# RD Sharma solutions for Mathematics for Class 9 chapter 3 - Rationalisation [Latest edition]

## Solutions for Chapter 3: Rationalisation

Below listed, you can find solutions for Chapter 3 of CBSE RD Sharma for Mathematics for Class 9.

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4
Exercise 3.1 [Pages 2 - 3]

### RD Sharma solutions for Mathematics for Class 9 Chapter 3 Rationalisation Exercise 3.1 [Pages 2 - 3]

Exercise 3.1 | Q 1.1 | Page 2

Simplify of the following:

root(3)4  xx root(3)16

Exercise 3.1 | Q 1.2 | Page 2

Simplify of the following:

root(4)1250/root(4)2

Exercise 3.1 | Q 2.1 | Page 2

Simplify the following expressions:

(4 + sqrt7)(3 + sqrt2)

Exercise 3.1 | Q 2.2 | Page 2

Simplify the following expressions:

(3 + sqrt3)(5 - sqrt2)

Exercise 3.1 | Q 2.3 | Page 2

Simplify the following expressions:

(sqrt5 - 2)(sqrt3 - sqrt5)

Exercise 3.1 | Q 3.1 | Page 2

Simplify the following expressions:

(11 + sqrt11)(11 - sqrt11)

Exercise 3.1 | Q 3.2 | Page 2

Simplify the following expressions:

(5 + sqrt7)(5 - sqrt7)

Exercise 3.1 | Q 3.3 | Page 2

Simplify the following expressions:

(sqrt8 - sqrt2)(sqrt8 + sqrt2)

Exercise 3.1 | Q 3.4 | Page 2

Simplify the following expressions:

(3 + sqrt3)(3 - sqrt3)

Exercise 3.1 | Q 3.5 | Page 2

Simplify the following expressions:

(sqrt5 - sqrt2)(sqrt5 + sqrt2)

Exercise 3.1 | Q 4.1 | Page 3

Simplify the following expressions:

(sqrt3 + sqrt7)^2

Exercise 3.1 | Q 4.2 | Page 3

Simplify the following expressions:

(sqrt5 - sqrt3)^2

Exercise 3.1 | Q 4.3 | Page 3

Simplify the following expressions:

(2sqrt5 + 3sqrt2)^2

Exercise 3.2 [Pages 14 - 15]

### RD Sharma solutions for Mathematics for Class 9 Chapter 3 Rationalisation Exercise 3.2 [Pages 14 - 15]

Exercise 3.2 | Q 1.1 | Page 14

Rationalise the denominator of each of the following

3/sqrt5

Exercise 3.2 | Q 1.2 | Page 14

Rationalise the denominator of the following:

3/(2sqrt5)

Exercise 3.2 | Q 1.3 | Page 14

Rationalise the denominator of each of the following

1/sqrt12

Exercise 3.2 | Q 1.4 | Page 14

Rationalise the denominator of the following

sqrt2/sqrt5

Exercise 3.2 | Q 1.5 | Page 14

Rationalise the denominator of the following

(sqrt3 + 1)/sqrt2

Exercise 3.2 | Q 1.6 | Page 14

Rationalise the denominator of the following

(sqrt2 + sqrt5)/3

Exercise 3.2 | Q 1.7 | Page 14

Rationalise the denominator of the following

(3sqrt2)/sqrt5

Exercise 3.2 | Q 2.1 | Page 14

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

2/sqrt3

Exercise 3.2 | Q 2.2 | Page 14

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

3/sqrt10

Exercise 3.2 | Q 2.3 | Page 14

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

(sqrt5 + 1)/sqrt2

Exercise 3.2 | Q 2.4 | Page 14

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

(sqrt10 + sqrt15)/sqrt2



Exercise 3.2 | Q 2.5 | Page 14

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

(2 + sqrt3)/3

Exercise 3.2 | Q 2.6 | Page 14

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

(sqrt2 - 1)/sqrt5

Exercise 3.2 | Q 3.1 | Page 14

Express the following with rational denominator:

1/(3 + sqrt2)

Exercise 3.2 | Q 3.2 | Page 14

Express of the following with rational denominator:

1/(sqrt6 - sqrt5)

Exercise 3.2 | Q 3.3 | Page 14

Express the following with rational denominator:

16/(sqrt41 - 5)

Exercise 3.2 | Q 3.4 | Page 14

Express the following with rational denominator:

30/(5sqrt3 - 3sqrt5)

Exercise 3.2 | Q 3.5 | Page 14

Express the following with rational denominator:

1/(2sqrt5 - sqrt3)

Exercise 3.2 | Q 3.6 | Page 14

Express the following with rational denominator:

