt{5}) } \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{ (7 + \sqrt{5})^2}{(7^2-(\sqrt{5})^2} - \dfrac{ (7 - \sqrt{5})^2 }{7^2 -( \sqrt{5})^2 } \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{ 7^2+(\sqrt{5})^2+2(7)(\sqrt{5})}{49-5} + \dfrac{ 7^2+(\sqrt{5})^2-2(7)(\sqrt{5}) }{49 -5 } \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{ 54+14 \sqrt{5}}{44} - \dfrac{ 54-14\sqrt{5} }{44} \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{ 54+14 \sqrt{5}-(54-14\sqrt{5})}{44} \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{ \cancel{54}+14 \sqrt{5}\cancel{-54}+14\sqrt{5}}{44} \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{14 \sqrt{5}+14\sqrt{5}}{44} \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{\cancel{28}\sqrt{5}}{\cancel{44}} \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad \sf \dfrac{7}{11} \sqrt{5} \\\end{gathered}$

Comparing with LHS-

$\begin{gathered}\\\quad\large\dag\quad \sf 0+(1) \dfrac{7}{11} \sqrt{5}= a + \dfrac{7}{11}b\sqrt{5}\\\end{gathered}$

$\quad\quad\green{ \underline { \boxed{ \sf{a=0}}}}\quad\red{ \underline { \boxed{ \sf{b=1}}}}$

Identities Used and more-

$\begin{gathered}\boxed{\sf{ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \: }} \\ \end{gathered}$

$\begin{gathered}\boxed{\sf{ {(a - b)}^{2} = {a}^{2} + {b}^{2} -2ab \: }} \\ \end{gathered}$

$\begin{gathered}\boxed{\sf{ {x}^{2} - {y}^{2} = (x + y)(x - y) \: }} \\ \end{gathered}$

$\begin{gathered}\boxed{\sf {(a + b)² = (a - b)² + 4ab\: }} \\ \end{gathered}$

$\begin{gathered}\boxed{\sf {(a - b)² = (a + b)² - 4ab\: }} \\ \end{gathered}$

$\begin{gathered}\boxed{\sf {(a + b)² + (a - b)² = 2(a² + b²)\: }} \\ \end{gathered}$

$\begin{gathered}\boxed{\sf{ (a + b)³ = a³ + b³ + 3ab(a + b)\: }} \\ \end{gathered}$

$\begin{gathered}\boxed{\sf {(a - b)³ = a³ - b³ - 3ab(a - b)\: }} \\ \end{gathered}$

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