Law of Indices

Law of Indices:

Introduction:

Let us learn the meaning of Law of Indices in general,Indices is the concept of Algebra which is one of the major branch of mathematics.The Index of a number is the number of times multiplied by itself. If X and Y are positive integers and also X [!=] 0 then

X Y = X * X * X * X * X ........ Y times it is multiplied by same number

X is called the base and Y is the power. We read it as " X raised to the power Y " . The power is also called the " Index" or "Exponent".

A few examples of Powers : (i) 25 = 2 x 2 x 2 x 2 x 2 = 32 where index (or power or exponent) = 5 and base = 2 .

Its read as base 2 raised to the index 5 equals to 32

(ii) 23 = 2 x 2 x 2 = 8 where index = 3 and base = 2.

Its read as base 2 raised to the index 3 equals to 8

(iii) 33 = 3 x 3 x 3 = 27 where index = 3 and base = 3

Its read as base 3 raised to the power 5 equals to 27

(iv) 34 = 3x3x3x3 = 81 where index = 4 and base = 3

Its read as base 3 raised to the index 4 equals to 81

(v) 42 = 4x4 = 16 where index = 2 and base = 4

Its read as base 4 raised to the index 2 equals to 16

Indices or powers or Exponents are used to write statements involving repeated multiplication in shorthand.

Now let us learn about the specifics of the Law of Indices:For any expression like numbers , variables or functions having same base but different indices (or power or exponent) and also for the expressions consists of same index (or power or exponent) but different bases can be solved using Indices concept involving algebraic operations by following the below set of Laws of Indices.

Rule (1) Product Law for Indices : am x an = a m+n

consider an example, (a) 53 x 52 = 5 3+2 = 55 = 5 x 5 x 5 x 5 x 5 = 3125

(b) e 3t e -7t = e 3t x e -7t = e [3t + (-7t) ] = e [3t - 7t] = e -4t

(c) -4 4 -4 -1 = -4 [4 + (-1) ] = -4 [ 4 - 1] = -4 3

(d) (5/2) -3 (5/2) -9 = (5/2) -3 + (-9) = (5/2) -3 - 9 = (5/2) -12

(e) (q-2) (q 1/3) = q -2 + (1/3) = q (-6+1) / 3 = q -5/3

(f) (3) -1 (3) 5 = 3 -1+5 = 3 4

Rule (2) Quotient Law for Indices : am [-:] an =(a m / a n ) = a m-n

For instance, (a) 68 [-:] 65 = 6 8-5 = 6 3 = 6 x 6 x 6 = 216

(b) (4/3) -3 [-:] (4/3) 9 = (4/3) [-3 - 9] = (4/3) -12 = 1 / (4/3) 12 = (3/4) 12

(c) 5 -2 [-:] 5 -2 = 5 [-2 -(-2)] = 5 [-2 + 2] = 5 0 = 1

(d) g 6 [-:] g -7 = g [6 - (-7)] = g [6 + 7] = g 13

(e) Y (-7/2) [-:] Y (-1) = Y [ (-7/2) - (-1)] = Y [ (-7/2 + 1)] = Y [(-7+2)/2] = Y -5/2

Rule (3) Power Law for Indices : (a m) n = a mn

Example (a) (7 4) 2 = 7 4x2 = 7 8

(b) (2 -5 )4 = 2 (-5x4) = 2 -20

(c) (X -5) -3 = X (-5 x -3) = X -15

(d) [t (1/2)] 3 = t [(1/2)x3] = t (3/2)

Hope you like the above example of Law of Indices.Please leave your comments, if you have any doubts.