Express the following with rational denominator:

$(v i) \frac{\sqrt{3} + 1}{2 \sqrt{2} - \sqrt{3}}$

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Solution

Given: $\frac{\sqrt{3} + 1}{2 \sqrt{2} - \sqrt{3}}$
Multiplying and dividing given number by
$2 \sqrt{2} + \sqrt{3}$
$= \frac{(\sqrt{3} + 1) (2 \sqrt{2} + \sqrt{3})}{(2 \sqrt{2} - \sqrt{3}) (2 \sqrt{2} + \sqrt{3})}$
$= \frac{2 \sqrt{6} + 3 + 2 \sqrt{2} + \sqrt{3}}{(2 \sqrt{2})^{2} - (\sqrt{3})^{2}}$
Using identity $(a + b) (a - b) = a^{2} - b^{2}$
$= \frac{2 \sqrt{6} + 3 + 2 \sqrt{2} + \sqrt{3}}{8 - 3}$
$= \frac{2 \sqrt{6} + 3 + 2 \sqrt{2} + \sqrt{3}}{5}$

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Similar questions

Express each one of the following with rational denominator :

(i) $\frac{1}{3 + \sqrt{2}}$

(ii) $\frac{1}{\sqrt{6} - \sqrt{5}}$

(iii) $\frac{16}{\sqrt{41} - 5}$

(iv) $\frac{30}{5 \sqrt{3} - 3 \sqrt{5}}$

(v) $\frac{1}{2 \sqrt{5} + \sqrt{3}}$

(vi) $\frac{3 \sqrt{3} + 1}{2 \sqrt{2} - \sqrt{3}}$

(vii) $\frac{6 - 4 \sqrt{2}}{6 + 4 \sqrt{2}}$

(viii) $\frac{3 \sqrt{2} + 1}{2 \sqrt{5} - 3}$

(ix) $\frac{b^{2}}{\sqrt{a^{2} + b^{2} + a}}$

(v i i i) \frac{3 \sqrt{2} + 1}{2 \sqrt{5} - 3}

(i) \frac{1}{3 + \sqrt{2}}

\frac{\sqrt{2} . \sqrt[3]{3}}{\sqrt[3]{3} + \sqrt{2}}

(v) \frac{1}{2 \sqrt{5} - \sqrt{3}}

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