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370 Chapter 9 Exponents and Scientific Notation
STATE
STANDARDS
MA.8.A.6. MA.8.A.6. MA.8.A.6.
S
9.4^ Zero and Negative Exponents
How can you define zero and negative exponents?
Work with a partner. a. Talk about the following notation.
4327 = (^4) ⋅ 103 + (^3) ⋅ 102 + (^2) ⋅ 10 1 + (^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 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 fi rst 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 32 =^3
34 — 32 =^3
33 — 32
= 3 3 −^2 = =
(^3) —^2 32 =^3
What patterns do you see in the fi rst four rows of the table? How would you define 3 0? Explain.
11 ACTIVITY: Finding Patterns and Writing Definitions
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, find its value, and describe what it means.
2 r
r
r
2 r r
Cylinder Cone Sphere
a. 2 π r^3 ÷ (^2) — 3
π r^3 =
b.^4 — 3
π r^3 ÷ (^) —^2 3
π r^3 =
c. 2 π r^3 ÷ (^4) — 3
π r^3 =
22 ACTIVITY: Comparing Volumes
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
3 2 — 3 5
(^3) ⋅ 3 —— (^3) ⋅ (^3) ⋅ (^3) ⋅ (^3) ⋅ 3
3 2 — 3 5
= 32 −^^5
1 — 33
= 3 −^3
33 ACTIVITY: Writing a Defi nition
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 define zero and negative exponents? Give two examples of each.
Section 9.4 Zero and Negative Exponents 373
EXAMPLE 22 Simplifying Expressions
a. − 5 x^0 = −5(1) Defi nition of zero exponent = − 5 Multiply.
b. 9 y −^3 — y^5 = 9 y −^3 −^5 Subtract the exponents.
= 9 y −^8 Simplify.
= 9 — y^8 Defi nition of negative exponent
Simplify. Write the expression using only positive exponents.
7. 8 x −^2 8. b^0 ⋅ b −^10 9. z^6 — 15 z^9
Exercises 20–
EXAMPLE 33 Real-Life Application
A drop of water leaks from a faucet every second. How many liters of water leak from the faucet in 1 hour? Convert 1 hour to seconds.
1 h × 60 min — 1 h
×
60 sec — 1 min = 3600 sec
Water leaks from the faucet at a rate of 50−^2 liter per second. Multiply the time by the rate.
(^3600) ⋅ 50 −^2 = (^3600) ⋅ 1 — 502 Defi nition of negative exponent
= (^3600) ⋅ 1 — 2500 Evaluate power.
3600 — 2500 Multiply.
11 — 25 = 1.44 Simplify.
So, 1.44 liters of water leak from the faucet in 1 hour.
10. WHAT IF? In Example 4, the faucet leaks water at a rate of 5 −^5 liter per second. How many liters of water leak from the faucet in 1 hour?
C
W
th
Drop of water: 50−^2 L
9.4 Exercises
9 +(-6)=3 3 +(-3)= 4 +(-9)= 9 +(-1)=
374 Chapter 9 Exponents and Scientific Notation
1. VOCABULARY If a is a nonzero number, does the value of a^0 depend on the value of a? Explain. 2. WRITING Explain how to evaluate 10−^3. 3. NUMBER SENSE Without evaluating, order 5 0 , 5^4 , and 5−^5 from least to greatest. 4. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
Rewrite (^) —^1
using a negative exponent. (^) Write 3 to the negative third power.
Write 1 — 3
cubed as a power. Write (−3)^ ⋅ (−3)^ ⋅ (−3) as a power.
5. Use the Quotient of Powers Property to copy and complete the table. 6. What patterns do you see? 7. How would you defi ne 5^0? Why? 8. How can you rewrite 5−^1 as a fraction?
Evaluate the expression.
9. 6 −^2 10. 1580 11. 43 — 45
− 3 — (−3)^2
13. (−2)−^8 ⋅ (−2)^8 14. 3 −^3 ⋅ 3 −^2 15.
1 —
5 −^3 ⋅^
—^1 56
(1.5)^2 ——
(1.5)−^2 ⋅ (1.5)^4
17. ERROR ANALYSIS Describe and correct the error in evaluating the expression. 18. SAND The mass of a grain of sand is about 10 −^3 gram. About how many grains of sand are in the bag of sand? 19. CRITICAL THINKING How can you write the number 1 as 2 to a power? 10 to a power?
Help with Homework
n 4 3 2 1
—^5 n 52
(4) −^3 = ( − 4)( − 4)( − 4)
