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Power Rule of Exponents
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Introduction

In this lesson, you will learn about the power rule for exponents.

When you have an exponential expression, or in other words, a base raised to a power, and then that whole exponential expression is raised to another power, it is equal to the base raised to both powers multiplied together.

\begin{align*} &(a^b)^n\rightarrow{a}^{b\times n}\rightarrow{a}^{bn} &\color{red}\small\text{where \(b\times n\) is the new exponent}\end{align*}

This video illustrates the lesson material below. Watching the video is optional.

Power Rule of Exponents

\begin{align*}\large\text{Power Rule of Exponents: }(a^b)^c\rightarrow{a}^{b\times c}\rightarrow{a}^{bc} \end{align*}

Example 1

\begin{align*} &({a}^{2})^{3} &\color{red}\small\text{Simplify this expression}\\\\ &(a^2)(a^2)(a^2) &\color{red}\small\text{Write \(a^2\) three times}\\\\ &(aa)(aa)(aa)&\color{red}\small\text{Expand the base a's}\\\\ & a^6 &\color{red}\small\text{Multiplication rule of exponents} \end{align*}
You can also do this with the Power Rule of Exponents:

\begin{align*} & ({a}^{2})^{3} &\color{red}\small\text{Simplify this expression}\\\\ & {a}^{2\times3} &\color{red}\small\text{Power rule of exponents}\\\\ & {a}^{6} &\color{red}\small\text{Multiply exponents} \end{align*}

Example 2
\begin{align*} & ({3}^{4})^{2}&\color{red}\small\text{Simplify this expression}\\\\ & {3}^{4\times2} &\color{red}\small\text{Power rule of exponents}\\\\ & {3}^{8} &\color{red}\small\text{Multiply exponents} \end{align*}

Power Rule with Negative Exponents

Example 3
\begin{align*} &({2}^{-1})^{3}&\color{red}\small\text{Simplify this expression}\\\\ &(2^{-1})(2^{-1})(2^{-1})&\color{red}\small\text{Write \(2^{-1}\) three times}\\\\ &(\frac{1}{2})(\frac{1}{2})(\frac{1}{2}) &\color{red}\small\text{Negative exponent rule}\\\\ & \frac{1}{2^3} &\color{red}\small\text{Rewrite using exponents} \end{align*}

Applying what you know about negative exponents, this is the same as \({2}^{-3}\). If you use the power rule of exponents with this same problem, you will get the same answer.

\begin{align*} &({2}^{-1})^{3}&\color{red}\small\text{Simplify this expression}\\\\ &2^{(-1)( 3)} &\color{red}\small\text{Power rule of exponents}\\\\ & 2^{-3} &\color{red}\small\text{Multiply exponents}\\\\ &\frac{1}{2^3} &\color{red}\small\text{Negative exponent rule} \end{align*}

Example 4
\begin{align*} &({3}^{2})^{-4}&\color{red}\small\text{Simplify this expression}\\\\ &3^{(2)(-4)} &\color{red}\small\text{Power rule of exponents}\\\\ & 3^{-8} &\color{red}\small\text{Multiply exponents}\\\\ & \frac{1}{3^{8}} &\color{red}\small\text{Negative exponent rule} \end{align*}

Example 5
\begin{align*} &({4}^{-3})^{-2}&\color{red}\small\text{Simplify this expression}\\\\ &4^{(-3)(-2)} &\color{red}\small\text{Power rule of exponents}\\\\ & 4^{6} &\color{red}\small\text{Multiply exponents} \end{align*}

Example 6
Simplify this expression: \(((-3)^{2})^{4}\).

This has a negative number as the base. It is always a good idea to use parentheses with negative numbers to show that the new exponent is applying to the entire number \((-3)\) and not just the \(3\) itself.

\begin{align*} &((-3)^{2})^{4}&\color{red}\small\text{Simplify this expression}\\\\ &(-3)^{(2)(4)} &\color{red}\small\text{Power rule of exponents}\\\\ & (-3)^{8} &\color{red}\small\text{Multiply exponents} \end{align*}

This is an example of having a negative base raised to 2 exponents, and the rules for exponents still applies.

