ADDITION AND SUBTRACTION OF SURDS WORKSHEET

Addition and subtraction of surds worksheet - Practice questions

(1)  Simplify 10√2 - 2√2 + 4√32

(2)  Simplify √48 - 3√72 - √27 + 5√18

(3)  Simplify ∛16 + 8∛54 - ∛128

(4)  Simplify 7∛2 + 6∛16 - ∛54

The following concepts will help you to solve the problems given above.

Some examples of like radicals

(i)  5√2 and -7√2 

(ii)  2 ∛7 and 3∛7

The numbers inside the radical sign is same and the order of the radicals is also same. Hence they are like radicals.

Some examples of unlike radicals

(i)  5√2 and -7 ∛2

(ii)  7 ∛2 and ∜8

In first example, the numbers inside the radical is same but order is not same. In the second example, the numbers inside the radical and order of the radicals are not same. 

Hence they are unlike radicals.

Question 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 + 162

  =  (10 + 16 - 2)√2

  =  24 √2

Hence the answer is 24√2.

Question 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.

Question 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. 

Question 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

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