**Index Form of Surd :**

The index form of a surd ^{n}√a is a^{1/n}

For example, ∛5 can be written in index form as

∛5 = 5 ^{1/3}

**What is surd ?**

If ‘a’ is a positive rational number and n is a positive integer such that ^{n}√a is an irrational number, then ^{n}√a is called a ‘surd’ or a ‘radical’.

The general form of a surd is ^{n}√a is "√" is called a radical sign n is called the order of the radical and "a" is called radicand.

In the following table, the index form, order and radicand of some surds are given.

Surd |
Index form |
Order |
Radicand |

√5 ∛14 ∜7 √50 |
5 14 7 50 |
2 3 4 2 |
5 14 7 50 |

Let us look into some examples based on the above concept.

**Example 1 :**

Convert the following surds into index form

√7

**Solution :**

**Radical form = **√7

Index form = 7^{1/2}

**Example 2 :**

Convert the following surds into index form

∜8

**Solution :**

**Radical form = **∜8

Index form = 8^{1/4}

**Example 3 :**

Convert the following surds into index form

∛6

**Solution :**

**Radical form = **∛6

Index form = 6^{1/3}

**Example 4 :**

Convert the following surds into index form

^{8}√7

**Solution :**

**Radical form = **^{8}√7

Index form = 7^{1/8}

(1) (^{n}√a)^{n } = a

(2) ^{n}√a x ^{n}√b = ^{n}√(a x b)

**(3) **^{n}√a / ^{n}√b = ^{n}√(a / b)

**(4) (**^{n}√a)^{m } = a^{m/n}

Let us look into some examples based on above formulas.

**Example 1 :**

Simplify the following

√5 ⋅ √18

**Solution :**

**We have two radicals with same order, so we may take one radical and multiply the terms**

**√5 ⋅ √18 = **√(5 ⋅ 18)

= √(5 ⋅ 3 ⋅ 3 ⋅ 2)

= 3 √(5 ⋅ 2)

= 3√10

**Example 2 :**

Simplify the following

3√35 ÷ 2√7

**Solution :**

**We have two radicals with same order, so we may take one radical and divide the terms**

**3√35 ÷ 2√7 = (3/2) **⋅ √(35/7)

= (3/2) ⋅ √5

= 3√5/2

**Example 3 :**

Simplify the following

^{4}√8 ÷ ^{4}√12

**Solution :**

**We have two radicals with same order, so we may take one radical and divide the terms**

^{4}√8 ÷ ^{4}√12 = ^{4}√(8/12)

= ^{4}√(2/3)

= ^{4}√2 / ^{4}√3

- 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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