Question

If x=9 + 4 square root of 5, find the value of square root of x - 1/root x

Answer 1

$$\begin{gathered} \mathrm{x}=9+4\sqrt{5}=5+4+4\sqrt{5}={\left(\sqrt{5}\right)}^2+{\left(2\right)}^2+2\left(2\right)\left(\sqrt{5}\right)={\left(\sqrt{5}+2\right)}^2 \\ \mathrm{Now},\sqrt{\mathrm{x}}=\sqrt{{\left(\sqrt{5}+2\right)}^2}=\sqrt{5}+2 \\ \mathrm{Now},\frac{1}{\sqrt{\mathrm{x}}}=\frac{1}{\sqrt{5}+2}=\frac{1}{\sqrt{5}+2}\times \frac{\sqrt{5}-2}{\sqrt{5}-2}=\frac{\sqrt{5}-2}{5-4}=\sqrt{5}-2 \\ \mathrm{Now},\sqrt{\mathrm{x}}-\frac{1}{\sqrt{\mathrm{x}}}=\sqrt{5}+2-\sqrt{5}+2=4 \end{gathered}$$