Simplify:

Solution:

(i) We have:;

$\sqrt{59.29}=\sqrt{\frac{5929}{100}}=\frac{\sqrt{7 \times 7 \times 11 \times 11}}{10}=\frac{7 \times 11}{10}=7.7$

$\sqrt{5.29}=\sqrt{\frac{529}{100}}=\frac{\sqrt{529}}{\sqrt{100}}=\frac{23}{10}=2.3$

$\frac{\sqrt{59.29}-\sqrt{5.29}}{\sqrt{59.29}+\sqrt{5.29}}=\frac{7.7-2.3}{7.7+2.3}=\frac{5.4}{10}=0.54$

(ii) We have:

$\sqrt{0.2304}=\sqrt{\frac{2304}{10000}}$

$=\frac{\sqrt{2} \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3}{\sqrt{10000}}$

$=\frac{2 \times 2 \times 2 \times 2 \times 3}{100}$

$=0.48$

$\sqrt{0.1764}=\sqrt{\frac{1764}{10000}}$

$=\frac{\sqrt{2 \times 2 \times 3 \times 3 \times 7 \times 7}}{\sqrt{10000}}$

$=\frac{2 \times 3 \times 7}{100}$

$=0.42$

$\frac{\sqrt{0.2304}+\sqrt{0.1764}}{\sqrt{0.2304}-\sqrt{0.1764}}=\frac{0.48+0.42}{0.48-0.42}=\frac{0.9}{0.06}=15$