Question

Find the value of 1 / root 3 - root 2 - 1 correct to three places of decimal if root 2=1.414 and root 6=2.45

Answer 1

$$\begin{gathered} \frac{1}{\sqrt{3}-\sqrt{2}-1} \\ =\frac{1}{\sqrt{3}-\sqrt{2}-1}\times \frac{\sqrt{3}+\sqrt{2}-1}{\sqrt{3}+\sqrt{2}-1} \\ =\frac{\sqrt{3}+\sqrt{2}-1}{{\left(\sqrt{3}\right)}^2-{\left(\sqrt{2}-1\right)}^2} \\ =\frac{\sqrt{3}+\sqrt{2}-1}{3-2-1+2\sqrt{2}} \\ =\frac{\sqrt{3}+\sqrt{2}-1}{2\sqrt{2}} \\ =\frac{\sqrt{3}+\sqrt{2}-1}{2\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}} \\ =\frac{\sqrt{6}+2-\sqrt{2}}{4} \\ =\frac{2.45+2-1.414}{4} \\ =\frac{3.036}{4} \\ =0.759 \end{gathered}$$