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Chapter 3: Rationalisation Exercise – 3.2

Quesiton: 1

Rationalize the denominator of each of the following:

Solution:

For rationalizing the denominator, multiply both numerator and denominator with √5

For rationalizing the denominator, multiply both numerator and denominator with √5

For rationalizing the denominator, multiply both numerator and denominator with √12

For rationalizing the denominator, multiply both numerator and denominator with √3

For rationalizing the denominator, multiply both numerator and denominator with √2

For rationalizing the denominator, multiply both numerator and denominator with 

For rationalizing the denominator, multiply both numerator and denominator with √5

Quesiton: 2

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

Solution:

Given,

Rationalizing the denominator by multiplying both numerator and denominator with

Rationalizing the denominator by multiplying both numerator and denominator with √10

Rationalizing the denominator by multiplying both numerator and denominator with √2

Rationalizing the denominator by multiplying both numerator and denominator with √2

Rationalizing the denominator by multiplying both numerator and denominator with√5

Quesiton: 3

Express each one of the following with rational denominator: 

Solution:

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know,

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know,

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know,

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a2 - b2)

As we know, (a - b)2 = (a2 – 2 × a × b + b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b)(a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

 As we know, (a - b)2 = (a2 – 2 × a × b + b2)

Quesiton: 4

Rationalize the denominator and simplify:

Solution:

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b)(a - b) = (a2 - b2)

As we know, (a - b)2 = (a2 – 2 × a × b + b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b)(a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b)(a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b)(a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b)(a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

Quesiton: 5

Simplify:

Solution:

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factorand the rationalizing factor

Now, (a + b)(a - b) = (a2 - b2)

As we know, (a - b)2 = (a2 – 2 × a × b + b2

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factorand the rationalizing factor

Now as we know, (a + b)(a - b)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factorand the rationalizing factor

Now as we know, (a + b)(a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factorthe rationalizing factorand the rationalizing factor

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor the rationalizing factorand the rationalizing factor

Since, (a + b) (a - b) = (a2 - b2)

Quesiton: 6

In each of the following determine rational numbers a and b:

Solution:

Given,

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a2 - b2)

On comparing the rational and irrational parts of the above equation, we get, a = 2 and b = 1

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b)(a - b)= (a2 - b2)

On comparing the rational and irrational parts of the above equation, we get, a = 3 and b = 2

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b)(a - b) = (a2 - b2)

On comparing the rational and irrational parts of the above equation, we get,

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b)(a - b)= (a2 - b2)

On comparing the rational and irrational parts of the above equation, we get, a = -1 and b = 1

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a2 - b2)

On comparing the rational and irrational parts of the above equation, we get, a = 92 and b = 12

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a2 - b2)

On comparing the rational and irrational parts of the above equation, we get,

Quesiton: 7

If x = 2+√3, find the value of

Solution:

Given, x = 2 + √3,

To find the value of

We have, x = 2 + √3,

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

Since, (a + b) (a - b) = (a2 - b2)

We know that, (a3 + b3) = (a + b)(a2 − ab + b2)

Putting the value of x+1x in the above equation, we get,

Quesiton: 8

If x = 3+√8, find the value of

Solution:

Given, x = 3 + √8,

We have, x = 3 + √8,

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

Since, (a + b) (a - b) = (a2 - b2)

Quesiton: 9

Find the value of it being given that √3 = 1.732 and √5 = 2.236.

Solution:

Given,

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

Since, (a + b) (a - b) = (a2 - b2)

= 3(2.236 + 1.732) = 3(3.968) = 11.904 

Quesiton: 10

Find the values of each of the following correct to three places of decimals, it being given that

Solution:

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

Since, (a + b) (a - b) = (a2 - b2)

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a2 - b2)

= 7 + 7.07 = 14.07

Quesiton: 11

Iffind the value of 4x+ 2x2- 8x + 7. 

Solution:

Given,and given to find the value of 4x3 + 2x2 − 8x + 7

2x = √3 + 1

2x – 1 = √3

Now, squaring on both the sides, we get, (2x − 1)2 = 3

4x2 − 4x + 1 = 3

4x2 − 4x + 1 − 3 = 0

4x2 − 4x − 2 = 0

2x2 − 2x − 1 = 0

Now taking 4x3 + 2x2 − 8x + 7

2x(2x2 − 2x − 1) + 4x2 + 2x + 2x2 − 8x + 7

2x(2x2 − 2x − 1) + 6x2 − 6x + 7

As, 2x2 − 2x − 1 = 0

2x(0) + 3(2x2 − 2x − 1)) + 7 + 3

0 + 3(0) + 10

10

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