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Surd and indices in Mathematics

  • Difficulty Level : Basic
  • Last Updated : 15 Jun, 2021

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

Example –

√2, √29, etc.

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

Indices :

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

Example – 
Let a number 23= 2×2×2= 8, then 2 is the base and 3 is 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 surds :
When a surd is multiplied by a rational number then it is known as a mixed surd.

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

S.No.Rules for surds  Example
1.n √x = x1/n √2 = 21/2
2.n√(x ×y) =n √x × n √x                                   √(2×3)= √2 × √3 
3.n√(x ÷y)=n √x ÷ n √y3√(5÷3) =  3√5 ÷ 3√3
4.(n √x)n = x(√2)2 = 2
5.(n√ x)m =  n√(x m   (3√27)2  =  3√(272) = 9
6.m√(n√ x) = m × n √x 2√(3√729)=  2×3√729 = 6√729 = 3 

Rules of indices :       

S.No.Rules for indicesExample
1.x0 = 1                                             20 = 1
2x m × x n = x m +n22 ×23= 25 = 32  
3x m ÷ x n  = x m-n23 ÷ 22 = 23-2 = 2 
4(x m)n = x m ×n  (23)2 = 23×2 = 64
5(x × y)n = x n × y n(2 × 3)2= 22 × 32 =36 
6(x ÷ y)n = x n ÷ y n(4 ÷ 2) 2=  42 ÷ 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) x m = x n then m=n and a≠ 0,1,-1.
8) x m = y m then 
   x = y if m is even 
    x= y, if m is odd

Basic problems based on surds and indices :

Question-1
Which of the following is a surd?

a)  2√36              b)  5√32      c)  6√729          d) 3√25

Solution – 
An answer is an option (d)

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

Question-2 :
Find √√√3



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 × 1/2 ×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 
⇒ (4/5)-3 = (4/5)2x-1
⇒ 2x-1 = -3
⇒ 2x = -2
⇒  x = -1

Question-4 : 

34x+1 = 1/27, then x is

Solution –

34x+1 = (1/3)3
⇒34x+1 = 3-3
⇒4x+1 = -3
⇒4x= -4  
⇒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 infractions, therefore multiply each exponent by LCM of all the exponents. The LCM of all numbers is 72.

2(1/12 × 72) = 26 = 64
3(1/72 ×72) = 3
4(1/24 ×72) = 43 = 64
6 (1/36 ×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.

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