6.2: Use Multiplication Properties of Exponents

Learning Objectives
Note

Before you get started, take this readiness quiz.

  1. Simplify: \(\frac\cdot \frac\)
    If you missed this problem, review Example 1.6.13.
  2. Simplify: \((−2)(−2)(−2)\).
    If you missed this problem, review Example 1.5.13.

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, \(2^4\) means the product of \(4\) factors of \(2\), so \(2^4\) means \(2·2·2·2\).

Let’s review the vocabulary for expressions with exponents.

EXPONENTIAL NOTATION

This figure has two columns. In the left column is a to the m power. <a href=The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m power means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue." />

This is read \(a\) to the \(m^\) power.

In the expression \(a^\), the exponent \(m\) tells us how many times we use the base a as a factor.

This figure has two columns. The left column contains 4 cubed. Below this is 4 times 4 times 4, with “3 factors” written below in blue. The right column contains negative 9 to the fifth power. Below this is negative 9 times negative 9 times negative 9 times negative 9 times negative 9, with “5 factors” written below in blue.

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

Example \(\PageIndex\)

Solution