# RATIONALISING FACTOR

Rationalising Factor :

A number that is needed to multiply in order to make an irrational number as rational number.

For example, to make the irrational number √3 as rational, we need to multiply √3 by √3.

√3 ⋅ √3  =  3

So, √3 is the rationalising factor of √3

The Simplest Rationalising Factor :

Write the given irrational number in simplest form and identify the simplest rationalising factor of the given irrational number.

Consider the irrational number √50.

Write √50 in the simplest form.

√(2 ⋅ 5 ⋅ 5)  =  5√2

To make 5√2 as rational, we need to multiply √2.

5√2 ⋅ √2  =  ⋅ 2  =  10

So, √2 is the simplest rationalising factor of √50

Other Forms of Rationalising Factors :

1.  (a + √x) and (a - √x) are rationalizing factors to each other.

2.  (a + b√x) and (a - b√x) are rationalizing factors to each other.

3.  (√x + y) and (√x y) are rationalizing factors of each other.

## Solved Problems

Write the rationalising factor of each of the following irrational numbers :

Problem 1 :

3√2

Solution :

3√2 is already in simplest form.

=  3√2  √2

=  3(√2  2)

=  3 ⋅ 2

=  6

6 is a rational number.

So, the rationalizing factor of 3√2 is √2.

Problem 2 :

43√25

Solution :

43√25 is already in simplest form.

=  43√25  3√5

=  4(3√25  3√5)

=  43√(25  5)

=  43(5 ⋅ 5 ⋅ 5)

=  4 ⋅ 5

=  20

20 is a rational number.

So, the rationalizing factor of 23√25 is 3√5.

Problem 3 :

3√432

Solution :

Write 3√432 in the simplest form.

3√432  =  3√(2 ⋅ 6  6 ⋅ 6)

=  63√2

The simplest form of 3√412 is 63√2.

=  63√2 ⋅ 3√4

=  63√(2 ⋅ 4)

=  63√(2 ⋅ 2 ⋅ 2)

=  6 ⋅ 2

=  12

12 is a rational number.

So, the rationalizing factor of 3√432 is 3√4.

Problem 4 :

2 + √3

Solution :

(2 + √3) is already in simplest form.

=  (2 + √3)(2 - √3)

Use the algebraic identity a2 - b2 = (a + b)(a - b).

=  22 - (√3)2

=  22 - (√3)2

=  4 - 3

=  1

1 is a rational number.

So, the rationalizing factor of (2 + √3) is (2 - 4√3).

Problem 5 :

5 - 4√3

Solution :

(5 - 4√3) is already in simplest form.

=  (5 - 4√3)(5 + 4√3)

Use the algebraic identity a2 - b2 = (a + b)(a - b).

=  52 - (4√3)2

=  25 - 42(√3)2

=  25 - 16(3)

=  25 - 48

=  -23

-23 is a rational number.

So, the rationalizing factor of (5 - 4√3) is (5 + 4√3).

Problem 6 :

√3 + √2

Solution :

(√3 + √2) is already in simplest form.

=  (√3 + √2)(√3 - √2)

Use the algebraic identity a2 - b2 = (a + b)(a - b).

=  (√3)2 - (√2)2

=  3 - 2

=  1

1 is a rational number.

So, the rationalizing factor of (√2 + √3) is (√2 - √3).

Problem 7 :

√75

Solution :

Write √75 in simplest form.

√75  =  √(3 ⋅ 5 ⋅ 5)

The simplest form of √75 is 5√3.

=  5√3  √3

=  5(√3  √3)

=  5 ⋅ 3

=  15

15 is a rational number.

So, the rationalizing factor of √75 is √3.

Problem 8 :

√125 + 2

Solution :

Write (√125 + 2) in the simplest form.

√125 + 2  √(5 ⋅ 5 ⋅ 5) + 2

=  5√5 + 2

The simplest form of (√125 + 2) is (5√5 + 2).

=  (5√5 + 2)(5√5 - 2)

Use the algebraic identity a2 - b2 = (a + b)(a - b).

=  (5√5)2 - 22

=  52(√5)2 - 4

25(5) - 4

=  125 - 4

=  121

121 is a rational number.

So, the rationalizing factor of (√125 + 2) is (5√5 - 2)

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