(sqrt3 + 1)/(2sqrt2 - sqrt3)

Exercise 3.2 | Q 3.7 | Page 14

Express the following with rational denominator:

(6 - 4sqrt2)/(6 + 4sqrt2)

Exercise 3.2 | Q 3.8 | Page 14

Express the following with rational denominator:

(3sqrt2 + 1)/(2sqrt5 - 3)

Exercise 3.2 | Q 3.9 | Page 14

Express each one of the following with rational denominator:

(b^2)/(sqrt(a^2 + b^2) + a)

Exercise 3.2 | Q 4.1 | Page 14

Rationales the denominator and simplify:

(3 - sqrt2)/(3 + sqrt2)

Exercise 3.2 | Q 4.2 | Page 14

Rationales the denominator and simplify:

(5 + 2sqrt3)/(7 + 4sqrt3)

Exercise 3.2 | Q 4.3 | Page 14

Rationales the denominator and simplify:

(1 + sqrt2)/(3 - 2sqrt2)

Exercise 3.2 | Q 4.4 | Page 14

Rationales the denominator and simplify:

(2sqrt6 - sqrt5)/(3sqrt5 - 2sqrt6)

Exercise 3.2 | Q 4.5 | Page 14

Rationales the denominator and simplify:

(4sqrt3 + 5sqrt2)/(sqrt48 + sqrt18)

Exercise 3.2 | Q 4.6 | Page 14

Rationales the denominator and simplify:

(2sqrt3 - sqrt5)/(2sqrt2 + 3sqrt3)

Exercise 3.2 | Q 5.1 | Page 14

Simplify:

(5 + sqrt3)/(5 - sqrt3) + (5 - sqrt3)/(5 + sqrt3)

Exercise 3.2 | Q 5.2 | Page 14

Simplify

1/(2 + sqrt3) + 2/(sqrt5 - sqrt3) + 1/(2 - sqrt5)

Exercise 3.2 | Q 5.3 | Page 14

Simplify

2/(sqrt5 + sqrt3) + 1/(sqrt3 + sqrt2) + 3/(sqrt5 + sqrt2)

Exercise 3.2 | Q 6.1 | Page 14

In the following determine rational numbers a and b:

(sqrt3 - 1)/(sqrt3 + 1) = a - bsqrt3

Exercise 3.2 | Q 6.2 | Page 14

In the following determine rational numbers a and b:

(4 + sqrt2)/(2 + sqrt2) = n - sqrtb

Exercise 3.2 | Q 6.3 | Page 14

In the following determine rational numbers a and b:

(3 + sqrt2)/(3 - sqrt2) = a + bsqrt2

Exercise 3.2 | Q 6.4 | Page 14

In the following determine rational numbers a and b:

(5 + 3sqrt3)/(7 + 4sqrt3) = a + bsqrt3

Exercise 3.2 | Q 6.5 | Page 14

In the following determine rational numbers a and b:

(sqrt11 - sqrt7)/(sqrt11 + sqrt7) = a - bsqrt77

Exercise 3.2 | Q 6.6 | Page 14

In the following determine rational numbers a and b:

(4 + 3sqrt5)/(4 - 3sqrt5) = a + bsqrt5

Exercise 3.2 | Q 7 | Page 15

Find the value of 6/(sqrt5 - sqrt3) it being given that sqrt3 = 1.732 and  sqrt5 = 2.236

Exercise 3.2 | Q 8.1 | Page 15

Find the values the following correct to three places of decimals, it being given that sqrt2 = 1.4142, sqrt3 = 1.732, sqrt5 = 2.2360, sqrt6 = 2.4495 and sqrt10 = 3.162

(3 - sqrt5)/(3 + 2sqrt5)

Exercise 3.2 | Q 8.2 | Page 15

Find the values the following correct to three places of decimals, it being given that sqrt2 = 1.4142, sqrt3 = 1.732, sqrt5 = 2.2360, sqrt6 = 2.4495 and sqrt10 = 3.162

(1 + sqrt2)/(3 - 2sqrt2)

Exercise 3.2 | Q 9.1 | Page 15

Simplify: $\frac{3\sqrt{2} - 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}} + \frac{\sqrt{12}}{\sqrt{3} - \sqrt{2}}$

Exercise 3.2 | Q 9.2 | Page 15

Simplify: $\frac{7 + 3\sqrt{5}}{3 + \sqrt{5}} - \frac{7 - 3\sqrt{5}}{3 - \sqrt{5}}$

