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Study Guides > College Algebra

Rules for Exponents

Learning Objectives

  • Use the product rule for exponents
  • Use the quotient rule for exponents
  • Use the power rule for exponents
Consider the product [latex]{x}^{3}\cdot {x}^{4}[/latex]. Both terms have the same base, x, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.
[latex]\begin{array}\text{ }x^{3}\cdot x^{4}\hfill&=\stackrel{\text{3 factors } \text{ 4 factors}}{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x} \\ \hfill& =\stackrel{7 \text{ factors}}{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x} \\ \hfill& =x^{7}\end{array}[/latex]
The result is that [latex]{x}^{3}\cdot {x}^{4}={x}^{3+4}={x}^{7}[/latex]. Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.
[latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]
Now consider an example with real numbers.
[latex]{2}^{3}\cdot {2}^{4}={2}^{3+4}={2}^{7}[/latex]
We can always check that this is true by simplifying each exponential expression. We find that [latex]{2}^{3}[/latex] is 8, [latex]{2}^{4}[/latex] is 16, and [latex]{2}^{7}[/latex] is 128. The product [latex]8\cdot 16[/latex] equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

A General Note: The Product Rule of Exponents

For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], the product rule of exponents states that
[latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]

Example: Using the Product Rule

Write each of the following products with a single base. Do not simplify further.
  1. [latex]{t}^{5}\cdot {t}^{3}[/latex]
  2. [latex]\left(-3\right)^{5}\cdot \left(-3\right)[/latex]
  3. [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}[/latex]

Answer: Use the product rule to simplify each expression.

  1. [latex]{t}^{5}\cdot {t}^{3}={t}^{5+3}={t}^{8}[/latex]
  2. [latex]{\left(-3\right)}^{5}\cdot \left(-3\right)={\left(-3\right)}^{5}\cdot {\left(-3\right)}^{1}={\left(-3\right)}^{5+1}={\left(-3\right)}^{6}[/latex]
  3. [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}[/latex]
At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.
[latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}=\left({x}^{2}\cdot {x}^{5}\right)\cdot {x}^{3}=\left({x}^{2+5}\right)\cdot {x}^{3}={x}^{7}\cdot {x}^{3}={x}^{7+3}={x}^{10}[/latex]
Notice we get the same result by adding the three exponents in one step.
[latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[/latex]

Try It

Write each of the following products with a single base. Do not simplify further.
  1. [latex]{k}^{6}\cdot {k}^{9}[/latex]
  2. [latex]{\left(\frac{2}{y}\right)}^{4}\cdot \left(\frac{2}{y}\right)[/latex]
  3. [latex]{t}^{3}\cdot {t}^{6}\cdot {t}^{5}[/latex]

Answer:

  1. [latex]{k}^{15}[/latex]
  2. [latex]{\left(\frac{2}{y}\right)}^{5}[/latex]
  3. [latex]{t}^{14}[/latex]

The following video gives more examples of using the power rule to simplify expressions with exponents. https://youtu.be/VjcKU5rA7F8

Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

  • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
  • Product Rule for Exponents. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Quotient Rule for Exponents. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Using the Power Rule to Simplify Expressions With Exponents. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Question ID 109745, 109748. Authored by: Lumen Learning. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
  • Question ID 93370, 93399, 93402. Authored by: Michael Jenck. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
  • Question ID 1961. Authored by: David Lippman. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.

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