**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.

(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.

(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.

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

- 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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