Question

your question refer to attachment

plz fast it's urgent ​

Answer 1

$\scriptsize\colorbox{lightyellow} {\text{ \bf♕ Brainliest answer }}$

$\rule{300pt}{0.1pt}$

given :-$\rm \: a = 5 - \sqrt{24} = 5 - 2 \sqrt{6}$

To find:-$\rm \: a + \dfrac{1}{a}$

Solution :-

$\begin{array}{l} \displaystyle \rm \frac{1}{a} = \frac{1}{5 - \sqrt{24} } \\ \\ \displaystyle \rm = \frac{1}{5 - 2 \sqrt{6} } \times \frac{5 +2 \sqrt{6} }{5 + 2 \sqrt{6} } \\ \\ \displaystyle \rm = \frac{5 + 2 \sqrt{6} }{ {5}^{2} -(2 \sqrt{6})^{2} } \\ \\ \displaystyle \rm = \frac{5+2\sqrt{6}}{25-{(2\sqrt{6})}^{2}} \cdot \frac{25+{(2\sqrt{6})}^{2}}{25+{(2\sqrt{6})}^{2}} \\ \\ \displaystyle \rm =\frac{125+120+50\sqrt{6}+48\sqrt{6}}{{25}^{2}-{({(2\sqrt{6})}^{2})}^{2}} \\ \\ \displaystyle \rm = \frac{245 + 98 \sqrt{6} }{625 - (2 \sqrt{6} )^{4} } \\ \\ \displaystyle \rm = \frac{49(5 + 2 \sqrt{6} )}{ {25}^{2} - {24}^{2} } \\ \\ \displaystyle \rm = \frac{49(5 + 2 \sqrt{6} )}{625 - 576} \\ \\ \displaystyle \rm = \frac{ \cancel{49}(5 + 2 \sqrt{6} )} {\cancel{49} } \\ \\ \boxed{\red{ \displaystyle \rm = 5 + 2 \sqrt{6} }}\end{array}$

$\\ \rm \therefore \: \: a + \frac{1}{a} = 5 - \sqrt{24} + 5 + 2 \sqrt{6}$

$= (5+5)+(-2\sqrt{6}+2\sqrt{6})=10$

Hence,

$\\ \boxed{\color{darkcyan} \rm a + \frac{1}{a} = 10}$