Find the value of:

Solution:

(i) We have:

$\frac{\sqrt{80}}{\sqrt{405}}=\sqrt{\frac{80}{405}}=\sqrt{\frac{16}{81}}=\frac{\sqrt{16}}{\sqrt{81}}=\frac{4}{9}$

(ii) Computing the square roots:

$\sqrt{441}=\sqrt{(3 \times 3) \times(7 \times 7)}=3 \times 7=21$

$\sqrt{625}=\sqrt{(5 \times 5) \times(5 \times 5)}=5 \times 5=25$

$\therefore$ $\frac{\sqrt{441}}{\sqrt{625}}=\frac{21}{25}$

(iii) We have:

$\frac{\sqrt{1587}}{\sqrt{1728}}=\sqrt{\frac{529}{576}}$       (by dividing both numbers by 3)

Computing the square roots of the numerator and the denominator:

$\sqrt{529}=\sqrt{23 \times 23}=23$

$\sqrt{576}=\sqrt{24 \times 24}=24$

$\therefore$$\frac{\sqrt{1587}}{\sqrt{1728}}=\frac{23}{24}$

(iv) We have:

$\sqrt{72} \times \sqrt{338}=\sqrt{72 \times 338}=\sqrt{2 \times 2 \times 2 \times 3 \times 3 \times 2 \times 13 \times 13}$

$=\sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 13 \times 13}=2 \times 2 \times 3 \times 13$

= 156

(v) We have:

$\sqrt{45} \times \sqrt{20}=\sqrt{3 \times 3 \times 5 \times 2 \times 2 \times 5}$

$=\sqrt{3} \times 3 \times 2 \times 2 \times 5 \times 5$

= 30