Languages
Back
Languages
Product Rule
> ... Math > Rules of Exponents > Product Rule

The word product means to multiply. The product rule of exponents helps us remember what we do when two numbers with exponents are multiplied together. Here are some math vocabulary words that will help you to understand this lesson better:

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

The following video will explain, with some examples, what the product rule is; the word product in math means what you get when you multiply things together.
\(\text{x}^3\,\text{x}^4 = \text{x}^7 \)

Rules of Exponents-Product Rule

Video Source (07:19 mins) | Transcript

Qualification for the product rule: bases must be the same. If the bases are the same, then the product rule says that you add the exponents. Also 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.

Additional Resources

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:

    (Video Source | 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 there 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.

    We 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 ms in the expression with the text: Factors with base m. These factors are tinted green.

    Since everything is being multiplied together, we 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.

    Our 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 we 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.

    We 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.

    Our 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:

    (Video Source | Transcript)
    )

Need More Help?

  1. Study other Math Lessons in the Resource Center.
  2. Visit the Online Tutoring Resources in the Resource Center.
  3. Contact your Instructor.
  4. If you still need help, Schedule a Tutor.
Quotient Rule