Question

If x = 9+4 square root 5.Find square root x - square root 1/x

Answer 1

$$\begin{gathered} \mathrm{x}=9+4\sqrt{5} \\ \Rightarrow \frac{1}{\mathrm{x}}=\frac{1}{9+4\sqrt{5}}\times \frac{9-4\sqrt{5}}{9-4\sqrt{5}}=\frac{9-4\sqrt{5}}{81-80}=9-4\sqrt{5} \\ \mathrm{Now},\mathrm{x}+\frac{1}{\mathrm{x}}=9+4\sqrt{5}+9-4\sqrt{5} \\ \mathrm{Now},\mathrm{x}+\frac{1}{\mathrm{x}}=18 \\ \mathrm{Now},{\left(\sqrt{\mathrm{x}}-\frac{1}{\sqrt{\mathrm{x}}}\right)}^2=\mathrm{x}+\frac{1}{\mathrm{x}}-2=18-2=16 \\ \Rightarrow \boxed{\sqrt{\mathbf{x}}\mathbf{}\mathbf{-}\mathbf{}\frac{\mathbf{1}}{\sqrt{\mathbf{x}}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\pm }\mathbf{4}} \end{gathered}$$