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Power Rule
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The power rule is about a base raised to a power, all raised to another power. What does this mean? This means everything raised to the interior exponent is then multiplied together the number of times of the exterior exponent.

Example:

\(\left (2^{3} \right )^{4} = 2^{(3\cdot4)} = 2^{12}\)

Rules of Exponents - Power Rule

Video Source (08:23 mins) | Transcript

Things to remember: with the power rule, the powers multiply. Everything within the parentheses is affected by the power.

Additional Resources

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, we can multiply the \(2 \cdot 4\) to get the exponent for m. We can also multiply the \(5 \cdot 2\) to get the exponent for y.

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

    Our 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 we 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.

    We now see there are 8 m’s being multiplied together and 10 y’s being multiplied together. If we 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.

    Our final answer is \({\text{m}}^{8}{\text{y}}^{10}\), which is the same answer as when we used 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:

    (Video Source | Transcript)
    )
  5. \((4^{2})^{3}(2^{5})^{1}\) (
    Video Solution
    x
    Solution:
    \(4^{6}2^{5} = 4096 \cdot 32 = 131,072\)
    Details:

    (Video Source | 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