8.2
Apply Exponent Properties
Involving Quotients
You used properties of exponents involving products.
Before
You will use properties of exponents involving quotients.
Now
So you can compare magnitudes of earthquakes, as in Ex. 53.
Why?
Key Vocabulary
• power, p. 3
• exponent, p. 3
• base, p. 3
Notice what happens when you divide powers with the same base.
apapapapa
a5
5 a p a 5 a2 5 a 5 2 3
}3 5 }}
apapa
a
The example above suggests the following property of exponents, known as
the quotient of powers property.
For Your Notebook
KEY CONCEPT
Quotient of Powers Property
Let a be a nonzero real number, and let m and n be positive integers such
that m > n.
Words To divide powers having the same base, subtract exponents.
7
am
m2n
Algebra }
,a?0
n 5a
4
Example }2 5 47 2 2 5 45
a
EXAMPLE 1
4
Use the quotient of powers property
(23)9
810
a. }
5 810 2 4
4
SIMPLIFY
EXPRESSIONS
b. }3 5 (23) 9 2 3
8
When simplifying
powers with numerical
bases only, write
your answers using
exponents, as in parts
(a), (b), and (c).
✓
58
(23)
6
5 (23) 6
4
512
p 58
c. 5}
5
}
7
7
5
x6
1
6
d. }
p
x
5
}
4
4
5
x
x
12 2 7
55
5 x6 2 4
5 55
5 x2
GUIDED PRACTICE
for Example 1
Simplify the expression.
611
1. }
5
6
(24) 9
2. }2
(24)
4
p 93
3. 9}
2
9
1
4. }
p y8
5
y
8.2 Apply Exponent Properties Involving Quotients
495
POWER OF A QUOTIENT Notice what happens when you raise a quotient to
a power.
a 4 5 a p a p a p a 5 a p a p a p a 5 a4
} } } }
}
}4
b b b b
bpbpbpb
b
b
}
1 2
The example above suggests the following property of exponents, known as
the power of a quotient property.
For Your Notebook
KEY CONCEPT
Power of a Quotient Property
Let a and b be real numbers with b Þ 0, and let m be a positive integer.
Words To find a power of a quotient, find the power of the numerator and
the power of the denominator and divide.
a m
am
Algebra 1 }
2 5 }m , b Þ 0
b
b
7
3
37
Example 1 }
2 5}
7
2
2
EXAMPLE 2
SIMPLIFY
EXPRESSIONS
When simplifying
powers with numerical
and variable bases,
evaluate the
numerical power,
as in part (b).
Use the power of a quotient property
x 3 x3
a. }
y 2 5 }3
1
1 7x 2
b. 2}
y
EXAMPLE 3
2 3
1 4x5y 2
(4x2)3
Power of a quotient property
(5y)
43 p (x2)3
5}
3 3
5y
6
64x
5}
3
2 5
1 ab 2
Power of a product property
Power of a power property
125y
(a2)5
1
1
p}
5}
p}
5
2
2
2a
b
2a
10
a
1
5}
p}
5
2
b
2a
10
a
5}
2 5
2a b
8
a
5}
5
2b
496
Chapter 8 Exponents and Exponential Functions
2
(27)2
27
49
5 1}
5}
2 5}
2
2
Use properties of exponents
a. } 5 }
3
b. }
2
Power of a quotient property
Power of a power property
Multiply fractions.
Quotient of powers property
x
x
x
✓
GUIDED PRACTICE
for Examples 2 and 3
Simplify the expression.
a 2
5. }
2
1 52
6. 2}
y
1b
EXAMPLE 4
3
2 2
3
t5
2s
8. }
p }
x
7. }
1 4y 2
1 3t 2 1 16 2
Solve a multi-step problem
FRACTAL TREE To construct what is known as a fractal tree, begin with a
single segment (the trunk) that is 1 unit long, as in Step 0. Add three shorter
1
segments that are }
unit long to form the first set of branches, as in Step 1.
2
Then continue adding sets of successively shorter branches so that each new
set of branches is half the length of the previous set, as in Steps 2 and 3.
