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Study Guides > Intermediate Algebra

Read: The Power Rule for Exponents

Learning Objectives

  • Use the power rule to simplify expressions with exponents raised to powers
 

Raise powers to powers

Another word for exponent is power.  You have likely seen or heard an example such as [latex]3^5[/latex] can be described as [latex]3[/latex] raised to the [latex]5[/latex]th power. In this section we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent.  We will also learn what to do when numbers or variables that are divided are raised to a power.  We will begin by raising powers to powers. Let’s simplify [latex]\left(5^{2}\right)^{4}[/latex]. In this case, the base is [latex]5^2[/latex] and the exponent is [latex]4[/latex], so you multiply [latex]5^{2}[/latex] four times: [latex]\left(5^{2}\right)^{4}=5^{2}\cdot5^{2}\cdot5^{2}\cdot5^{2}=5^{8}[/latex] (using the Product Rule—add the exponents). [latex]\left(5^{2}\right)^{4}[/latex] is a power of a power. It is the fourth power of [latex]5[/latex] to the second power. And we saw above that the answer is [latex]5^{8}[/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\cdot4=8[/latex]. So, [latex]\left(5^{2}\right)^{4}=5^{2\cdot4}=5^{8}[/latex] (which equals 390,625, if you do the multiplication). Likewise, [latex]\left(x^{4}\right)^{3}=x^{4\cdot3}=x^{12}[/latex] This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\left(2^{3}\right)^{5}=2^{15}[/latex].

The Power Rule for Exponents

For any positive number x and integers a and b: [latex]\left(x^{a}\right)^{b}=x^{a\cdot{b}}[/latex]. Take a moment to contrast how this is different from the product rule for exponents found on the previous page.

Example

Write each of the following products with a single base. Do not simplify further.
  1. [latex]{\left({x}^{2}\right)}^{7}[/latex]
  2. [latex]{\left({\left(2t\right)}^{5}\right)}^{3}[/latex]
  3. [latex]{\left({\left(-3\right)}^{5}\right)}^{11}[/latex]

Answer: Use the power rule to simplify each expression.

  1. [latex]{\left({x}^{2}\right)}^{7}={x}^{2\cdot 7}={x}^{14}[/latex]
  2. [latex]{\left({\left(2t\right)}^{5}\right)}^{3}={\left(2t\right)}^{5\cdot 3}={\left(2t\right)}^{15}[/latex]
  3. [latex]{\left({\left(-3\right)}^{5}\right)}^{11}={\left(-3\right)}^{5\cdot 11}={\left(-3\right)}^{55}[/latex]

 In the following video you will see more examples of using the power rule to simplify expressions with exponents. https://youtu.be/VjcKU5rA7F8 Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.
Product Rule Power Rule
[latex]5^{3}\cdot5^{4}[/latex] =  [latex]5^{3+4}[/latex] = [latex]5^{7}[/latex] but [latex]\left(5^{3}\right)^{4}[/latex] = [latex]5^{3\cdot4}[/latex] = [latex]5^{12}[/latex]
[latex]x^{5}\cdot x^{2}[/latex] = [latex]x^{5+2}[/latex] = [latex]x^{7}[/latex] but [latex]\left(x^{5}\right)^{2}[/latex] =  [latex]x^{5\cdot2}[/latex] = [latex]x^{10}[/latex]
[latex]\left(3a\right)^{7}\cdot\left(3a\right)^{10} [/latex] = [latex]\left(3a\right)^{7+10} [/latex] = [latex]\left(3a\right)^{17}[/latex] but [latex]\left(\left(3a\right)^{7}\right)^{10} [/latex] = [latex]\left(3a\right)^{7\cdot10} [/latex] = [latex]\left(3a\right)^{70}[/latex]

Licenses & Attributions

CC licensed content, Original

  • Simplify Expressions Using the Power Rule of Exponents (Basic). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.

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