Indices


   
 
Indices
We know that 64 = 2 x 2 x 2 x 2 x 2 x 2 = 26
 
Here 2 is called the base and 6 is called the power (or index or exponent).
 
We say that "64 is equal to base 2 raised to the power 6".
 
Similarly, if m is a positive integer and
 
then a a a …m times = am.
 
Laws of Indices
 
If m and n are positive integers, and then
 
(i) am an = am + n [Product Law]
 
(ii) [Quotient Law]
 
(iii) (am)n = amn [Power Law]
 
(iv) (ab)m = am . bm
 
(v)
 
(vi) ao = 1
 
(vii)
 
Surd
 
An irrational root of a positive rational number is called a surd. Consider a number with base 'a' as a positive rational number with power of a fraction, say then
 
 
Since is an nth root, it is called a surd of order n, if it is irrational.
 
e.g., (i) is a surd of order 3.
 
(ii) is a surd of order 2.
 
(iii) is NOT a surd because and 3 is NOT an irrational number.
 
 
Without using tables, simplify:
 
(i) 36-1/2 (ii)
 
 
(i) 36-1/2
 
= (62)-1/2 = 6-1
 
 
(ii)
 
 
 
 
Evaluate:
 
 
Given expression
 
 
 
 
Simplify:
 
(i)
 
(ii) If 2x+2 = 128, find the value of x.
 
(iii) Simplify:
 
(iv) Simplify:
 
(v) Show that (xp - q)p + q (xq - r)q + r (xr - s)r+s = 1
 
(vi) 49 7x = (343)2x - 5 find 'x'.
 
 
(i) Given expression:
 
 
(ii) Since 128 = 2 2 2 2 2 2 2
 
= 27
 
We have 2x+2 = 27 [ bases are equal ]
 
x + 2 = 7 [ powers are equal ]
 
x = 5
 
(iii)
 
 
 
(iv)
 
= (x3a y6)1/4 (x2/3 y-1)a
 
= x3a/4 y3/2 x2a/3 y-a [ (x3a)1/4 = x3a/4 (power law)]
 
 
 
(v) LHS = x(p - q) (p + q) x(q - r) (q + r) x(r - s) (r +s)
 
 
 
= x0 = 1 = R.H.S
 
(vi) 49 7x = (343)2x - 5
 
72 7x = (73)2x - 5
 
72+x = 76x - 15
 
As the bases are equal, the powers are also equal.
 
2 + x = 6x - 15
 
or 5x = 17
 
 
  
     
   
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