Exercise 3.2 | Q 10 | Page 15

if x = 2 +  sqrt3,find the value of x^2 + 1/x^2

Exercise 3.2 | Q 11 | Page 15

if   x= 3 + sqrt8, find the value of x^2 + 1/x^2

Exercise 3.2 | Q 12 | Page 15

if x =  (sqrt3 + 1)/2 find the value of 4x^2 +2x^2 - 8x + 7

Exercise 3.3 [Page 16]

### RD Sharma solutions for Mathematics for Class 9 Chapter 3 Rationalisation Exercise 3.3 [Page 16]

Exercise 3.3 | Q 1 | Page 16

Write the value of $\left( 2 + \sqrt{3} \right) \left( 2 - \sqrt{3} \right) .$

Exercise 3.3 | Q 2 | Page 16

Write the reciprocal of $5 + \sqrt{2}$.

Exercise 3.3 | Q 3 | Page 16

Write the rationalisation factor of $7 - 3\sqrt{5}$.

Exercise 3.3 | Q 4 | Page 16

If$\frac{\sqrt{3} - 1}{\sqrt{3} + 1} = x + y\sqrt{3},$  find the values of and y.

Exercise 3.3 | Q 5 | Page 16

If x= $\sqrt{2} - 1$, then write the value of $\frac{1}{x} .$

Exercise 3.3 | Q 6 | Page 16

If $a = \sqrt{2} + 1$,then find the value of  $a - \frac{1}{a}$.

Exercise 3.3 | Q 7 | Page 16

If $x = 2 + \sqrt{3}$ ,  find the value of $x + \frac{1}{x}$.

Exercise 3.3 | Q 8 | Page 16

Write the rationalisation factor of $\sqrt{5} - 2$.

Exercise 3.3 | Q 9 | Page 16

Simplify $\sqrt{3 + 2\sqrt{2}}$.

Exercise 3.3 | Q 10 | Page 16

Simplify $\sqrt{3 - 2\sqrt{2}}$.

Exercise 3.3 | Q 11 | Page 16

If $x = 3 + 2\sqrt{2}$,then find the value of $\sqrt{x} - \frac{1}{\sqrt{x}}$.

Exercise 3.4 [Pages 16 - 18]

