Consider the following;

a*a*a=a³

a*a*a*a=a⁴

a*a*a*a*a=a⁵

a*a*a*…*a (n times) =aⁿ

The plural of an index is indices. Index form means in power form, ex: a^3, a⁴, etc…

Ex: write the following in index form a)16, b)125, c)7*7*7*9*9

Solution: a)16=2*2*2*2=2⁴, b)125=5*5*5=5³, c)7*7*7*9*9=7³*9²

Laws of indices

1. Multiplication law

It states that a^m * aⁿ=a^{m + n}   where a≠0 and m and n are any number.

Ex: evaluate the following

a) a⁵*a³=a⁵⁺³=a⁸, b) 5⁹*5³=5¹²

2. Division law

It states that a^m / aⁿ = a^{m - n}   where a≠0

Ex: a) a⁵*a³=a^5-3=a², b) 5⁹/5³=5⁶

3. Law of exponents (powers)

It states that (a^m)ⁿ = a^{m * n} where a≠0.

Ex: a) (a²)²=a^2*2=a⁴, b) (m^-4)⁵=m^-4*5=m^-20

4. Negative law

It states that a^-m = 1/(a^m)

Ex: a) a³/a⁵=a^-2=1/(a²)   b) 2^-4=1/ (2⁴) =1/16

5. Zero law of indices

It states that a⁰ = 1 where a≠0

Ex: 5⁰=1