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Surd and Indices in Mathematics

Last Updated : 9 Mar, 2026

Surds are irrational numbers that are expressed in root form, such as โˆš2 or โˆš5, and cannot be simplified into whole numbers.

Indices are used to represent repeated multiplication of a number by itself. They show the power or exponent of a number, such as 2ยณ, where 2 is the base, and 3 is the index.

Note: Relationship between Surds and Indices

  • Surds and Indices are linked to powers and roots concepts in mathematics.
  • A surd can be represented using indices with fractional powers.
  • For example - โˆ›x can be represented as x1/3

Surd

Let x be a rational number(i.e., can be expressed in p/q form where q โ‰  0) and n is any positive integer such that x1/n = n โˆšx is irrational(i.e. can't be expressed in p/q form where q โ‰  0), then that n โˆšx is known as a surd of nth order.

Example: โˆš2, โˆš29, etc.

โˆš2 = 1.414213562..., which is non-terminating and non-repeating, therefore โˆš2 is an irrational number. And โˆš2= 21/2, where n=2, therefore โˆš2 is a surd. In simple words, a surd is a number whose power is an infraction and can not be solved completely(i.e., we can not get a rational number).

Rules of surds

When a surd is multiplied by a rational number, then it is known as a mixed surd.

Example: 2โˆš2, where 2 is a rational number, and โˆš2 is a surd. Here, x and y used in the rules are decimal numbers as follows.

S.No.Rules for surds  Example
1.n โˆšx = x1/nโˆš2 = 21/2
2.nโˆš(x ร—y) =n โˆšx ร— n โˆšx                                   โˆš(2ร—3)= โˆš2 ร— โˆš3 
3.nโˆš(x รทy)=n โˆšx รท n โˆšy3โˆš(5รท3) =  3โˆš5 รท 3โˆš3
4.(n โˆšx)n = x(โˆš2)2 = 2
5.(nโˆš x)m =  nโˆš(x m) (3โˆš27)2  =  3โˆš(272) = 9
6.mโˆš(nโˆš x) = m ร— n โˆšx2โˆš(3โˆš729)=  2ร—3โˆš729 = 6โˆš729 = 3 

Indices

  • It is also known as power or exponent.
  • X p, where x is a base and p is the power(or index)of x. where p, x can be any decimal number.

Example

Let a number 23= 2ร—2ร—2= 8, then 2 is the base and 3 is the indices.

  • An exponent of a number represents how many times a number is multiplied by itself.
  • They are used to representing roots, fractions.

Rules of Indices

When a number is expressed in exponential (power) form, it is said to be written using indices.

Example
23, where 2 is the base and 3 is the index.

Here, x and y used in the rules represent numbers, and m and n represent integers.

S.No.Rules for indicesExample
1.x0 = 1  20 = 1
2x m ร— x n = x m +n22 ร—23= 25 = 32  
3x m รท x n  = x m-n23 รท 22 = 23-2 = 2 
4(x m)n = x m ร—n(23)2 = 23ร—2 = 64
5(x ร— y)n = x n ร— y n(2 ร— 3)2= 22 ร— 32 =36 
6(x รท y)n = x n รท y n(4 รท 2) 2=  42 รท 22 = 4   

Other Rules

Some other rules are used in solving surds and indices problems as follows.

From 1 to 6 rules covered in table.
7) If xแต = xโฟ, then m = n (when x โ‰  0, 1, -1)
8) x m = y m then
x = y, if m is odd
x = ยฑy if m is even

Basic problems based on surds and indices

Question 1: Which of the following is a surd?

a) 2โˆš36
b) 5โˆš32
c) 6โˆš729
d) 3โˆš25

Solution:

An answer is an option (d)

Explanation: 3โˆš25= (25)1/3 = 2.92401773821... which is irrational So it is surd.

Question 2: Find โˆšโˆšโˆš3

a) 31/3
b) 31/4
c) 31/6
d) 31/8

Solution:

An answer is an option (d) 

Explanation: ((3 1/2)1/2) 1/2) = 31/2 ร— 1/2 ร—1/2 = 3 1/8 according to rule number 5 in indices.

Question 3: If (4/5)3 (4/5)-6= (4/5)2x-1, the value of x is

a) -2
b) 2
c) -1
d) 1

Solution: 

The answer is option (c)

Explanation: LHS = (4/5)3 (4/5)-6= (4/5)3-6 = (4/5)-3 RHS = (4/5)2x-1 According to question LHS = RHS โ‡’ (4/5)-3 = (4/5)2x-1 โ‡’ 2x-1 = -3 โ‡’ 2x = -2 โ‡’ x = -1

Question 4: 34x+1 = 1/27, then x is

Solution:

34x+1 = (1/3)3 โ‡’34x+1 = 3-3 โ‡’4x+1 = -3 โ‡’4x= -4 โ‡’x = -1

Question 5: Find the smallest among 2 1/12 , 3 1/72, 41/24, 61/36.

Solution:

The answer is 31/72

Explanation:
As the exponents of all numbers are in fractions, therefore multiply each exponent by LCM of all the exponents. The LCM of all numbers is 72.

2(1/12 ร— 72) = 26 = 64 3(1/72 ร—72) = 3 4(1/24 ร—72) = 43 = 64 6 (1/36 ร—72) = 62 = 36

Question 6: The greatest among 2400, 3300,5200,6200.

a) 2400
b)3300
c)5200
d)6200

Solution:

An answer is an option (d)

Explanation:
As the power of each number is large, and it is very difficult to compare them, therefore we will divide each exponent by a common factor(i.e. take HCF of each exponent).

The HCF of all exponents is 100. 2400/100 = 24 = 8. 3300/100 = 33 = 27 5200/100 = 52 = 25 6200/100= 62 = 36 So 6200 is largest among all.

Practice Problems on Surd and indices

1. Which of the following is a surd?

a)

b)

c)

d)

2. Simplify the following expression .

3. Find the value of .

4. Simplify .

5. Simplify the expression .

6. Simplify the expression .

7. Simplify the Indices .

8. Simplify the Indices .

9. Simplify the Mixed Surd .

10. Simplify the Mixed Surd .

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