Find the value to three places of decimals of each of the following. It is given that

Solution:

Given,

$\sqrt{2}=1.414, \sqrt{3}=1.732, \sqrt{5}=2.236, \sqrt{10}=3.162$

(i) $\frac{2}{\sqrt{3}}$

Rationalizing the denominator by multiplying both numerator and denominator with

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

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

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

$=\frac{2 \times 1.732}{3}$

$=\frac{3.464}{3}=1.154666666$

(ii) $\frac{3}{\sqrt{10}}$

Rationalizing the denominator by multiplying both numerator and denominator with $\sqrt{10}$

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

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

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

$=\frac{9.486}{10}=0.9486$

(iii) $\frac{\sqrt{5}+1}{\sqrt{2}}$

Rationalizing the denominator by multiplying both numerator and denominator with $\sqrt{2}$

$=\frac{(\sqrt{5} \times \sqrt{2})+\sqrt{2}}{\sqrt{2} \times \sqrt{2}}$

$=\frac{\sqrt{10}+\sqrt{2}}{2}$

$=\frac{4.576}{2}=2.288$

(iv) $\frac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}$

Rationalizing the denominator by multiplying both numerator and denominator with $\sqrt{2}$

$=\frac{(\sqrt{10} \times \sqrt{2})+(\sqrt{15} \times \sqrt{2})}{\sqrt{2} \times \sqrt{2}}$

$=\frac{\sqrt{20}+\sqrt{30}}{2}$

$=\frac{(\sqrt{10} \times \sqrt{2})+(\sqrt{10} \times \sqrt{3})}{2}$

$=\frac{(3.162 \times 1.414)+(3.162 \times 1.732)}{2}$

$=\frac{(4.471068)+(5.476584)}{2}$

$=\frac{9.947652}{2}$

$=4.973826$

(v) $\frac{2+\sqrt{3}}{3}$

$=\frac{2+1.732}{3}$

$=\frac{3.732}{3}$

$=1.244$

(vi) $\frac{\sqrt{2}-1}{\sqrt{5}}$

Rationalizing the denominator by multiplying both numerator and denominator with $\sqrt{5}$

$=\frac{(\sqrt{2} \times \sqrt{5})-\sqrt{5}}{\sqrt{5} \times \sqrt{5}}$

$=\frac{\sqrt{10}-\sqrt{5}}{5}$

$=\frac{3.162-2.236}{5}$

$=\frac{0.926}{5}=0.1852$