Question

find the value of each of the following correct to 3 places of decimals, it being given that root 2 = 1.4142 , root 3 = 1.732, root 5 = 2.2360, root 6 = 2.4495 & root 10 = 3.162 . i) 3 - root 5 / 3 + 2 root 5, ii) 1 + root 2 / 3 - 2 root 2

Answer 1

Dear Student,

$$\begin{gathered} \left(i\right)\frac{3-\sqrt{5}}{3+2\sqrt{5}} \\ =\frac{3-\sqrt{5}}{3+2\sqrt{5}}\times \frac{3-2\sqrt{5}}{3-2\sqrt{5}} \\ =\frac{9-3\sqrt{5}-6\sqrt{5}+10}{3^2-{\left(2\sqrt{5}\right)}^2} \\ =\frac{19-9\sqrt{5}}{9-20} \\ =\frac{9\sqrt{5}-19}{11} \\ =\frac{9\left(2.2360\right)-19}{11} \\ =\frac{20.124-19}{11} \\ =\frac{1.124}{11} \\ =0.102\overline{18}\approx 0.102 \\ \left(ii\right)\frac{1+\sqrt{2}}{3-2\sqrt{2}} \\ =\frac{1+\sqrt{2}}{3-2\sqrt{2}}\times \frac{3+2\sqrt{2}}{3+2\sqrt{2}} \\ =\frac{3+2\sqrt{2}+3\sqrt{2}+4}{3^2-{\left(2\sqrt{2}\right)}^2} \\ =\frac{7+5\sqrt{2}}{9-8} \\ =7+5\sqrt{2} \\ =7+5\left(1.4142\right) \\ =7+7.071 \\ =14.071 \end{gathered}$$