Surd and indices in Mathematics

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.        
Rules of indices :       
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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