The index form of a surd ^{n}√a is

a^{1/n}

For example, ^{3}√5 can be written in index form as shown below.

^{3}√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.

For example,

^{3}√5 is a surd of order '3'

^{5}√10 is a surd of order '5'

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

Surd |
Index form |
Order |
Radicand |

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

**Question 1 :**

Convert the following surd to index form.

√7

**Answer :**

**Surd = **√7

Index form = 7^{1/2}

**Question 2 :**

Convert the following surd to index form.

^{4}√8

**Answer :**

**Radical form = **^{4}√8

Index form = 8^{1/4}

**Question 3 :**

Convert the following surd to index form.

^{3}√6

**Answer :**

**Radical form = **^{3}√6

Index form = 6^{1/3}

**Question 4 :**

Convert the following surd to index form.

^{8}√7

**Answer :**

**Radical form = **^{8}√7

Index form = 7^{1/8}

**Law 1 : **

^{n}√a = a^{1/n}

**Law 2 : **

^{n}√(ab) = ^{n}√a x ^{n}√b

**Law 3 : **

^{n}√(a/b) = ^{n}√a / ^{n}√b

**Law 4 : **

(^{n}√a)^{n} = a

**Law 5 : **

^{m}√(^{n}√a) = ^{mn}√a

**Law 6 : **

(^{n}√a)^{m} = ^{n}√a^{m}

**Law 1 : **

x^{m} ⋅ x^{n} = x^{m+n}

**Law 2 : **

x^{m} ÷ x^{n} = x^{m-n}

**Law 3 : **

(x^{m})^{n} = x^{mn}

**Law 4 : **

(xy)^{m} = x^{m} ⋅ y^{m}

**Law 5 : **

(x / y)^{m} = x^{m} / y^{m}

**Law 6 : **

x^{-m} = 1 / x^{m}

**Law 7 : **

x^{0} = 1

**Law 8 : **

x^{1} = x

**Law 9 : **

x^{m/n} = y -----> x = y^{n/m}

**Law 10 : **

(x / y)^{-m} = (y / x)^{m}

**Law 11 : **

a^{x} = a^{y} -----> x = y

**Law 12 : **

x^{a} = y^{a} -----> x = y

**Problem 1 :**

Simplify the following :

√5 ⋅ √18

**Solution :**

**We have two radicals with same order. So, we can take the radical once and multiply the values inside the radicals. **

**√5 ⋅ √18 = **√(5 ⋅ 18)

= √(5 ⋅ 3 ⋅ 3 ⋅ 2)

= 3 √(5 ⋅ 2)

= 3√10

**Problem 2 :**

Simplify the following :

3√35 ÷ 2√7

**Solution :**

**We have two radicals with same order. So, we can take the radical once and divide the values inside the radicals. **

**3√35 ÷ 2√7 = (3/2) **⋅ √(35/7)

= (3/2) ⋅ √5

= 3√5/2

**Problem 3 :**

Simplify the following :

^{4}√8 ÷ ^{4}√12

**Solution :**

**We have two radicals with same order. So, we can take the radical once and divide the values inside the radicals. **

^{4}√8 ÷ ^{4}√12 = ^{4}√(8/12)

= ^{4}√(2/3)

= ^{4}√2 / ^{4}√3

**Problem 4 :**

Simplify the following :

x^{2 }⋅ x^{3}

**Solution :**

**We have the same base 'x' for both the terms. Because the terms are multiplied, we can take the base once and add the exponents. **

x^{2 }⋅ x^{3}** = x ^{2+3}**

x^{2 }⋅ x^{3}** = x ^{5}**

**Problem 5 :**

Simplify the following :

x^{7} ÷ x^{5}

**Solution :**

**We have the same base 'x' for both the terms. Because the terms are in division, we can take the base once and subtract the exponents. **

x^{7} ÷ x^{5} = x^{7-5}

x^{7} ÷ x^{5} = x^{2}

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