Step 0
Step 1
Step 2
Step 3
a. Make a table showing the number of new branches at each step for
Steps 124. Write the number of new branches as a power of 3.
b. How many times greater is the number of new branches added at Step 5
than the number of new branches added at Step 2?
Solution
a.
Step
Number of new branches
1
3 5 31
2
9 5 32
3
27 5 33
4
81 5 34
b. The number of new branches added at Step 5 is 35. The number of new
branches added at Step 2 is 32. So, the number of new branches added at
5
3
Step 5 is }
5 33 5 27 times the number of new branches added at Step 2.
2
3
✓
GUIDED PRACTICE
for Example 4
9. FRACTAL TREE In Example 4, add a column to the table for the length of
the new branches at each step. Write the lengths of the new branches as
1
powers of }
. What is the length of a new branch added at Step 9?
2
8.2 Apply Exponent Properties Involving Quotients
497
EXAMPLE 5
Solve a real-world problem
ASTRONOMY The luminosity (in watts) of a star is the
total amount of energy emitted from the star per unit
of time. The order of magnitude of the luminosity of
the sun is 1026 watts. The star Canopus is one of the
brightest stars in the sky. The order of magnitude of
the luminosity of Canopus is 1030 watts. How many
times more luminous is Canopus than the sun?
Solution
Luminosity of Canopus (watts)
1030
}}} 5 } 5 1030 2 26 5 104
Luminosity of the sun (watts)
1026
Canopus
c Canopus is about 104 times as luminous as the sun.
✓
GUIDED PRACTICE
for Example 5
10. WHAT IF? Sirius is considered the brightest star in the sky. Sirius is less
luminous than Canopus, but Sirius appears to be brighter because it is
much closer to Earth. The order of magnitude of the luminosity of Sirius
is 1028 watts. How many times more luminous is Canopus than Sirius?
8.2
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 33 and 51
★ 5 STANDARDIZED TEST PRACTICE
Exs. 2, 19, 37, 46, and 54
5 MULTIPLE REPRESENTATIONS
Ex. 49
SKILL PRACTICE
1. VOCABULARY Copy and complete: In the power 43, 4 is the ? and 3 is
the ? .
2.
★
WRITING Explain when and how to use the quotient of powers
property.
EXAMPLES
1 and 2
on pp. 4952496
for Exs. 3220
SIMPLIFYING EXPRESSIONS Simplify the expression. Write your answer
using exponents.
56
3. }
2
5
(24)7
(24)
1
11. }
132
5
(212)
4
122
1
15. 79 p }
2
7
498
(212)9
8. }3
3
12. }
6. }5
3
2
7. }4
(26)8
39
5. }
5
211
4. }
6
1
16. }
p 911
5
9
Chapter 8 Exponents and Exponential Functions
(26)
5
p 105
9. 10
}
4
10
1
17. }
132
4
6
4
2
14. 2}
p 312
1
18. 49 p 2}
1 54 2
13. 2}
7
p 64
10. 6}
6
1 52
5
1 42
5
19.
★
MULTIPLE CHOICE Which expression is equivalent to 166 ?
164
A }
2
1612
B }
2
16
C
16
20. ERROR ANALYSIS Describe and correct
5
p 93
the error in simplifying 9}
.
94
EXAMPLES
1, 2, and 3
on pp. 495–496
for Exs. 21–37
6 2
16
1}
16 2
D
3
95 p 93
9
98
9
12
j 11
k
1 2
1 2
p 4
1
26. 2}
1 2
1
4c
29. }
2
1d 2
5
1
1 2y 2
2x
34. }
y
1 2
x
a
28. 2}
4
2 2
3x5
32. }
2
3
x
2
1 b2
x
31. }
3
1 2b 2
3 3
24. }
3
4
27. 2}
a
30. }
1
3x
33. }
p}
2
★
2
7 5
3
3 2
37.
x
1 3y 2
1 7y 2
4 3
1
p}
3
m
n
1 5x 2
3
35. }
p }
2
5
6x
6
5 }4 5 9
}
4
SIMPLIFYING EXPRESSIONS Simplify the expression.
