ADDITION AND SUBTRACTION OF SURDS

Here we are going to see how to add and subtract surds.

Two or more like surds can be added or subtracted.

Like surds means the number inside the radical sign and order of the radical terms must be same.

For example 5√2 and -7√2 are like surds.

Let us look into some examples based on addition and subtraction of surds.

Example 1 :

Simplify 10√2 - 2√2 + 4√32

Solution :

10√2 - 2√2 + 4√32  =  10√2 - 2√2 + 4√(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

  =  10√2 - 2√2 + 4(2 ⋅ 2)√2

  =  10√2 - 2√2 + 16√2

  =  (10 + 16 - 2)√2

  =  24 √2

Hence the answer is 24√2.

Example 2 :

Simplify √48 - 3√72 - √27 + 5√18

Solution :

√48 - 3√72 - √27 + 5√18

Let us find the factors the numbers inside the radicals

√48  =  √(2 ⋅ 2 ⋅ 2 ⋅ 2⋅ 3)  =  (2 ⋅ 2)√3  =  4 √3

3√72  =  3√(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3)  =  (3 ⋅ 2 ⋅ 3)√2  =  18 √2

√27  =  √(3 ⋅ 3 ⋅ 3)  =  3 √3

5√18  =  5√(3 ⋅ 3 ⋅ 2)  =  (5 ⋅ 3) √2  =  15√2

√48 - 3√72 - √27 + 5√18  =  4√3 - 18√2 - 3√3 + 15√2

  =  4√3 - 3√3 - 18√2 + 15√2

  =  1√3 - 3√2

Hence the answer is 1√3 - 3√2.

Example 3 :

Simplify ∛16 + 8∛54 - ∛128

Solution :

∛16 + 8∛54 - ∛128

Let us find the factors the numbers inside the radicals

∛16  =  ∛(2 ⋅ 2 ⋅ 2 ⋅ 2)  =  2∛2

8∛54  =  8 ∛(2 ⋅ 3 ⋅ 3 ⋅ 3)  =  (8 ⋅ 3) ∛2  =  24∛2

∛128  =  ∛(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)   =  (2 ⋅ 2)∛2  =  4∛2

∛16 + 8∛54 - ∛128  =  2∛2 + 24∛2 - 4∛2

  =  (2 + 24 - 4)∛2

  =  22∛2

Hence the answer is 22∛2. 

Example 4 :

Simplify 7∛2 + 6∛16 - ∛54

Solution :

7∛2 + 6∛16 - ∛54

Let us find the factors the numbers inside the radicals

6∛16  =  6∛(2 ⋅ 2 ⋅ 2 ⋅ 2)  =  (6 ⋅ 2) ∛2  =  12∛2

∛54  =  ∛(2 ⋅ 3 ⋅ 3 ⋅ 3)  =  3 ∛2  =  3∛2

7∛2 + 6∛16 - ∛54  =  7∛2 + 12∛2 - 3∛2

  =  (7 + 12 - 3)∛2

  =  16∛2

Related topics

• Rationalization of surds
• Comparison of surds
• Operations with radicals
• Ascending and descending  order of surds
• Simplifying radical expression
• Exponents and powers

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