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Understanding Multiplication and Division of Powers with the Same Base

This guide explores the properties of multiplying and dividing powers with the same base. We provide clear examples demonstrating the multiplication and division of powers, establishing that for a real number ( b ) and positive integers ( m ) and ( n ), specific rules apply. The properties for the multiplication and division of powers are illustrated with practical examples, reinforcing the concept that both operations can be simplified when the bases are identical. Engage with practical problems to solidify your understanding.

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Understanding Multiplication and Division of Powers with the Same Base

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  1. Multiplication and Division of Powers • We will often want to be able to multiply and divide powers that have the same base. • An example of Multiplication of Powers • An example of Division of Powers

  2. Consider the following multiplication of powers: • This suggests the property for the Multiplication of Powers. Let b represent a real number and m and n represent positive integers. Then …

  3. Consider the following division of powers: • This suggests the property for the Division of Powers. Let b represent a real number and m and n represent positive integers. Then …

  4. Example 1 Both bases are the same, so apply the property of multiplication of powers:

  5. Example 2 Since the bases are not the same, we cannot use the property of multiplication of powers.

  6. Example 3 Both bases are the same, so apply the property of multiplication of powers:

  7. Example 4 Both bases are the same, so apply the property of division of powers:

  8. Example 5 Both bases are the same, so apply the property of division of powers:

  9. Example 6 This problem uses both of the properties. Note that all of the bases are the same. First apply the multiplication property in the numerator:

  10. Now apply the division property:

  11. Example 7 Note that all of the bases are the same. First apply the multiplication property in the denominator:

  12. Now apply the division property:

  13. END OF PRESENTATION

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