Question

$$\begin{gathered} 20.Ifx=3+2\sqrt{2},findthevalueof\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) \\ 21.Ifa=2+\sqrt{3},findthevalueofa+\frac{1}{a}and{a}^2+\frac{1}{a^2} \end{gathered}$$

Answer 1

Dear student
$$\begin{gathered} 21.\mathrm{Given}:\mathrm{a}=2+\sqrt{3} \\ \mathrm{Consider}, \\ \frac{1}{\mathrm{a}}=\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}=\frac{2-\sqrt{3}}{2^2-{\left(\sqrt{3}\right)}^2}=\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3} \\ \mathrm{So},\mathrm{a}+\frac{1}{\mathrm{a}}=2+\sqrt{3}+2-\sqrt{3}=4 \\ \mathrm{i}.\mathrm{e}\mathrm{a}+\frac{1}{\mathrm{a}}=4 \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides}\mathrm{we}\mathrm{get} \\ {\left(\mathrm{a}+\frac{1}{\mathrm{a}}\right)}^2=4^2 \\ {\mathrm{a}}^2+\frac{1}{{\mathrm{a}}^2}+2\times \mathrm{a}\times \frac{1}{\mathrm{a}}=16 \\ {\mathrm{a}}^2+\frac{1}{{\mathrm{a}}^2}=16-2=14 \\ \mathrm{Note}: \\ {\left(\mathrm{x}+\mathrm{y}\right)}^2={\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{xy} \end{gathered}$$
For remaining queries we request you to post them in separate threads to have rapid assistance from our experts.
Regards