Question

If $\sqrt{2}=1.41\mathrm{and}\sqrt{5}=2.24$, find the value of $\frac{3}{\left(8\sqrt{2}+5\sqrt{5}\right)}+\frac{2}{\left(8\sqrt{2}-5\sqrt{5}\right)}$.

Answer 1

$$\begin{gathered} \frac{3}{\left(8\sqrt{2}+5\sqrt{5}\right)}+\frac{2}{\left(8\sqrt{2}-5\sqrt{5}\right)}=\frac{3}{\left(8\sqrt{2}+5\sqrt{5}\right)}\times \frac{\left(8\sqrt{2}-5\sqrt{5}\right)}{\left(8\sqrt{2}-5\sqrt{5}\right)}+\frac{2}{\left(8\sqrt{2}-5\sqrt{5}\right)}\times \frac{\left(8\sqrt{2}+5\sqrt{5}\right)}{\left(8\sqrt{2}+5\sqrt{5}\right)} \\ =\frac{3\times \left(8\sqrt{2}-5\sqrt{5}\right)}{64\times 2-25\times 5}+\frac{2\times \left(8\sqrt{2}+5\sqrt{5}\right)}{64\times 2-25\times 5} \\ =\frac{3\times \left(8\sqrt{2}-5\sqrt{5}\right)}{3}+\frac{2\times \left(8\sqrt{2}+5\sqrt{5}\right)}{3} \\ =\left(8\sqrt{2}-5\sqrt{5}\right)+\frac{2\times \left(8\sqrt{2}+5\sqrt{5}\right)}{3} \\ =8\sqrt{2}+\frac{2}{3}8\sqrt{2}+\frac{2}{3}5\sqrt{5}-5\sqrt{5} \\ =8\sqrt{2}\left(1+\frac{2}{3}\right)+5\sqrt{5}\left(\frac{2}{3}-1\right) \\ =8\sqrt{2}\left(\frac{5}{3}\right)+5\sqrt{5}\left(\frac{-1}{3}\right) \\ =\frac{40}{3}\left(1.41\right)-\frac{5}{3}\left(2.24\right) \\ =18.8-3.73=15.07 \end{gathered}$$