### RD Sharma solutions for Mathematics for Class 9 Chapter 3 Rationalisation Exercise 3.4 [Pages 16 - 18]

Exercise 3.4 | Q 1 | Page 16

$\sqrt{10} \times \sqrt{15}$ is equal to

• 5$\sqrt{6}$

• 6$\sqrt{5}$

• $\sqrt{30}$

• $\sqrt{25}$

Exercise 3.4 | Q 2 | Page 16

$\sqrt[5]{6} \times \sqrt[5]{6}$ is equal to

• $\sqrt[5]{36}$

• $\sqrt[5]{6 \times 0}$

• $\sqrt[5]{6}$

• $\sqrt[5]{12}$

Exercise 3.4 | Q 3 | Page 16

The rationalisation factor of $\sqrt{3}$ is

• $- \sqrt{3}$

• $\frac{1}{\sqrt{3}}$

• $2\sqrt{3}$

• $- 2\sqrt{3}$

Exercise 3.4 | Q 4 | Page 17

The rationalisation factor of $2 + \sqrt{3}$ is

• $2 - \sqrt{3}$

• $2 + \sqrt{3}$

• $\sqrt{2} - 3$

• $\sqrt{3} - 2$

Exercise 3.4 | Q 5 | Page 17

If x = $\sqrt{5} + 2$,then $x - \frac{1}{x}$ equals

• $2\sqrt{5}$

• 4

• 2

• $\sqrt{5}$

Exercise 3.4 | Q 6 | Page 17

If $\frac{\sqrt{3 - 1}}{\sqrt{3} + 1}$ =$a - b\sqrt{3}$ then

• a = 2, b =1

• a = 2, b =−1

• a = −2, b = 1

• a = b = 1

Exercise 3.4 | Q 7 | Page 17

The simplest rationalising factor of  $\sqrt[3]{500}$ is

• $\sqrt[3]{2}$

• $\sqrt[3]{5}$

• $\sqrt{3}$

• none of these

Exercise 3.4 | Q 8 | Page 17

The simplest rationalising factor of $\sqrt{3} + \sqrt{5}$ is

• $\sqrt{3} - 5$

• $3 - \sqrt{5}$

• $\sqrt{3} - \sqrt{5}$

• $\sqrt{3} + \sqrt{5}$

Exercise 3.4 | Q 9 | Page 17

The simplest rationalising factor of $2\sqrt{5}-$$\sqrt{3}$ is

• $2\sqrt{5} + 3$

• $2\sqrt{5} + \sqrt{3}$

• $\sqrt{5} + \sqrt{3}$

• $\sqrt{5} - \sqrt{3}$

Exercise 3.4 | Q 10 | Page 17

If x = $\frac{2}{3 + \sqrt{7}}$,then (x−3)2 =

• 1

• 3

• 6

• 7

Exercise 3.4 | Q 11 | Page 17

If $x = 7 + 4\sqrt{3}$ and xy =1, then $\frac{1}{x^2} + \frac{1}{y^2} =$

• 64

• 134

• 194

• 1/49

Exercise 3.4 | Q 12 | Page 17

If $x + \sqrt{15} = 4,$ then $x + \frac{1}{x}$ =

• 2

• 4

• 8

• 1

Exercise 3.4 | Q 13 | Page 17

If $x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ and $y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ then x + y +xy=

• 9

• 5

• 17

• 7

Exercise 3.4 | Q 14 | Page 17

If x= $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$ and y = $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ , then x2 + y +y2 =

• 101

• 99

• 98

• 102

Exercise 3.4 | Q 15 | Page 17

$\frac{1}{\sqrt{9} - \sqrt{8}}$ is equal to

• $3 + 2\sqrt{2}$

• $\frac{1}{3 + 2\sqrt{2}}$

• $3 - 2\sqrt{2}$

• $\frac{3}{2} - \sqrt{2}$

Exercise 3.4 | Q 16 | Page 17

The value of $\frac{\sqrt{48} + \sqrt{32}}{\sqrt{27} + \sqrt{18}}$ is

• $\frac{4}{3}$

• 4

• 3

• 3/4`

Exercise 3.4 | Q 17 | Page 17

If $\frac{5 - \sqrt{3}}{2 + \sqrt{3}} = x + y\sqrt{3}$ , then

•  x = 13, y = −7

• x = −13, y = 7

• x = −13, y =- 7

• x = 13, y = 7

Exercise 3.4 | Q 18 | Page 17

If x = $\sqrt[3]{2 + \sqrt{3}}$ , then $x^3 + \frac{1}{x^3} =$

• 2

• 4

• 8

• 9

Exercise 3.4 | Q 19 | Page 17

The value of $\sqrt{3 - 2\sqrt{2}}$ is

• $\sqrt{2} - 1$

• $\sqrt{2} + 1$

• $\sqrt{3} - \sqrt{2}$

• $\sqrt{3} + \sqrt{2}$

Exercise 3.4 | Q 20 | Page 18

The value of $\sqrt{5 + 2\sqrt{6}}$ is

• $\sqrt{3} - \sqrt{2}$

• $\sqrt{3} + \sqrt{2}$

• $\sqrt{5} + \sqrt{6}$

• none of these

Exercise 3.4 | Q 21 | Page 18

If $\sqrt{2} = 1 . 4142$ then $\sqrt{\frac{\sqrt{2} - 1}{\sqrt{2} + 1}}$ is equal to

• 0.1718

•  5.8282

•  0.4142

• 2.4142

Exercise 3.4 | Q 22 | Page 18

If $\sqrt{2} = 1 . 414,$ then the value of $\sqrt{6} - \sqrt{3}$ upto three places of decimal is

•  0.235

• 0.707

• 1.414

• 0.471

Exercise 3.4 | Q 23 | Page 18

The positive square root of $7 + \sqrt{48}$ is

• $7 + 2\sqrt{3}$

• $7 + \sqrt{3}$

• $\sqrt{3}+2$

• $3 + \sqrt{2}$

Exercise 3.4 | Q 24 | Page 18

If $x = \sqrt{6} + \sqrt{5}$,then $x^2 + \frac{1}{x^2} - 2 =$

• $2\sqrt{6}$

• $2\sqrt{5}$

• 24

• 20

Exercise 3.4 | Q 25 | Page 18

If $\sqrt{13 - a\sqrt{10}} = \sqrt{8} + \sqrt{5}, \text { then a } =$

• −5

• −6

• −4

• −2

## Solutions for Chapter 3: Rationalisation

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4

## RD Sharma solutions for Mathematics for Class 9 chapter 3 - Rationalisation

Shaalaa.com has the CBSE Mathematics Mathematics for Class 9 CBSE solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. RD Sharma solutions for Mathematics Mathematics for Class 9 CBSE 3 (Rationalisation) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

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Concepts covered in Mathematics for Class 9 chapter 3 Rationalisation are Introduction of Real Number, Concept of Irrational Numbers, Real Numbers and Their Decimal Expansions, Representing Real Numbers on the Number Line, Operations on Real Numbers, Laws of Exponents for Real Numbers.

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