Given that $\sqrt2, = 1.414, \sqrt3 = 1.732, \sqrt5 = 2.236$ and $\sqrt7 = 2.646$, find the square roots of the following:
(i) $ \frac{196}{75} $
(ii) $ \frac{400}{63} $
(iii) $ \frac{150}{7} $
(iv) $ \frac{256}{5} $
(v) $ \frac{27}{50} $

To do: 

We have to find the square roots of the given numbers.

Solution:

(i) $\sqrt{\frac{196}{75}}=\sqrt{\frac{196}{25 \times 3}}$

$=\frac{\sqrt{196}}{\sqrt{25 \times 3}}$

$=\frac{\sqrt{196}}{5 \sqrt{3}}$

$=\frac{14 \times \sqrt{3}}{5 \sqrt{3} \times \sqrt{3}}$

$=\frac{14 \sqrt{3}}{5 \times 3}$

$=\frac{14 \sqrt{3}}{15}$

$=\frac{14 \times 1.732}{15}$

$=\frac{24.248}{15}$

$=1.6165$

$=1.617$

(ii) $\sqrt{\frac{400}{63}}=\sqrt{\frac{400}{9 \times 7}}$

$=\frac{20}{3 \sqrt{7}}$

$=\frac{20 \times \sqrt{7}}{3 \times \sqrt{7} \times \sqrt{7}}$

$=\frac{20 \sqrt{7}}{3 \times 7}$

$=\frac{20 \sqrt{7}}{21}$

$=\frac{20 \times 2.646}{21}$

$=\frac{52.92}{21}$

$=2.52$

(iii) $\sqrt{\frac{150}{7}}=\frac{\sqrt{150}}{\sqrt{7}}$

$=\frac{\sqrt{150} \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}}$

$=\frac{\sqrt{150} \times \sqrt{7}}{7}$

$=\frac{\sqrt{25 \times 2 \times 3 \times \sqrt{7}}}{7}$

$=\frac{5 \times \sqrt{2} \times \sqrt{3} \times \sqrt{7}}{7}$

$=\frac{5 \times(1.414)(1.732)(2.646)}{7}$

$=\frac{5 \times 6.4801}{7}$

$=\frac{32.4009}{7}$

$=4.6287$

$=4.629$

(iv) $\sqrt{\frac{256}{5}}=\frac{\sqrt{256}}{\sqrt{5}}$

$=\frac{16}{\sqrt{5}}$

$=\frac{16\times\sqrt5}{\sqrt{5}\times\sqrt5}$

$=\frac{16\times2.236}{5}$

$=\frac{35.776}{5}$

$=7.155$

(v) $ \sqrt{\frac{27}{50}}=\frac{\sqrt{27}}{\sqrt{50}}$

$=\sqrt{\frac{9 \times 3}{25 \times 2}}$

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

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

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

$=\frac{3 \times \sqrt{3} \times \sqrt{2}}{10}$

$=\frac{3 \times(1.732)(1.414)}{10}$

$=\frac{3 \times 2.449048}{10}$

$=\frac{7.3471}{10}$

$=0.7347$

$=0.735$

Tutorialspoint

Updated on: 10-Oct-2022

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