The Five Categories of Exponent Rules

Terms that have exponents can be added, subtracted, multiplied, divided, and raised to a power. There is an exponent rule for each of these elementary math operations.

A term with an exponent is generally notated as:

aⁿ

Where a is the base and n is the exponent. This is the form of writing an exponent term that we will use throughout the lesson and for the exponent rule formulas.

Exponent Addition Rule

We can add exponent terms as long as they have the same base a and exponent n. The rule is given as:

Caⁿ + Daⁿ = (C + D)aⁿ

Example:
2x³ + 3x³ = 5(x³)

Exponent Subtraction Rule

We can subtract exponent terms as long as they have the same base a and exponent n. The rule is given as:

Caⁿ - Daⁿ = (C - D)aⁿ

Example:
6x³ - 3x³ = 3(x³)

Exponent Multiplication Rule

We can multiply exponent terms as long as they have the same base a. The rule is given as:

(aⁿ)(a^m) = a^{(n + m)}

Example:
(5²)(5⁴) = 5⁶

Exponent Division Rule

We can divide exponent terms as long as they have the same base a. The rule is given as:

(aⁿ)/(a^m) = a^{(n - m)}

Example:
(5⁵)/(5²) = 5³

Exponent Power Rule

We can raise an exponent term to a power. The rule is given as:

(aⁿ)^m = a^(n·m)

Example:
(x³)⁴ = x¹²