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Operations on Irrational Number

Express the f...

Question

Express the following with rational denominator:

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

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Solution

Given: $\frac{6 - 4 \sqrt{2}}{6 + 4 \sqrt{2}}$
Multiplying and dividing given number by
$6 - 4 \sqrt{2}$
$= \frac{(6 - 4 \sqrt{2}) (6 - 4 \sqrt{2})}{(6 + 4 \sqrt{2}) (6 - 4 \sqrt{2})}$
$= \frac{(6 - 4 \sqrt{2})^{2}}{6^{2} - (4 \sqrt{2})^{2}}$
Using identity $(a + b) (a - b) = a^{2} - b^{2}$
$= \frac{36 - 48 \sqrt{2} + 32}{36 - 32}$
$= \frac{68 - 48 \sqrt{2}}{4}$
$= \frac{4 (17 - 12 \sqrt{2})}{4}$
$= 17 - 12 \sqrt{2}$

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

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

Express each of the following as a rational number in the form $\frac{p}{q} :$

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Q. Add the rational expression:

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Q. Express the following with rational denominator:

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Operations on Irrational Number

Standard IX Mathematics

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