Chapter 1 Review

WHY is it helpful to write numbers in
different ways?
The Number System
Course 3, Lesson 1-1

To
• write fractions as decimals
• write decimals as fractions
The Number System

• rational number
• repeating decimal
• terminating decimal
The Number System

The Number System
Words A rational number is a number that can be written as the
ratio of two integers in which the denominator is not zero.
Symbols , where a and b are integers and b ≠ 0
Model
a
b

1
Need Another Example?
2
Step-by-Step Example
1. Write the fraction as a decimal.
means 5 ÷ 8.
0.6
5.000
– 48
20
– 16
40
– 40
0
8 Divide 5 by 8.
25

Answer
Write as a decimal.
0.1875

1
2
3
2. Write the mixed number as a decimal.
can be rewritten as .
Divide 5 by 3 and add a
negative sign.
The mixed number can be written as –1.6.

Answer
Write as a decimal.
–3.18

1
2
3
4
3. In a recent season, St. Louis Cardinals first
baseman Albert Pujols had 175 hits in 530 at bats.
To the nearest thousandth, find his batting average.
To find his batting average, divide the number of hits, 175,
by the number of at bats, 530.
175
Look at the digit to the right of the thousandths place.
Since 1 < 5, round down.
530 0.3301886792
Albert Pujols’s batting average was 0.330.

Answer
When Juliana went strawberry picking, 28 of
the 54 strawberries she picked weighed less
than 2 ounces. To the nearest thousandth, find
the fraction of the strawberries that weighed
less than 2 ounces.
0.519

1
2
4. Write 0.45 as a fraction.
0.45 =
=
0.45 is 45 hundredths.
Simplify.

Answer
Write 0.32 as a fraction in simplest form.

1
2
3
4
5
6
5. Write 0.5 as a fraction in simplest form.
Assign a variable to the value 0.5. Let N = 0.555... . Then
perform operations on N to determine its fractional value.
N = 0.555...
Simplify.
Multiply each side by 10 because 1 digit repeats.
7
10(N) = 10(0.555...)
Multiplying by 10 moves the decimal point
1 place to the right.
10N = 5.555...
–N = 0.555... Subtract N = 0.555... to eliminate the repeating part.
Divide each side by 9.
9N = 5
N =
The decimal 0.5 can be written as .

Answer
Write 2.7 as a mixed number in simplest
form.

1
2
3
4
5
6
6. Write 2.18 as a mixed number in simplest form.
Assign a variable to the value 2.18. Let N = 2.181818….
Then perform operations on N to determine its fractional value.
N = 2.181818...
Simplify.
Multiply each side by 100 because 2 digits repeat.100(N) = 100(2.181818...)
Multiplying by 100 moves the decimal point
2 places to the right.
100N = 218.181818
–N = 2.181818… Subtract N = 2.181818… to eliminate the repeating part.
Divide each side by 99. Simplify.
99N = 216
N =
7 The decimal 2.18 can be written as .

Answer
Write 5.45 as a mixed number in simplest form.

• To write and evaluate
expressions using exponents
Course3, Lesson 1-2
The Number System

• power
• base
• exponent
Course3, Lesson 1-2
The Number System

1
2
1. Write the expression using exponents.
(–2) • (–2) • (–2) • 3 • 3 • 3 • 3
The base –2 is a factor 3 times, and the base 3
is a factor 4 times.
(–2) • (–2) • (–2) • 3 • 3 • 3 • 3 = (–2)3 • 34

Answer
Write (–9) • (–9) • (–9) • (–9) • (–9) • 4 • 4 • 4 as
an expression using exponents.
(–9)5 • 43

1
2
3
2. Write the expression using exponents.
a • b • b • a • b
Use the properties of operations to rewrite and group
like bases together. The base a is a factor 2 times,
and the base b is a factor 3 times.
a • b • b • a • b = a • a • b • b • b
= a2 • b3

Answer
Write x • y • x • y • x • x as an expression
using exponents.
x4 • y2

1
2
3. Evaluate .
=
Write the power as a product.
Multiply

Answer
Evaluate .

1
2
3
4
4. The deck of a skateboard has an area of about
25 • 7 square inches. What is the area of the
skateboard deck?
25 • 7 = 2 • 2 • 2 • 2 • 2 • 7 Write the power as a product.
Associative Property= (2 • 2 • 2 • 2 • 2) • 7
= 32 • 7 or 224
The area of the skateboard deck is about
224 square inches.
Multiply.

