Using the Negative Rule of Exponents
Another useful result occurs if we relax the condition that in the quotient rule even further. For example, can we simplify ? When —that is, where the difference is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal. Divide one exponential expression by another with a larger exponent. Use our example, .
If we were to simplify the original expression using the quotient rule, we would have
Putting the answers together, we have . This is true for any nonzero real number, or any variable representing a nonzero real number.
A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.
We have shown that the exponential expression is defined when is a natural number, 0, or the negative of a natural number. That means that is defined for any integer . Also, the product and quotient rules and all of the rules we will look at soon hold for any integer .
A General Note: The Negative Rule of Exponents
For any nonzero real number and natural number , the negative rule of exponents states thatExample 5: Using the Negative Exponent Rule
Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.Solution
Try It 5
Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.a. b. c.
SolutionExample 6: Using the Product and Quotient Rules
Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.Solution
Try It 6
Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.Licenses & Attributions
CC licensed content, Specific attribution
- College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface. License: CC BY: Attribution.