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

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

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

$(i) 6^{- 1} (i i) (- 7)^{- 1} (i i i) {(\frac{1}{4})}^{- 1} (i v) (- 4)^{- 1} \times {(\frac{- 3}{2})}^{- 1} (v) {(\frac{3}{5})}^{- 1} \times {(\frac{5}{2})}^{- 1}$

Q. Add the rational expression:

\frac{n}{n^{2} - 4} + \frac{2}{3 n + 6}

Q. Express the following with rational denominator:

(i i) \frac{1}{\sqrt{6} - \sqrt{5}}

Q. Simplify the following by rationalizing the denominator.

\frac{6 - 4 \sqrt{2}}{6 + 4 \sqrt{2}}

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Operations on Irrational Number

Standard IX Mathematics

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Express the following with rational denominator: (vii) 6−4√26+4√2

Given: 6−4√26+4√2 Multiplying and dividing given number by 6−4√2 =(6−4√2)(6−4√2)(6+4√2)(6−4√2) =(6−4√2)262−(4√2)2 Using identity (a+b)(a−b)=a2−b2 =36−48√2+3236−32 =68−48√24 =4(17−12√2)4 =17−12√2

Express the following with rational denominator:

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