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Product Rule of Exponents
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Introduction

In this lesson, you will learn how to use the product rule of exponents to simplify expressions. The word product means to multiply. The product rule of exponents helps you remember what to do when two numbers with exponents are multiplied together.


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


The Product Rule of Exponents

Review these math vocabulary words to help you understand this lesson better:

  • Base: the number or variable that is being multiplied by itself.
  • Power: the number of the exponent, meaning how many times the base is multiplied by itself.

In English, the word product has several different meanings. In math, product refers to what you get when you multiply things together. For example, the product of \( 2\times4\) is 8. In other words, the product is the result of multiplication.

The product rule states that if the bases are the same, then you can add the exponents. Notice the qualification for the product rule is the bases must be the same or you cannot add the exponents.

\begin{align*}\color{black}\large\text{Product Rule of Exponents:} \;\frac{a^x} {a^y} =a^{x - y}\\\end{align*}

Remember that you can multiply in any order, so \( (a)(b) = (b)(a) \). This means that if there are multiple bases, you can rearrange the order and add the exponents of any of the bases that are the same.

When talking about exponents x and y, the product rule says that if you have two terms with the same base—for example, base b—being multiplied together, you can add the exponents of the same base.

Example 1
Simplify \(2^{3}\times2^{4}\).

According to the product rule, this is equal to \( 2^3\cdot 2^4 = 2^{3+4} = 2^{7}\).

Another way of looking at this example:
\begin{align*}2^{3}&\cdot2^{4} &\color{red}\small\text{Simplify this expression}\\\\ 2\times2\times2\cdot 2&\times2\times2\times2 &\color{red}\small\text{Product rule of exponents}\\\\
&2^{7} &\color{red}\small\text{There are seven 2's}\\\\ \end{align*}

Remember, the key with the product rule is that the bases of these terms have to be the same. If you had \( 2^{3}\times4^{4}\), the product rule won't work because it would be two multiplied together three times, multiplied by four multiplied together four times. Even if the exponents are the same, such as in \( 2^{7}\times4^{7}\), the product rule still wouldn’t work because the bases, the bottom numbers in each term, 2 and 4, are not the same.

Example 2
Simplify \( x^{3}x^{4}\).

This example involves variables, but it is performed in the same way as before.

\begin{align*}
&x^{3}x^{4} &\color{red}\small\text{Simplify this expression}\\\\
&x^{3 + 4} &\color{red}\small\text{Product rule of exponents}\\\\
&x^7 &\color{red}\small\text{Add the exponents with common base}\\\
\end{align*}

Example 3
Simplify \(x^{2}x\).

\begin{align*}
&x^{2}x &\color{red}\small\text{Simplify this expression}\\\\ &x^{2 + 1} &\color{red}\small\text{Product rule of exponents}\\\\ &x^3 &\color{red}\small\text{Add the exponents with common base}\\\ \end{align*}

Remember: A variable without a power, such as the second \(x\) in this example, is the same as \(x^{1}\).

Example 4
Simplify \(3^{2}\cdot 5^{4}\cdot 3^{7}\).

This example includes different bases.

\begin{align*}
&3^{2}\cdot 5^{4}\cdot3^{7} &\color{red}\small\text{Simplify this expression}\\\\
&3^{2}\cdot 3^{7}\cdot 5^{4} &\color{red}\small\text{Rearrange into like bases}\\\\
&3^{(2+7)}\cdot 5^{4} &\color{red}\small\text{Product rule of exponents}\\\\
&3^{9}\cdot 5^{4} &\color{red}\small\text{Simplify and add exponents}\\
\end{align*}

You cannot do anything with the base of 5 because it is different from the base of 3. The simplest form for this expression is \(3^{9}5^{4}\).

Example 5
Simplify \(m^{2}\cdot x^{3}\cdot m^{6}\cdot x\).

This example also includes different bases, but in the form of different variables.

\begin{align*}&m^{2}\cdot x^{3}\cdot m^{6}\cdot x&\color{red}\small\text{Simplify this expression}\\\\ &m^{2}\cdot m^{6}\cdot x^{3}\cdot x^1&\color{red}\small\text{Rearrange into like bases}\\\\ &m^{(2+6)}\cdot x^{(3+1)} &\color{red}\small\text{Product rule of exponents}\\\\ &m^{8}\cdot x^{4} &\color{red}\small\text{Simplify and add exponents}\\\\ \end{align*}

The simplest form for this expression is \(m^{8}\cdot x^{4}\).

Example 6
Simplify \((x)(x^{3})\).

\begin{align*}&(x)(x^3) &\color{red}\small\text{Simplify this expression}\\\\ &x^ {(1 + 3)} &\color{red}\small\text{Product rule of exponents}\\\\&x^4 &\color{red}\small\text{Simplify and add exponents}\\\\\end{align*}

Remember that \(x\) has an implied exponent of 1, making it equal to \(x^{1}\) which will make the answer \(x^4\).

