Rules for calculating with powers

Multiplying powers with the same base

When multiplying two powers with the same base, we add the exponents together:

a^n × a^m = a^(n+m)

Example: 2³ × 2² = 2^(3+2) = 2⁵ = 32

Dividing powers with the same base

We subtract the exponents (where a is not equal to 0):

a^n / a^m = a^(n-m)

Example: 5^4 / 5^2 = 5^(4-2) = 5^2 = 25

Power of a power

Multiply the exponents together:

(a^n)^m = a^(n × m)

Example: (3^2)^3 = 3^(2 × 3) = 3^6

Power of a product

(a × b)^n = a^n × b^n

Example: (2 × 5)^3 = 2^3 × 5^3 = 8 × 125 = 1000

Negative exponents

A negative exponent indicates an inverse:

a^(-n) = 1 / a^n, for a ≠ 0

Example: 2^(-3) = 1/2^3 = 1/8

Rule	Formula
Product	a^n × a^m = a^(n+m)
Quotient	a^n / a^m = a^(n-m)
Power of a power	(a^n)^m = a^(n × m)
Negative exponent	a^(-n) = 1/a^n

Common pitfalls

• You only add exponents if the bases are the same. 2³ × 3² cannot be simplified to a single term; you must calculate 8 × 9 = 72.
• (a^n)^m gives a^(n × m) and not a^(n+m): do not confuse the two rules.
• a^n + a^m can never be simplified by adding the exponents: there is no general rule for simplifying the sum of powers (unlike the product).