text: Back
  Power Rule
  > ... Math > Rules of Exponents > Power Rule
  
  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
  
  Things to remember: with the power rule, the powers multiply. Everything within the parentheses is affected by the power.
  
  Practice Problems
  
  Simplify the following expressions:
  1. $$({\text{x}}^{\text{m}})^{\text{n}}$$ (
  Solution
  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
  Solution:
  $$3^{6} = 729$$
  )
  3. $$({\text{m}}^{2})^{4}({\text{y}}^{5})^{2}$$ (
  Solution
  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:
  
  $${\text{m m m m m m m m y y y y y y y y y y}}$$
  
  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 $${\text{m}}^{8}{\text{y}}^{10}$$.
  
  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
  Solution:
  $${\text{x}}^{8}{\text{y}}^{8}$$
  Details:
  
  (Video Source | Transcript)
  )
  5. $$(4^{2})^{3}(2^{5})^{1}$$ (
  Video Solution
  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
  Solution:
  $${\text{a}}^{2{\text{x}}}{\text{b}}^{6}$$
  )
  
  Need More Help?
  
  1. Study other Math Lessons in the Resource Center.
  2. Visit the Online Tutoring Resources in the Resource Center.
explanation: The document is an educational resource that explains the power rule of exponents, which involves a base raised to a power, all raised to another power. It provides an example and several practice problems with detailed solutions. Each practice problem is a math question with an answer and explanation.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
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distilabel_metadata.content: Back
  Power Rule
  > ... Math > Rules of Exponents > Power Rule
  
  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
  
  Things to remember: with the power rule, the powers multiply. Everything within the parentheses is affected by the power.
  
  Practice Problems
  
  Simplify the following expressions:
  1. $$({\text{x}}^{\text{m}})^{\text{n}}$$ (
  Solution
  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
  Solution:
  $$3^{6} = 729$$
  )
  3. $$({\text{m}}^{2})^{4}({\text{y}}^{5})^{2}$$ (
  Solution
  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:
  
  $${\text{m m m m m m m m y y y y y y y y y y}}$$
  
  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 $${\text{m}}^{8}{\text{y}}^{10}$$.
  
  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
  Solution:
  $${\text{x}}^{8}{\text{y}}^{8}$$
  Details:
  
  (Video Source | Transcript)
  )
  5. $$(4^{2})^{3}(2^{5})^{1}$$ (
  Video Solution
  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
  Solution:
  $${\text{a}}^{2{\text{x}}}{\text{b}}^{6}$$
  )
  
  Need More Help?
  
  1. Study other Math Lessons in the Resource Center.
  2. Visit the Online Tutoring Resources in the Resource Center.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is an educational resource that explains the power rule of exponents, which involves a base raised to a power, all raised to another power. It provides an example and several practice problems with detailed solutions. Each practice problem is a math question with an answer and explanation.
  
  Label: multiple-question-and-answer