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