1. Integer Exponents and Scientific Notation
5.1
Integer Exponents and Scientific Notation
Use the product rule for exponents.
Define 0 and negative exponents.
Use the quotient rule for exponents.
Use the power rules for exponents.
Simplify exponential expressions.
Use the rules for exponents with scientific notation.

2. Integer Exponents and Scientific Notation
We use exponents to write products of repeated factors. For example,
25 is defined as 2 • 2 • 2 • 2 • 2 = 32.
The number 5, the exponent, shows that the base 2 appears as a factor five times. The quantity 25 is called an exponential or a power. We read 25 as “2 to the fifth power” or “2 to the fifth.”
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3. Use the product rule for exponents.
Objective 1
Use the product rule for exponents.
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4. Product Rule for Exponents
Use the product rule for exponents.
Product Rule for Exponents
If m and n are natural numbers and a is any real number, then
am • an = am + n.
That is, when multiplying powers of like bases, keep the same base and add the exponents.
Be careful not to multiply the bases. Keep the same base and add the exponents.
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5. m8 • m6 m5 • p4 (–5p4) (–9p5) (–3x2y3) (7xy4)
CLASSROOM EXAMPLE 1
Using the Product Rule for Exponents
Apply the product rule, if possible, in each case.
Solution:
m8 • m6
m5 • p4
(–5p4) (–9p5)
(–3x2y3) (7xy4)
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6. Define 0 and negative exponents.
Objective 2
Define 0 and negative exponents.
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7. Zero Exponent If a is any nonzero real number, then a0 = 1.
Define 0 and negative exponents.
Zero Exponent
If a is any nonzero real number, then
a0 = 1.
The expression 00 is undefined.
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8. 290 (–29)0 –290 80 – 150 Using 0 as an Exponent Evaluate.
CLASSROOM EXAMPLE 2
Using 0 as an Exponent
Evaluate.
290
(–29)0
–290
80 – 150
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9. Define 0 and negative exponents.
For any natural number n and any nonzero real number a,
A negative exponent does not indicate a negative number; negative exponents lead to reciprocals.
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10. 6-5 (2x)-4 –7p-4 Using Negative Exponents
CLASSROOM EXAMPLE 3
Using Negative Exponents
Write with only positive exponents.
6-5
(2x)-4
–7p-4
Evaluate: 4-1 – 2-1
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11. Using Negative Exponents
CLASSROOM EXAMPLE 4
Using Negative Exponents
Evaluate.
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12. Special Rules for Negative Exponents
Define 0 and negative exponents.
Special Rules for Negative Exponents
If a ≠ 0 and b ≠ 0 , then
and
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13. Use the quotient rule for exponents.
Objective 3
Use the quotient rule for exponents.
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14. Quotient Rule for Exponents
Use the quotient rule for exponents.
Quotient Rule for Exponents
If a is any nonzero real number and m and n are integers, then
That is, when dividing powers of like bases, keep the same base and subtract the exponent of the denominator from the exponent of the numerator.
Be careful when working with quotients that involve negative exponents in the denominator. Write the numerator exponent, then a subtraction symbol, and then the denominator exponent. Use parentheses.
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15. Using the Quotient Rule for Exponents
CLASSROOM EXAMPLE 5
Using the Quotient Rule for Exponents
Apply the quotient rule, if possible, and write each result with only positive exponents.
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16. Use the power rules for exponents.
Objective 4
Use the power rules for exponents.
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17. Power Rule for Exponents
Use the power rules for exponents.
Power Rule for Exponents
If a and b are real numbers and m and n are integers, then
and
That is,
a) To raise a power to a power, multiply exponents.
b) To raise a product to a power, raise each factor to that power.
c) To raise a quotient to a power, raise the numerator and the denominator to that power.
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18. Using the Power Rules for Exponents
CLASSROOM EXAMPLE 6
Using the Power Rules for Exponents
Simplify, using the power rules.
Solution:
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19. Special Rules for Negative Exponents, Continued
Use the power rules for exponents.
Special Rules for Negative Exponents, Continued
If a ≠ 0 and b ≠ 0 and n is an integer, then
and
That is, any nonzero number raised to the negative nth power is equal to the reciprocal of that number raised to the nth power.
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20. Using Negative Exponents with Fractions
CLASSROOM EXAMPLE 7
Using Negative Exponents with Fractions
Write with only positive exponents and then evaluate.
Solution:
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21. Use the power rules for exponents. Definition and Rules for Exponents
For all integers m and n and all real numbers a and b, the following rules apply.
Product Rule
Quotient Rule
Zero Exponent
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22. Use the power rules for exponents.
Definition and Rules for Exponents, Continued
Negative Exponent
Power Rules
Special Rules
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23. Simplify exponential expressions.
Objective 5
Simplify exponential expressions.
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24. Using the Definitions and Rules for Exponents
CLASSROOM EXAMPLE 8
Using the Definitions and Rules for Exponents
Simplify. Assume that all variables represent nonzero real numbers.
Solution:
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25. Simplify. Assume that all variables represent nonzero real numbers.
CLASSROOM EXAMPLE 8
Using the Definitions and Rules for Exponents (cont’d)
Simplify. Assume that all variables represent nonzero real numbers.
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26. Use the rules for exponents with scientific notation.
Objective 6
Use the rules for exponents with scientific notation.
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27. Use the rules for exponents with scientific notation.
In scientific notation, a number is written with the decimal point after the first nonzero digit and multiplied by a power of 10.
This is often a simpler way to express very large or very small numbers.
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28. Use the rules for exponents with scientific notation.
A number is written in scientific notation when it is expressed in the form
a × 10n
where 1 ≤ |a| < 10 and n is an integer.
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29. Converting to Scientific Notation
Use the rules for exponents with scientific notation.
Converting to Scientific Notation
Step 1 Position the decimal point. Place a caret, ^, to the right of the first nonzero digit, where the decimal point will be placed.
Step 2 Determine the numeral for the exponent. Count the number of digits from the decimal point to the caret. This number gives the absolute value of the exponent on 10.
Step 3 Determine the sign for the exponent. Decide whether multiplying by 10n should make the result of Step 1 greater or less. The exponent should be positive to make the result greater; it should be negative to make the result less.
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30. CLASSROOM EXAMPLE 9
Writing Numbers in Scientific Notation
Write the number in scientific notation.
29,800,000
346,100,000,000
102,000,000,000,000
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31. CLASSROOM EXAMPLE 9
Writing Numbers in Scientific Notation (cont’d)
Write the number in scientific notation.
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32. Converting a Positive Number from Scientific Notation
Use the rules for exponents with scientific notation.
Converting a Positive Number from Scientific Notation
Multiplying a positive number by a positive power of 10 makes the number greater, so move the decimal point to the right if n is positive in 10n.
Multiplying a positive number by a negative power of 10 makes the number less, so move the decimal point to the left if n is negative.
If n is 0, leave the decimal point where it is.
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33. CLASSROOM EXAMPLE 10
Converting from Scientific Notation to Standard Notation
Write each number in standard notation.
2.51 ×103
5.07 ×10−7
–6.8 ×10−4
When converting from scientific notation to standard notation, use the exponent to determine the number of places and the direction in which to move the decimal point.
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34. Using Scientific Notation in Computation
CLASSROOM EXAMPLE 11
Using Scientific Notation in Computation
Evaluate
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35. CLASSROOM EXAMPLE 12
Using Scientific Notation to Solve Problems
The distance to the sun is 9.3 × 107 mi. How long would it take a rocket traveling at 3.2 × 103 mph to reach the sun?
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