Product Rule, Quotient Rule, and Power Rules

Exponents are used to show repeated multiplication. For example, 43 means 4 · 4 · 4 = 64.

In this section, we will review basic rules of exponents.

Product Rule of Exponents aman = am + n

When multiplying exponential expressions that have the same base, add the exponents.

Example:

Multiply: 4x3 · −6x2

Solution:

Multiply coefficients: 4 · −6 = −24

Use the product rule to multiply variables : x3 · x2 = x3 + 2 = x5

4x3 · −6x2 = −24x5

Quotient Rule of Exponents \frac{a^m}{a^n} = a^{m-n}

When dividing exponential expressions that have the same base, subtract the exponents.

Example:

Simplify: \frac{8x^6}{2x^3} = 4x^3

Solution:

Divide coefficients: 8 ÷ 2 = 4

Use the quotient rule to divide variables : \frac{x^6}{x^3} = x^{6-3} = x^3

\frac{8x^6}{2x^3} = 4x^3

Power Rule of Exponents (am)n = amn

When raising an exponential expression to a new power, multiply the exponents.

Example:

Simplify: (7a4b6)2

Solution:

Each factor within the parentheses should be raised to the 2nd power:

(7a4b6)2 = 72(a4)2(b6)2

Simplify using the Power Rule of Exponents :

(7a4b6)2 = 72(a4)2(b6)2 = 49a8b12

6 3 · 6 5 =
\frac{2 \times 10^3}{5 \times 10^{-3}}
(3x^2y^3)^3 =