1
1
a 9
21. }
p y15
22. z8 p }
23. }
7
8
y
z
y
25. }
q
9 3
16
1}
16 2
1 2
8m
2
2x4 2
y
1 2
36. 2} p }
3
7x3 2
MULTIPLE CHOICE Which expression is equivalent to }4 ?
2y
1 2
5
6
7x
A }
6
5
7x
B }
8
2y
49x6
D }
8
49x
C }
6
2y
4y
4y
SIMPLIFYING EXPRESSIONS Find the missing exponent.
(28)7
?
p 72
39. 7}
5 76
4
38. }? 5 (28) 3
7
(28)
2c3 ?
16c12
41. }
5}
2
8
1
40. }
p p ? 5 p9
5
1d 2
p
d
SIMPLIFYING EXPRESSIONS Simplify the expression.
1
2f 2g 3 4
3fg
42. }
46.
★
3
3 3
(3st)
t
43. 2s
p}
}
2
2
2
st
OPEN – ENDED
are equivalent to 147.
st
2m5n
44. }
2
1 4m
2
mn4
p }
2 1 5n 2
2
3x3y 3
y 2x4 2
5y
1x 2 1 2
45. }
2
p }
Write three expressions involving quotients that
47. REASONING Name the definition or property that justifies each step to
m
a
1
show that }
n 5}
n 2 m for m < n.
a
a
Let m < n.
Given
1
}
1 2
am
am am
}
}
n 5}
1
a
an }
?
m
a
1
5}
n
?
1
5}
n2m
?
a
a
}
m
a
bx
b
48. CHALLENGE Find the values of x and y if you know that }y 5 b 9 and
b x p b2
5 b13. Explain how you found your answer.
}
b3y
8.2 Apply Exponent Properties Involving Quotients
499
PROBLEM SOLVING
EXAMPLES
4 and 5
on pp. 4972498
for Exs. 49251
49.
MULTIPLE REPRESENTATIONS Draw a square with side lengths that
are 1 unit long. Divide it into four new squares with side lengths that are
one half the side length of the original square, as shown in Step 1. Keep
dividing the squares into new squares, as shown in Steps 2 and 3.
Step 0
Step 1
Step 2
Step 3
a. Making a Table Make a table showing the number of new squares
and the side length of a new square at each step for Steps 1–4. Write
the number of new squares as a power of 4. Write the side length of a
1
new square as a power of }
.
2
b. Writing an Expression Write and simplify an expression to find by how
many times the number of squares increased from Step 2 to Step 4.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
50. GROSS DOMESTIC PRODUCT In 2003 the gross domestic product (GDP)
for the United States was about 11 trillion dollars, and the order of
magnitude of the population of the U.S. was 108. Use order of magnitude
to find the approximate per capita (per person) GDP?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
51. SPACE TRAVEL Alpha Centauri is the closest star system to Earth. Alpha
Centauri is about 1013 kilometers away from Earth. A spacecraft leaves
Earth and travels at an average speed of 104 meters per second. About
how many years would it take the spacecraft to reach Alpha Centauri?
52. ASTRONOMY The brightness of one star
relative to another star can be measured
by comparing the magnitudes of the stars.
For every increase in magnitude of 1, the
relative brightness is diminished by a
factor of 2.512. For instance, a star of
magnitude 8 is 2.512 times less bright
than a star of magnitude 7.
The constellation Ursa Minor (the Little
Dipper) is shown. How many times less
bright is Eta Ursae Minoris than Polaris?
Ursa Minor
Polaris
(magnitude 2)
Eta Ursae
Minoris
(magnitude 5)
53. EARTHQUAKES The energy released by one earthquake relative to
another earthquake can be measured by comparing the magnitudes (as
determined by the Richter scale) of the earthquakes. For every increase
of 1 in magnitude, the energy released is multiplied by a factor of about
31. How many times greater is the energy released by an earthquake of
magnitude 7 than the energy released by an earthquake of magnitude 4?
500
5 WORKED-OUT SOLUTIONS
Chapter 8 Exponents
on p. WS1and Exponential Functions
★ 5 STANDARDIZED
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
54.
