Express each one of the following with rational denominator - i 1-3-2 ii 1-6 - -5 iii 16-41 - 5 iv 30-5 -3 - 3 -5 v 1-2 -5 - -3 vi 3 -3 - 1-2 -2 - -3 vii 6 - 4 -2-6 - 4 -2 viii 3 -2 - 1-2 -5 - 3 ix b2-a2 - b2 - a

The correct option is (i) 1/3+√(2) = 1(3 √(2))/(3+√(2))(3 √(2)) =3 √(2)/(3)^2 (√(2))^2 {∵ (a+b)(a b) = a^2 b^2} =3 √(2)/9 2=3 √(2)/7 (ii) 1/√(6) √(5) = ...

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Byju's Answer

Standard IX

Mathematics

Rationalisation

Express each ...

Question

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

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Solution

(i) $\frac{1}{3 + \sqrt{2}} = \frac{1 (3 - \sqrt{2})}{(3 + \sqrt{2}) (3 - \sqrt{2})} = \frac{3 - \sqrt{2}}{(3)^{2} - (\sqrt{2})^{2}} {∵ (a + b) (a - b) = a^{2} - b^{2}} = \frac{3 - \sqrt{2}}{9 - 2} = \frac{3 - \sqrt{2}}{7}$

(ii) $\frac{1}{\sqrt{6} - \sqrt{5}} = \frac{1 (\sqrt{6} + \sqrt{5})}{(\sqrt{6} - \sqrt{5}) (\sqrt{6} + \sqrt{5})} = \frac{\sqrt{6} + \sqrt{5}}{(\sqrt{6})^{2} - (\sqrt{5})^{2}} ∵ {(a + b) (a - b) = a^{2} - b^{2}} = \frac{\sqrt{6} + \sqrt{5}}{6 - 5} = \frac{\sqrt{6} + \sqrt{5}}{1} = \sqrt{6} + \sqrt{5}$

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

${M u l t i p l y i n g a n d d i v i d i n g b y (\sqrt{41} + 5)}$

$= \frac{16 (\sqrt{41} + 5)}{(\sqrt{41})^{2} - (5)^{2}}$

${∵ (a + b) (a - b) = a^{2} - b^{2}}$

$= \frac{16 (\sqrt{41} + 5)}{41 - 25} = \frac{16 (\sqrt{41} + 5)}{16}$

$= \sqrt{41} + 5$

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

{Multiplying and dividing by $(5 \sqrt{3} + 3 \sqrt{5})$ }

$= \frac{30 (5 \sqrt{3} + 3 \sqrt{5})}{(5 \sqrt{3})^{2} - (3 \sqrt{5})^{2}}$

$= \frac{30 (5 \sqrt{3} + 3 \sqrt{5})}{25 \times 3 - 9 \times 5} = \frac{30 (5 \sqrt{3} + 3 \sqrt{5})}{75 - 45}$

$= \frac{30 (5 \sqrt{3} + 3 \sqrt{5})}{30} = 5 \sqrt{3} + 3 \sqrt{5}$

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

{Multiplying and dividing by $(2 \sqrt{5} + \sqrt{3})$ }

$= \frac{2 \sqrt{5} + \sqrt{3}}{(2 \sqrt{5})^{2} - (\sqrt{3})^{2}} = \frac{2 \sqrt{5} + \sqrt{3}}{20 - 3}$

$= \frac{2 \sqrt{5} + \sqrt{3}}{17}$

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

{Multiplying and dividing by $(2 \sqrt{2} + \sqrt{3})$ }

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

$= \frac{2 \sqrt{6} + 3 + 2 \sqrt{2} + \sqrt{3}}{8 - 3}$

$= \frac{2 \sqrt{6} + 2 \sqrt{2} + \sqrt{3} + 3}{5}$

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

{Multiplying and dividing by $6 - 4 \sqrt{2}$ }

$= \frac{(6 - 4 \sqrt{2})^{2}}{(6)^{2} - (4 \sqrt{2})^{2}}$

$= \frac{36 + 16 \times 2 - 2 \times 6 \times 4 \sqrt{2}}{36 - 32}$

$= \frac{36 + 32 - 48 \sqrt{2}}{4} = \frac{68 - 48 \sqrt{2}}{4}$

$= 17 - 12 \sqrt{2}$

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

{Multiplying and dividing by $(2 \sqrt{5} + 3)$ }

$= \frac{3 \sqrt{2} \times 2 \sqrt{5} + 3 \times 3 \sqrt{2} + 2 \sqrt{5} + 3}{(2 \sqrt{5})^{2} - (3)^{2}}$

$= \frac{6 \sqrt{10} + 9 \sqrt{2} + 2 \sqrt{5} + 3}{20 - 9}$

$= \frac{6 \sqrt{10} + 9 \sqrt{2} + 2 \sqrt{5} + 3}{11}$

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

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

{Dividing and multiplying by $(\sqrt{a^{2} + b^{2}} - a)$ }

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

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

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Similar questions

Q. 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{\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}

Add (i) $(2 \sqrt{3} - 5 \sqrt{2}) a n d (\sqrt{3} + 2 \sqrt{2})$

(ii) $(2 \sqrt{2} + 5 \sqrt{3} - 7 \sqrt{5}) a n d (3 \sqrt{3} - \sqrt{2} + \sqrt{5})$

(iii) $(\frac{2}{3} \sqrt{7} - \frac{1}{2} \sqrt{2} + 6 \sqrt{11}) a n d (\frac{1}{3} \sqrt{7} + \frac{3}{2} \sqrt{2} - \sqrt{11})$

Evaluate :

$(i) (3^{- 1} \times 9^{- 1}) \div 3^{- 2} (i i) (3^{- 1} \times 4^{- 1}) \div 6^{- 1} (i i i) (2^{- 1} + 3^{- 1})^{3} (i v) (3^{- 1} \div 4^{- 1})^{2} (v) (2^{2} + 3^{2}) \times {(\frac{1}{2})}^{2} (v i) (5^{2} - 3^{2}) \times {(\frac{2}{3})}^{- 3} (v i i) [{(\frac{1}{4})}^{- 3} - {(\frac{1}{3})}^{- 3}] \div {(\frac{1}{6})}^{- 3} (v i i i) {[{(- \frac{3}{4})}^{- 2}]}^{2} (i x) {{(\frac{3}{5})}^{- 2}}^{- 2} (x) (5^{- 1} \times 3^{- 1}) \div 6^{- 1}$

Q. Express the following rations in the simplest form.

1) 6 : 8

2) 21 : 24

3) 33 : 77

4) 25 : 125

5) \frac{2}{3} : \frac{3}{5}

6) \frac{4}{5} : \frac{3}{8}

7) 1 \frac{1}{2} : 4 \frac{1}{2}

8) \frac{1}{3} : 2

Q. Simplify each of the following by rationalizing the denominator

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

\frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}}

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

\frac{3 \sqrt{5} - \sqrt{7}}{3 \sqrt{3} + \sqrt{2}}

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Express each one of the following with rational denominator : (i) 13+√2 (ii) 1√6−√5 (iii) 16√41−5 (iv) 305√3−3√5 (v) 12√5+√3 (vi) 3√3+12√2−√3 (vii) 6−4√26+4√2 (viii) 3√2+12√5−3 (ix) b2√a2+b2+a