Answer
A racquetball court has an area of 2 • 42 • 52
square feet. What is the area of the
racquetball court?
800 ft2

1
2
3
5. Evaluate the expression if a = 3 and b = 5.
a2 + b4
a2 + b4 = 32 + 54 Replace a with 3 and b with 5.
Write the powers as products.= (3 • 3) + (5 • 5 • 5 • 5)
= 9 + 625 or 634 Add.

Answer
Evaluate x3 + y5 if x = 4 and y = 2.
96

1
2
3
6. Evaluate the expression if a = 3 and b = 5.
(a – b)2
(a – b)2 = (3 – 5)2 Replace a with 3 and b with 5.
Perform operations in the
parentheses first.
= (–2)2
= (–2) • (–2) or 4
Write the powers as products.
Then simplify.

Answer
Evaluate (x + y)2 if x = 4 and y = 2.
36

• To multiply and divide powers
The Number System

• monomial
The Number System

The Number System
Words To multiply powers with the same base, add their
exponents.
Numbers Algebra
Examples or4 3 4 3
2 2 2 
 7
2 m n m n
a a a 


1
2
3
4
1. Simplify using the Laws of Exponents.
52 • 5
52 • 5 = 52 • 51 5 = 51
The common base is 5.= 52 + 1
= 53 or 125 Add the exponents.
Simplify.
Check 52 • 5 = (5 • 5) • 5
= 5 • 5 • 5
= 53

Answer
Simplify 76 • 7 using the Laws of Exponents.
77

1
2
c3 • c5
c3 • c5 = c3 + 5 The common base is c.
Add the exponents.= c8

Answer
Simplify r4 • r6 using the Laws of Exponents.
r10

1
2
3
–3x2 • 4x5
–3x2 • 4x5 = (–3 • 4)( x2 • x5) Commutative and Associative Properties
The common base is x.= (–12)(x2 + 5)
= –12x7 Add the exponents.

Answer
Simplify −7x2 • 11x4 using the Laws of
Exponents.
−77x6

The Number System
Words To divide powers with the same base, subtract their
exponents.
Numbers Algebra
Examples
or
7
7 3
3
3
3
3

 4
3 , where 0
m
m n
n
a
a a
a

 

1
2
= 48 – 2 The common base is 4.
Simplify.= 46 or 4,096

Answer
Simplify using the Laws of Exponents.
610

1
2
= n9 – 4 The common base is n.
Simplify.= n5

Answer
Simplify using the Laws of Exponents.
a6

1
2
3
4
= Group by common base.
Subtract the exponents.= 23 • 31 • 51
23 = 8= 8 • 3 • 5
Simplify.= 120

Answer
Simplify .
450

1
2
3
7. Hawaii’s total shoreline is about 210 miles long.
New Hampshire’s shoreline is about 27 miles long.
About how many times longer is Hawaii’s shoreline
than New Hampshire’s?
To find how many times longer, divide 210 by 27.
Quotient of Powers= 210 – 7 or 23
Hawaii’s shoreline is about 23 or 8 times longer.

Answer
One centimeter is equal to 10 millimeters, and
one kilometer is equal to 106 millimeters. How
many centimeters are in one kilometer?
105 cm

To find the
• power of a power,
• power of a product
The Number System

The Number System
Words To find the power of a power, multiply the exponents.
Numbers Algebra
Examples
 
3
2 2 • 3 65 = 5 or 5   •=
n
m m na a

1
2
(84)3
(84)3 = 84 • 3
Simplify.= 812
Power of a Power

Answer
Simplify (52)8 .
516

1
2
(k7)5
(k7)5 = k7 • 5
Simplify.= k35
Power of a Power

Answer
Simplify (a3)7 .
a21

The Number System
Words To find the power of a product, find the power of each
factor and multiply.
Numbers Algebra
Examples
     
33 32 2 66 = 6 or 216x x x   =
m
m mab a b

1
2
(4p3)4
(4p3)4 = 44 • p3 • 4
Simplify.= 256p12
Power of a Power

Answer
Simplify (3c4)3.
27c12

1
2
(–2m7n6)5
(–2m7n6)5 = (–2)5m7 • 5n6 • 5
Simplify.= –32m35n30
Power of a Product

Answer
Simplify (−4p5q)2.
16p10q2

1
2
3
4
5
5. A magazine offers a special service to its
subscribers. If they scan the square logo
shown on a smartphone, they can receive
special offers from the magazine. Find the
area of the logo.
A = s2
Replace s with 7a4b.A = (7a4b)2
Area of a square
Power of a ProductA = 72(a4)2(b1)2
Simplify.A = 49a8b2
The area of the logo is 49a8b2 square units.

