370 Chapter 9 Exponents and Scientific Notation STATE STANDARDS MA.8.A.6.1 MA.8.A.6.3 MA.8.A.6.4 S Zero and Negative Exponents 9.4 How can you define zero and negative exponents? Work with a partner. a. Talk about the following notation. 4327 = 4 ⋅ 10 3 + 3 ⋅ 10 2 + 2 ⋅ 10 1 + 7 ⋅ 10 What patterns do you see in the first three exponents? Continue the pattern to find the fourth exponent. How would you define 10 0 ? Explain. b. Copy and complete the table. n 5 4 3 2 1 0 2 n What patterns do you see in the first six values of 2 n ? How would you define 2 0 ? Explain. c. Use the Quotient of Powers Property to complete the table. 3 5 — 3 2 = 3 5 − 2 = 3 3 = 27 3 4 — 3 2 = 3 4 − 2 = = 3 3 — 3 2 = 3 3 − 2 = = 3 2 — 3 2 = 3 2 − 2 = = What patterns do you see in the first four rows of the table? How would you define 3 0 ? Explain. ACTIVITY: Finding Patterns and Writing Definitions 1 1 Thousands Hundreds Tens Ones
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370 Chapter 9 Exponents and Scientifi c Notation
STATE STANDARDS
MA.8.A.6.1 MA.8.A.6.3 MA.8.A.6.4
S
Zero and Negative Exponents9.4
How can you defi ne zero and negative
exponents?
Work with a partner.
a. Talk about the following notation.
4327 = 4 ⋅ 103 + 3 ⋅ 102 + 2 ⋅ 101 + 7 ⋅ 10
What patterns do you see in the fi rst three exponents?
Continue the pattern to fi nd the fourth exponent.
How would you defi ne 100? Explain.
b. Copy and complete the table.
n 5 4 3 2 1 0
2n
What patterns do you see in the fi rst six values of 2n ?
How would you defi ne 20 ? Explain.
c. Use the Quotient of Powers Property to complete the table.
35
— 32 = 35 − 2 = 33 = 27
34
— 32 = 34 − 2 = =
33
— 32 = 33 − 2 = =
32
— 32 = 32 − 2 = =
What patterns do you see in the fi rst four rows of the table?
How would you defi ne 30 ? Explain.
ACTIVITY: Finding Patterns and Writing Defi nitions11
Thousands Hundreds Tens Ones
Section 9.4 Zero and Negative Exponents 371
Work with a partner.
The quotients show three ratios of the volumes of the solids. Identify each ratio, fi nd its value, and describe what it means.
2r
r
r
2r r
Cylinder Cone Sphere
a. 2π r 3 ÷ 2
— 3
π r 3 =
b. 4
— 3
π r 3 ÷ 2
— 3
π r 3 =
c. 2π r 3 ÷ 4
— 3
π r 3 =
ACTIVITY: Comparing Volumes22
Work with a partner.
Compare the two methods used to simplify 32
— 35 . Then describe how you can
rewrite a power with a negative exponent as a fraction.
Method 1 Method 2
32
— 35 =
3 ⋅ 3 ——
3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3
32
— 35 = 32 − 5
= 1
— 33 = 3−3
ACTIVITY: Writing a Defi nition33
1 1
1 1
Use what you learned about zero and negative exponents to complete Exercises 5 – 8 on page 374.
4. IN YOUR OWN WORDS How can you defi ne zero and negative exponents? Give two examples of each.
372 Chapter 9 Exponents and Scientifi c Notation
Lesson9.4Lesson Tutorials
Zero Exponents
Words Any nonzero number to the zero power is equal to 1. Zero to the zero power, 00, is undefi ned.
Numbers 40 = 1 Algebra a0 = 1, where a ≠ 0
Negative Exponents
Words For any integer n and any number a not equal to 0, a−n is equal to 1 divided by an.
Numbers 4−2 = 1
— 42 Algebra a−n =
1 —
an , where a ≠ 0
EXAMPLE Evaluating Expressions11
a. 3−4 = 1
— 34 Defi nition of negative exponent
= 1
— 81
Evaluate power.
b. (−8.5)−4 ⋅ (−8.5)4 = (−8.5)−4 + 4 Add the exponents.