Express the following with rational denominator:

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

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Solution

Given: $\frac{3 \sqrt{2} + 1}{2 \sqrt{5} - 3}$
Multiplying and dividing given number by
$2 \sqrt{5} + 3$
$= \frac{(3 \sqrt{2} + 1) (2 \sqrt{5} + 3)}{(2 \sqrt{5} - 3) (2 \sqrt{5} + 3)}$
$= \frac{6 \sqrt{10} + 9 \sqrt{2} + 2 \sqrt{5} + 3}{(2 \sqrt{5})^{2} - 3^{2}}$
Using identity $(a + b) (a - b) = a^{2} - b^{2}$
$= \frac{6 \sqrt{10} + 9 \sqrt{2} + 2 \sqrt{5} + 3}{20 - 9}$
$= \frac{6 \sqrt{10} + 9 \sqrt{2} + 2 \sqrt{5} + 3}{11}$

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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) \frac{\sqrt{3} + 1}{2 \sqrt{2} - \sqrt{3}}

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

(i v) \frac{30}{5 \sqrt{3} - 3 \sqrt{5}}

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

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