1. 4.2 Integer Exponents and the Quotient Rule

2. Integer Exponents and the Quotient Rule
In all earlier work, exponents were positive integers. Now, to develop a meaning for exponents that are not positive integers, consider the following list.
Each time the exponent is reduced by 1, the value is divided by 2 (the bases). Using this pattern, the list can be continued to smaller and smaller integers.
From the preceding list, it appears that we should define 20 as 1 and negative exponents as reciprocals.
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3. Objective 1
Use 0 as an exponent.
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4. Use 0 as an exponent. Zero Exponent
The definitions of 0 and negative exponents must satisfy the rules for exponents from Section 4.1. For example, if 60 = 1, then
and
so that the product rule is satisfied. Check that the power rules are also valid for a 0 exponent. Thus we define a 0 exponent as follows.
Zero Exponent
For any nonzero real number a, a0 = 1.
Example: = 1
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5. CLASSROOM EXAMPLE 1 Using Zero Exponents Evaluate. Solution:
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6. Use negative numbers as exponents.
Objective 2
Use negative numbers as exponents.
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7. Use negative numbers as exponents.
Since and we can deduce that 2−n should equal
Is the product rule valid in such a case? For example,
The expression 6−2 behaves as if it were the reciprocal of 62 since their product is 1. The reciprocal of 62 is also leading us to define 6−2 as
Negative Exponents
For any nonzero real number a and any integer n,
Example:
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8. Using Negative Exponents
CLASSROOM EXAMPLE 2
Using Negative Exponents
Simplify.
Solution:
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9. Changing from Negative to Positive Exponents
Use negative numbers as exponents. (cont’d)
Consider the following:
Therefore,
Changing from Negative to Positive Exponents
For any nonzero numbers a and b and any integers m and n,
and
Example: and
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10. Changing from Negative to Positive Exponents
CLASSROOM EXAMPLE 3
Changing from Negative to Positive Exponents
Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers.
Solution:
We cannot use this rule to change negative exponents to positive exponents if the exponents occur in a sum or difference of terms. For example,
would be written with positive exponents as
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11. Use the quotient rule for exponents.
Objective 3
Use the quotient rule for exponents.
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12. Use the quotient rule for exponents.
We know that
Notice that the difference between the exponents, 5 − 3 = 2, this is the exponent in the quotient. This example suggests the quotient rule for exponents.
Quotient Rule for Exponents
For any nonzero real number a and any integer m and n,
(Keep the same base; subtract the exponents.)
Example:
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13. Using the Quotient Rule
CLASSROOM EXAMPLE 4
Using the Quotient Rule
Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers.
Solution:
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14. Use the quotient rule for exponents.
The product, quotient, and power rules are the same for positive and negative exponents.
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15. Use combinations of rules.
Objective 4
Use combinations of rules.
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16. Using Combinations of Rules
CLASSROOM EXAMPLE 5
Using Combinations of Rules
Simplify. Assume that all variables represent nonzero real numbers.
Solution:
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