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 = 5 ⋅ 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)
- Rationalization of surds
- Comparison of surds
- Operations with radicals
- Ascending and descending order of surds
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