Basic Laws of Indices

A number of the form a^m where a is a real number, a is multiplied by itself m times

The number a is called the base and the super script m is called the index (plural indices) or exponent.

1. A ^m x A ⁿ = A^{m + n} ——————–Multiplication law

Example:  X³ x X²  =( X x X x X) x (X x X) = X ⁵

Or X³ x X² = X^{3 + 2} = X⁵

2. A^m ÷ Aⁿ=A^m-n  ———————Division law

Example: X⁶ ÷ X⁴ = X^6-4 = X²

3. (a ^m )ⁿ = a^mn —————-Power law

Example:  (x³)² = X³ x X³ = X³⁺³ =X⁶

Or X^3X2 = X⁶

4. a^m ÷ a^m = a^m-m = a⁰ =1

a^m ÷a^m = a^m/a^m = a^o = 1

a⁰………………………………….Zero Index

:. Any number raised to power zero is 1

Example: 3^o = 1, c^o = 1, y^o = 1

5. (ab)^m =a^mb^m   ————-Product power law

e.g. (2xy)²= 4x²y²

6. Negative index

a ^–m = 1/a^m        ————- Negative Index

Example: 2 ^-1 = ½, and 3 ^-2 = 1/3 ² = 1/9

7. a^1/n =ⁿ√a  ————–Root power law

Example : 9 ^½ =√9=3

27 ^1/3 =³√27 = 3 ie (3)³ = 3

8. a ^m/n = (a ^1/n) ^m = (ⁿ√a)^m ———–Fraction Index

or a ^m/n = (a^m) ^1/n =   (ⁿ√a)^m

Example : 27^2/3 = 3√27=3²=9.

EVALUATION

1. 27^5/3 2.   1000000000⁰