CLASS-8INTRODUCTION OF EXPONENTS

EXPONENTS

As we all know that, a X a = a², which is to be read as x squared or ‘x raised to the power 2 or ‘x to the power 2’. Hence, x is the ‘base’ and 2 is ‘exponents’ or ‘index’. An exponent ( or index ) is a number written to the right and a little above the base. It indicates the number of times the base occurs in a product.

In xᵐ = ( x, x, x, x, ………………m times ), read as x to the power m, the exponent is m

Examples.1)  In a⁵ = ( a X a X a X a X a ), read as ‘a’ to the power 5, the desired exponent is 5.

Examples.2)  In a⁶ = ( a X a X a X a X a X a ), read as ‘a’ to the power 6, the desired exponent is 6.

Examples.3)  In a² = ( a X a ), read as ‘a’ to the power 2, the desired exponent is 2.

There are some laws are given below –

1)   aᵐ X aⁿ = aᵐ⁺ⁿ      and    aᵐ X aⁿ X aᵖ  =  aᵐ⁺ⁿ⁺ᵖ

2)    aᵐ ÷  aⁿ =   aᵐ X a⁻ⁿ =  aᵐ⁻ⁿ

3)    (a X b)ᵐ = aᵐ X bᵐ   or   (a X b)ⁿ = aⁿ X bⁿ

4)    (aᵐ)ⁿ = aᵐⁿ

5)    aᵐ = aⁿ = 1  ( if m = n = 0 and a ≠ 0 )

6)    aᵐ = aⁿ = a  ( if m = n = 1 and a ≠ 0  )

7)  If  aᵐ = aⁿ    ( when m = n and a ≠ 0 )

a                aᵐ

8)   ( ------- )ᵐ  =  --------

b                bᵐ

9)  [{(a)ᵐ}ⁿ]ᵖ =  [{aᵐ}ⁿ]ᵖ  =  [aᵐⁿ]ᵖ  = aᵐⁿᵖ

10) If n is an even integers, (-1)ⁿ = (-1)²ᵐ = {(-1)²}ᵐ  =  {(-1) X (-1)}ᵐ =  1ᵐ  =  1

11)  If n is an odd integers, (-1)ⁿ = (-1)²ᵐ⁺¹= (-1) X (-1)²ᵐ = (-1) X (1) =  -1

12) n√a =   a¹∕ⁿ

13) aᵐ∕ⁿ = (aᵐ)¹∕ⁿ

14)  a¹∕ᵐ¹∕ⁿ = a¹∕ᵐⁿ