Things to Remember

  • Power Rule of Exponents: Multiply the exponents together to get the new exponent; simplify the power.

\begin{align*} ({b}^{x})^{y} =({b})^{x\cdot y}\end{align*}

Practice Problems

Simplify the following expressions:
  1. \(({\text{x}}^{\text{m}})^{\text{n}}\) (
    Solution
    x
    Solution: \({\text{x}}^{\text{mn}}\)
    Details:
    The power rule for exponents says that any base raised to an exponent that is also raised to an exponent is the same as that base raised to the multiplication of the two exponents.

    In this example, x is the base. It is being raised first to the power of m and then all of that is being raised to the power of n.

    According to the power rule for exponents, this is the same as: \(({\text{x}}^{\text{m}})^{\text{n}} = {\text{x}}^{\text{mn}}\), where m and n are being multiplied together.
    )
  2. \((3^{2})^{3}\) (
    Solution
    x
    Solution: \(3^{6} = 729\)
    )
  3. \(({\text{m}}^{2})^{4}({\text{y}}^{5})^{2}\) (
    Solution
    x
    Solution: \({\text{m}}^{8}{\text{y}}^{10}\)
    Details:
    Version 1: Applying the power rule for exponents

    \(\left ({\text{m}}^{2} \right )^{4}\left ({\text{y}}^{5} \right )^{2}\)

    According to the power rule for exponents, you can multiply the \(2 \cdot 4\) to get the exponent for m. You can also multiply the \(5 \cdot 2\) to get the exponent for y.

    \(\left ( {\text{m}}^{2} \right )^{4}\left ( {\text{y}}^{5} \right )^{2} = {\text{m}}^{\left ( 2\cdot4 \right )}{\text{y}}^{\left ( 5\cdot2 \right )} = {\text{m}}^{8}{\text{y}}^{10}\)

    The final answer is: \({\text{m}}^{8}{\text{y}}^{10}\).

    Version 2: Applying the rules of multiplication to show why the power rule for exponents works

    \(\left ({\text{m}}^{2} \right )^{4}\left ({\text{y}}^{5} \right )^{2}\)

    This means \({\text{m}}^{2}\) is being multiplied together 4 times and \({\text{y}}^{5}\) is the same as y multiplied 5 times.

    If you expand out each factor using the rules of multiplication it becomes the following:

    \({\text{m m m m m m m m y y y y y y y y y y}}\)

    This image shows two expressions: m squared m squared m squared m squared y to the fifth power y to the fifth power and m m m m m m m m y y y y y y y y y y. Each factor with exponents in the top expression has a line pointing down to its corresponding group in the bottom expression. Each m squared in the first equation corresponds to m m in the second equation. Each y to the fifth power in the first equation corresponds to y y y y y in the second equation.

    You now see there are 8 m’s being multiplied together and 10 y’s being multiplied together. If you write this in exponent form it is \({\text{m}}^{8}{\text{y}}^{10}\).

    This image is the same as the previous one, two expressions: m squared m squared m squared m squared y to the fifth power y to the fifth power and m m m m m m m m y y y y y y y y y y. Each factor with exponents in the top expression has a line pointing down to its corresponding group in the bottom expression. Each m squared in the first equation corresponds to m m in the second equation. Each y to the fifth power in the first equation corresponds to y y y y y in the second equation. But now we have a third expression below the second that represents the exponent form of the equation m m m m m m m m y y y y y y y y y y. That is m to the eighth power y to the tenth power.

    The final answer is \({\text{m}}^{8}{\text{y}}^{10}\), which is the same answer as when using the power rule.
    )
  4. \(({\text{x}}^{3})^{2}({\text{y}}^{4})^{2}(x^{2})^{1}\) (
    Video Solution
    x
    Solution: \({\text{x}}^{8}{\text{y}}^{8}\)
    Details:

    (Power Rule #4 (01:54 mins) | Transcript)
    ) | Transcript)
  5. \((4^{2})^{3}(2^{5})^{1}\) (
    Video Solution
    x
    Solution: \(4^{6}2^{5} = 4096 \cdot 32 = 131,072\)
    Details:

    (Power Rule #5 (02:39 mins) | Transcript)
     | Transcript)
  6. \(({\text{a}}^{\text{x}})^{2}({\text{b}}^{3})^{2}\) (
    Solution
    x
    Solution: \({\text{a}}^{2{\text{x}}}{\text{b}}^{6}\)
    )

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Negative Exponents
Exponents of 0 and 1