Question

Simplify:
(i) $\frac{\sqrt{59.29}-\sqrt{5.29}}{\sqrt{59.29}+\sqrt{5.29}}$
(ii) $\frac{\sqrt{0.2304}+\sqrt{0.1764}}{\sqrt{0.2304}-\sqrt{0.1764}}$

Answer 1

(i) We have:
$$\begin{gathered} \sqrt{59.29}=\sqrt{\frac{5929}{100}}=\frac{\sqrt{7\times 7\times 11\times 11}}{10}=\frac{7\times 11}{10}=7.7 \\ \sqrt{5.29}=\sqrt{\frac{529}{100}}=\frac{\sqrt{529}}{\sqrt{100}}=\frac{23}{10}=2.3 \\ \frac{\sqrt{59.29}-\sqrt{5.29}}{\sqrt{59.29}+\sqrt{5.29}}=\frac{7.7-2.3}{7.7+2.3}=\frac{5.4}{10}=0.54 \end{gathered}$$


(ii) We have:
$$\begin{gathered} \sqrt{0.2304}=\sqrt{\frac{2304}{10000}} \\ =\frac{\sqrt{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\times 3}}{\sqrt{10000}} \\ =\frac{2\times 2\times 2\times 2\times 3}{100} \\ =0.48 \\ \sqrt{0.1764}=\sqrt{\frac{1764}{10000}} \\ =\frac{\sqrt{2\times 2\times 3\times 3\times 7\times 7}}{\sqrt{10000}} \\ =\frac{2\times 3\times 7}{100} \\ =0.42 \\ \frac{\sqrt{0.2304}+\sqrt{0.1764}}{\sqrt{0.2304}-\sqrt{0.1764}}=\frac{0.48+0.42}{0.48-0.42}=\frac{0.9}{0.06}=15 \end{gathered}$$