Example 7
Simplify \(m^{2}m^{5}\).

\begin{align*}&m^2\cdot m^5 &\color{red}\small\text{Simplify this expression}\\\\&m^ {(2+5)} &\color{red}\small\text{Product rule of exponents}\\\\&m^7&\color{red}\small\text{Simplify and add exponents}\\\\\end{align*}

The answer for \(m^{2}m^{5}\) is \(m^7\).

Example 8
Simplify \(m^{3}x^{6}m\).

\begin{align*}&m^{3}x^{6}m &\color{red}\small\text{Simplify this expression}\\\\&m^{3}m^{1}x^{6} &\color{red}\small\text{Rearrange into like terms}\\\\&m^{3 + 1}x^6 &\color{red}\small\text{Product rule of exponents}\\\\&m^4x^6 &\color{red}\small\text{Simplify and add exponents}\\\\\end{align*}

Since you can’t use the product rule on \(m^{4}\) and \(x^{6}\) because the bases are different, you must leave it as is. Therefore, the answer is \(m^{4}x^{6}\).

Example 9
Simplify \(x^{2}y^{3}x^{4}\).

\begin{align*}&x^{2}y^{3}x^{4} &\color{red}\small\text{Simplify this expression}\\\\&x^{2}x^{4}y^{3} &\color{red}\small\text{Rearrange into like terms}\\\\&x^{2+4}y^{3} &\color{red}\small\text{Product rule of exponents}\\\\&x^{6}y^{3} &\color{red}\small\text{Simplify and add exponents}\\\\\end{align*}

Since you can’t use the product rule with different bases, such as \(x\) and \(y\), you must leave it as is. Therefore, the answer is \(x^{6}y^{3}\).

Example 10
Simplify \(mx^{2}m^{3}x^{7}\).

\begin{align*}&mx^2m^3x^7 &\color{red}\small\text{Simplify this expression}\\\\&m^1 m^3x^2x^7 &\color{red}\small\text{Rearrange so like terms are together}\\\\&m^{1+3}x^{2+7} &\color{red}\small\text{Product rule of exponents}\\\\&m^4x^9 &\color{red}\small\text{Add the exponents when bases are the same}\\\\\end{align*}

By using the product rule to combine like terms, you now have only two terms: \(m^{4}\) and \(x^{9}\). You can’t combine these two terms further, so the answer is \(m^{4}x^{9}\).

Example 11
Simplify \(x^{5}y^{4}x^{2}yz^{2}\).

\begin{align*}
&x^{5}y^{4}x^{2}yz^{2} &\color{red}\small\text{Simplify this expression}\\\\ &x^{5}x^{2}y^{4}yz^{2} &\color{red}\small\text{Rearrange so like bases are together}\\\\ &x^{5+2}y^{4+1}z^{2} &\color{red}\small\text{Product rule of exponents}\\\\ &x^{7}y^{5}z^{2} &\color{red}\small\text{Add exponents with like bases}\\
\end{align*}

As you apply the product rule of exponents to this problem, the answer is \(x^{7}y^{5}z^{2}\).


Things to Remember


  • The base is the number or variable that is being multiplied by itself.
  • The power is the number of the exponent, meaning how many times the base is multiplied by itself.
  • To use the product rule, the bases of the exponents must be the same.
  • When the bases are the same, add the exponents.
  • When the bases are different, you cannot add the exponents.

Practice Problems

Simplify the following expressions:
  1. \(\text{x}\,{\text{x}}^{3}\) (
    Solution
    x
    Solution:
    \({\text{x}}^{4}\)
    Details:
    The product rule states that if two factors raised to an exponent are being multiplied together, and they have the same base, we can add the exponents.

    In this example, \({\text{x}}\) and \({\text{x}}^{3}\) are our two factors. Factors are the numbers that multiply together to make another number or expression.

    This image contains the expression x x to the third power. The first x is tinted blue. The second x and its exponent are tinted green. Both x's represent factors with base x.

    A number without an exponent is the same as a number with exponent 1.

    \({\text{x}} = {\text{x}}^{1}\)

    \({\color{Blue} {\text{x}}}^{{\color{Red} ?}}\,{\color{DarkGreen} {\text{x}}^{3}}\rightarrow {\color{Blue} {\text{x}}^{1}}\,{\color{DarkGreen} {\text{x}}^{3}}\)

    We can rewrite our example problem as \({\text{x}}^{1}\,{\text{x}}^{3}\)

    So \({\text{xx}}^{3} = {\text{x}}^{1}\,{\text{x}}^{3}\)

    In this example, both factors have the same base x, so we can add the exponents together.