★
EXTENDED RESPONSE A byte is a unit used to measure computer
memory. Other units are based on the number of bytes they represent.
The table shows the number of bytes in certain units. For example, from
the table you can calculate that 1 terabyte is equivalent to 210 gigabytes.
a. Calculate How many kilobytes are there in
1 terabyte?
b. Calculate How many megabytes are there in
1 petabyte?
c. CHALLENGE Another unit used to measure
computer memory is a bit. There are 8 bits
in a byte. Explain how you can convert the
number of bytes per unit given in the table
to the number of bits per unit.
Unit
Number of bytes
Kilobyte
210
Megabyte
220
Gigabyte
230
Terabyte
240
Petabyte
250
MIXED REVIEW
PREVIEW
Solve the equation. Check your solution. (p. 134)
Prepare for
Lesson 8.3 in
Exs. 55–60.
3
55. }
k59
2
56. }
t 5 24
5
58. 2}y 5 235
2
7
14
59. 2}z 5 }
5
3
4
2
57. 2}
v 5 14
3
3
3
60. 2}z 5 2}
2
4
5
Write an equation of the line that passes through the given points. (p. 292)
61. (22, 1), (0, 25)
62. (0, 3), (24, 1)
63. (0, 23), (7, 23)
64. (4, 3), (5, 6)
65. (4, 1), (22, 4)
66. (21, 23), (23, 1)
QUIZ for Lessons 8.1–8.2
Simplify the expression. Write your answer using exponents.
1. 32 p 36 (p. 489)
2. (54) 3 (p. 489)
3. (32 p 14)7 (p. 489)
4. 72 p 76 p 7 (p. 489)
5. (24)(24) 9 (p. 489)
6. }
(p. 495)
4
7
p 34
8. 3}
(p. 495)
6
9. }
(29) 9
7. }7 (p. 495)
(29)
712
7
1 54 2
3
4
(p. 495)
Simplify the expression.
10. x2 p x5 (p. 489)
11. (3x3)2 (p. 489)
12. 2(7x)2 (p. 489)
13. (6x5) 3 p x (p. 489)
14. (2x5) 3 (7x 7)2 (p. 489)
1
15. }
p x21 (p. 495)
9
w 6
17. 1 }
v 2 (p. 495)
18. }
3
4
16. 1 2}
x 2 (p. 495)
x
3 2
1 x4 2
(p. 495)
19. MAPLE SYRUP PRODUCTION In 2001 the order of magnitude of the number
of maple syrup taps in Vermont was 106. The order of magnitude of the
number of gallons of maple syrup produced in Vermont was 105. About how
many gallons of syrup were produced per tap in Vermont in 2001? (p. 495)
EXTRA PRACTICE for Lesson 8.2, p. 945
ONLINE QUIZ at classzone.com
501
Investigating
g
g
Algebra
ACTIVITY Use before Lesson 8.3
8.3 Zero and Negative Exponents
M AT E R I A L S • paper and pencil
QUESTION
EXPLORE
How can you simplify expressions with zero or
negative exponents?
Evaluate powers with zero and negative exponents
STEP 1 Find a pattern
Copy and complete the tables for the powers of 2 and 3.
Exponent, n
Value of 2n
Exponent, n
Value of 3n
4
16
4
81
3
?
3
?
2
?
2
?
1
?
1
?
As you read the tables from the bottom up, you see that each time the
exponent is increased by 1, the value of the power is multiplied by the base.
What can you say about the exponents and the values of the powers as you
read the table from the top down?
STEP 2 Extend the pattern
Copy and complete the tables using the pattern you observed in Step 1.
Exponent, n
Power, 2n
Exponent, n
Power, 3n
3
8
3
27
2
?
2
?
1
?
1
?
0
?
0
?
21
?
21
?
22
?
22
?
DR AW CONCLUSIONS
Use your observations to complete these exercises
1. Find 2n and 3n for n 5 23, 24, and 25.
2. What appears to be the value of a0 for any nonzero number a?
3. Write each power in the tables above as a power with a positive
1
exponent. For example, you can write 321 as }
.
1
3
502
Chapter 8 Exponents and Exponential Functions