Arithmetic : Exponents, Powers and Roots of Numbers

Exponents

The power n is called the exponent and a is called the base in aⁿ.
For example in 7⁴, 4 is the exponent and 7 is the base.

The rules or laws of exponents are:

Here, a and b are any numbers, and n and m are any positive integers or zero.

Consider the following examples:

Rule 1 : 2² × 2⁵ = 4 × 32 = 128 and 2² × 2⁵ = 2⁷ = 128
Rule 2 : 2³ × 4³ = 8 × 64 = 512 and 2³ × 4³ = 8³ = 512
Rule 3 : (3²)³ = 9³ = 729 and (3²)³ = 3⁶ = 729

If n is a positive integer, then a negative exponent is defined as

MUST-KNOW : a⁻ⁿ = 1/aⁿ

This is obtained from the first law of exponents as follows:
a⁻ⁿ × aⁿ = a^{−n + n} = a⁰ = 1, so a⁻ⁿ = 1/aⁿ.
For example, 2⁻⁴ = 1/2⁴ = 1/16 and (¼)⁻¹ = 1/(¼) = 4