    \({\text{x}}^{1}\,{\text{x}}^{3} = {\text{x}}^{\left ( 1 + 3 \right )} = {\text{x}}^{4}\)

    Another way to look at this is to examine what the factors mean.

    \({\text{x}}^{3}\) Is the same as x multiplied together 3 times.

    \({\color{Blue} {\text{x}}}\,{\color{DarkGreen} {\text{x}}^{3}}= {\color{Blue} {\text{x}}}\,{\color{DarkGreen} {\text{xxx}}}\)

    So \({\text{xx}}^{3} = {\text{xxxx}}\)

    This is a total of 4 x’s multiplied together. We can rewrite that as \({\text{x}}^{4}\).
    )
  2. \({\text{m}}^{2}\,{\text{m}}^{5}\) (
    Solution
    x
    Solution:
    \({\text{m}}^{7}\)
    )
  3. \({\text{m}}^{3}\,{\text{x}}^{6}\,{\text{m}}\) (
    Video Solution
    x
    Solution: \({\text{m}}^{4}\,{\text{x}}^{6}\)
    Details:

    (Product Rule #3 (02:03 mins) | Transcript)
    | Transcript)
  4. \({\text{x}}^{2}\,{\text{y}}^{3}\,{\text{x}}^{4}\) (
    Solution
    x
    Solution:
    \({\text{x}}^{6}\,{\text{y}}^{3}\)
    )
  5. \({\text{mx}}^{2}\,{\text{m}}^{3}\,{\text{x}}^{7}\) (
    Solution
    x
    Solution: \({\text{m}}^{4}\,{\text{x}}^{9}\)
    Details:
    Below are two different ways to solve the same problem.

    Version 1:
    In this example, there are four factors: \({\text{m}}\), \({\text{x}}^{2}\), \({\text{m}}^{3}\), and \({\text{x}}^{7}\).

    This image contains the expression m x to the second power m to the third power and x to the seventh power. Arrows are pointing to the x's in the expression with the text: Factors with base x. These factors are tinted blue. Other arrows are pointing to the m's in the expression with the text: Factors with base m. These factors are tinted green.
    You don’t see an exponent on the factor m, but this actually means it has an exponent of 1.

    This image is the same as before, but the first m factor has an exponent of 1. The expression is now m to the first power x to the second power m to the third power and x to the seventh power. Still has arrows pointing to the x's in the expression with the text: Factors with base x. These factors are tinted blue. Other arrows are pointing to the m's in the expression with the text: Factors with base m. These factors are tinted green.
    Since everything is being multiplied together, you can rearrange the factors so the m factors are next to each other and the x factors are next to each other.

    \({\text{m}}^{1}\,{\text{x}}^{2}\,{\text{m}}^{3}\,{\text{x}}^{7} = {\text{m}}^{1}\,{\text{m}}^{3}\,{\text{x}}^{2}\,{\text{x}}^{7}\)

    This image has the same expression as the previous image:  m to the first power x to the second power m to the third power and x to the seventh power. But now there is another equation below the first one. It is the same expression, with the factors arranged in this way: m to the first power m to the third power x to the second power and x to the seventh power.
    The final answer is: \({\text{m}}^{4}\,{\text{x}}^{9}\).

    Version 2:
    Another way to look at this is to break all the factors with exponents into the multiplication of their bases.

    This image shows two expressions: m x to the second power m to the third power and x to the seventh power and m x x m m m x x x x x x x. Each factor with exponents has a line from it to the corresponding group in the second expression. m in the first expression corresponds to m in the second expression. x to the second power to x x, m to the third power to m m m , and x to the seventh power to x x x x x x x.
    m multiplied 1 time

    x multiplied 2 times

    m multiplied 3 times

    x multiplied 7 times

    \({\text{mxxmmmxxxxxxx}}\)

    Since all the factors are being multiplied together you can rearrange them so the factors with the same base are next to each other.

    \({\text{mxxmmmxxxxxxx = mmmmxxxxxxxxx}}\)

    This image shows to the expression m x x m m m x x x x x x x and below it the same expression, but rearranged to m m m m x x x x x x x x x. an arrow from x x in the top expression points to the x x x x x x x x x grouped on the right of the bottom expression. an arrow from m m m in the top expression points down to m m m m grouped on the left of the bottom expression.
    You see there are 4 m factors being multiplied together and 9 x factors being multiplied together. The final step is to rewrite this in exponent form.

    The final answer is: \({\text{m}}^{4}\,{\text{x}}^{9}\).
    )
  6. \({\text{x}}^{5}\,{\text{y}}^{4}\,{\text{x}}^{2}\,{\text{y}}\,{\text{z}}^{2}\) (
    Video Solution
    x
    Solution: \({\text{x}}^{7}\,{\text{y}}^{5}\,{\text{z}}^{2}\)
    Details:

    (Product Rule #6 (02:41 mins))
    | Transcript )

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