PURE AND MIXED SURDS

Pure Surd :

A surd which has unity only as a rational factor the other factor being irrational is called pure surd.

In other words, a surd having no rational factor except unity is called pure surd or complete surd.

Examples :

√3, √5, ³√7, ⁴√5

Mixed Surd :

A surd consisting of the product of a rational and irrational is called mixed surd.

Examples :

2√3, 7√5, 5³√7

Examples 1-4 : Express the following mixed surds as pure surds :

Example 1 :

16√2

Solution :

In the given surd 16√2, consider √2. The order of √2 is 2. So, write 16 as a surd with order 2.

16 = √16² = √256

16√2 = √256√2

= √(256 ⋅ 2)

= √512

Example 2 :

6√5

Solution :

In the given surd 6√7, consider √7. The order of √7 is 2. So, write 6 as a surd with order 2.

6 = √6² = √36

6√7 = √36√7

= √(36 x 7)

= √242

Example 3 :

4³√2

Solution :

In the given surd 4³√2, consider ³√2. The order of ³√2 is 3. So, write 4 as a surd with order 3.

4 = ³√4³ = ³√64

4³√2 = ³√64³√2

= ³√(64 x 2)

= ³√128

Example 4 :

2⁴√5

Solution :

In the given surd 2⁴√5, consider ⁴√5. The order of ⁴√5 is 4. So, write 2 as a surd with order 4.

2 = ⁴√2⁴ = ⁴√16

2⁴√5 = ⁴√16⁴√5

= ⁴√(16 x 5)

= ⁴√80

Examples 5-8 : Express the following pure surds as mixed surds, if possible :

Example 5 :

√40

Solution :

The number inside the radical is 40. If 40 is a multiple of a perfect square, then we can write the pure surd √40 as a mixed surd.

To check whether 40 is a multiple of a perfect square, decompose 40 into its prime factor.

40 = 2 x 2 x 2 x 5

= 2² x 2 x 5

= 2² x 10

40 is a multiple of a perfect square.

So, the given pure surd √40 can be written as a mixed surd.

√40 = √(2² x 10)

= 2√10

Example 6 :

√15

Solution :

The number inside the radical is 15. If 15 is a multiple of a perfect square, then we can write the pure surd √15 as a mixed surd.

To check whether 15 is a multiple of a perfect square, decompose 15 into its prime factor.

15 = 3 x 5

15 is not a multiple of a perfect square.

So, the given pure surd √15 can not be written as a mixed surd.

Example 7 :

√98

Solution :

The number inside the radical is 98. If 98 is a multiple of a perfect square, then we can write the pure surd √98 as a mixed surd.

To check whether 98 is a multiple of a perfect square, decompose 98 into its prime factor.

98 = 7 x 7 x 2

= 7² x 2

98 is a multiple of a perfect square.

So, the given pure surd √98 can be written as a mixed surd.

√98 = √(7² x 2)

= 7√2

Example 8 :

√51

Solution :

The number inside the radical is 51. If 51 is a multiple of a perfect square, then we can write the pure surd √51 as a mixed surd.

To check whether 51 is a multiple of a perfect square, decompose 51 into its prime factor.

51 = 3 x 17

51 is not a multiple of a perfect square.

So, the given pure surd √51 can not be written as a mixed surd.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.