INDEX FORM OF SURD

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’.

Order of a Surd

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.

Practice Questions

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

Laws of Surds

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

Laws of Indices

Law 1 : 

x^m ⋅ xⁿ  =  x^m+n

Law 2 : 

x^m ÷ xⁿ  =  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 : 

x⁰  =  1

Law 8 : 

x¹  =  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

Laws of Surds and Indices - Practice Problems

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 :

x² ⋅ x³

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. 

x² ⋅ x³  =  x²⁺³

x² ⋅ x³  =  x⁵

Problem 5 :

Simplify the following :

x⁷ ÷ x⁵

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. 

x⁷ ÷ x⁵  =  x^7-5

x⁷ ÷ x⁵  =  x²

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