Question

Question :-

Answer 1

$\qquad\qquad\huge\underline{{\sf Answer}}$

let's solve ~

We have been given the value of x as :

$\qquad \sf \:x = \dfrac{1}{ \sqrt{7} + \sqrt{5} }$

Now, let's solve the problem ~

$\qquad \sf \dashrightarrow \:(x + \dfrac{1}{x} ) {}^{2}$

$\qquad \sf \dashrightarrow \: {x}^{2} + \dfrac{1}{x {}^{2} } + 2 \cdot x \cdot \dfrac{1}{x}$

$\qquad \sf \dashrightarrow \: {x}^{2} + \dfrac{1}{x {}^{2} } + 2$

Now plug in the values ~

$\qquad \sf \dashrightarrow \: {\bigg(\dfrac{1}{ \sqrt{7} + \sqrt{5} } \bigg) }^{2} + {( \sqrt{7 } + \sqrt{5} ) }^{2} + 2$

$\qquad \sf \dashrightarrow \: \dfrac{1}{ {7} + {5} + 2 \sqrt{7 \cdot5} } + {{7 } + {5} + 2 \sqrt{7 \cdot5} } + 2$

$\qquad \sf \dashrightarrow \: \dfrac{1}{ 12+ 2 \sqrt{35} } + 14 + 2 \sqrt{35}$

$\qquad \sf \dashrightarrow \: \dfrac{168 + 24 \sqrt{35} + 28 \sqrt{35} + 140}{12 + 2 \sqrt{35} }$

$\qquad \sf \dashrightarrow \: \dfrac{308 + 52\sqrt{35} }{12 + 2 \sqrt{35} }$

$\qquad \sf \dashrightarrow \: \dfrac{2(154+ 26\sqrt{35}) }{2(6+ \sqrt{35} )}$

$\qquad \sf \dashrightarrow \: \dfrac{154+ 26\sqrt{35} }{6+ \sqrt{35} }$