Answer
Find the volume of a cube with side lengths
of 6mn7 . Express as a monomial.
216m3n21 cubic units

To
• use negative exponents,
• multiply and divide powers
using negative exponents
The Number System

The Number System
Words Any nonzero number to the zero power is 1. Any nonzero
number to the negative n power is the multiplicative inverse
of its nth power.
Numbers Algebra
Examples 05 1
  3
or
-3 1 1 1 17
7 7 7 7
 0 1, 0x x
, - 0
1
n
n xx x

1
1. Write the expression using a positive exponent.
6–3
6–3 = Definition of negative exponent.

Answer
Write 4−4 using a positive exponent.

1
2. Write the expression using a positive exponent.
a–5
a–5 = Definition of negative exponent.

Answer
Write c−7 using a positive exponent.

1
3. Write the fraction as an expression using a
negative exponent other than –1.
= 5–2 Definition of negative exponent.

Answer
Write as an expression using a negative
exponent.
f–5

1
2
4. Write the fraction as an expression using a
negative exponent other than –1.
= 6–2
Definition of exponent
Definition of negative exponent

Answer
Write as an expression using a negative
exponent.
3−2

1
2
3
4
5. One human hair is about 0.001 inch
in diameter. Write the decimal as a power of 10.
= 10–3
1,000 = 103
Definition of negative exponent
0.001 = Write the decimal as a fraction.
=
A human hair is 10–3 inch thick.

Answer
A grain of salt has a mass of about 0.0001
gram. Write the decimal as a power of 10.
10–4

1
2
3
6. Simplify the expression.
53 • 5–5
= or
Simplify.
Write using positive exponents. Simplify.
53 • 5–5 = 53 + (–5) Product of Powers
= 5–2

Answer
Simplify 4−5 · 4−3.

1
2
7. Simplify the expression.
Subtract the exponents.
= w–1 – (–4) Quotient of Powers
= w(–1) + 4 or w3

Answer
Simplify .
c2

To
• write numbers in standard form,
• write numbers in scientific notation
The Number System

• scientific notation
The Number System

The Number System
Words Scientific notation is when a number is written as the
product of a factor and an integer power of 10. When the
number is positive the factor must be greater than or equal
to 1 and less than 10.
Symbols , where 1 ≤ a < 10 and n is an integer
Example 425,000,000 =
×10na
84.25×10

1
1. Write the number in standard form.
5.34 × 104
5.34 × 104 = 53,400.

Answer
Write 9.62 × 105 in standard form.
962,000

1
2. Write the number in standard form.
3.27 × 10–3
3.27 × 10−3 = 0.00327

Answer
Write 2.85 × 10−5 in standard form.
0.0000285

1
2
3. Write the number in scientific notation.
3,725,000
3,725,000 = 3.725 × 1,000,000 The decimal point moves 6 places.
= 3.725 × 106 Since 3,725,000 > 1, the exponent is positive.

Answer
Write 931,500,000 in scientific notation.
9.315 × 108

1
2
4. Write the number in scientific notation.
0.000316
0.000316 = 3.16 × 0.0001 The decimal point moves 4 places.
= 3.16 × 10–4 Since 0 < 0.000316 < 1, the exponent is negative.

Answer
Write 0.0044 in scientific notation.
4.4 × 10−3

1
2
5. Refer to the table at the right.
Order the countries according
to the amount of money
visitors spent in the United
States from greatest to least.
1.06 × 107
Canada and
United Kingdom
↓
1.06 > 1.03
Group the numbers
by their power of 10.
Mexico
and India
↓
1.03 × 107
7.15 × 106
1.83 × 106
>
7.15 > 1.83
Compare the decimals.↑
United
Kingdom
↑
Canada
↑
Mexico
↑
India

Answer
The following table lists the maximum
frequency for the colors of the visible light
spectrum. List the colors from greatest to
least frequency.
violet, blue, green, orange, red

1
6. If you could walk at a rate of 2 meters per
second, it would take you 1.92 × 108 seconds to walk to
the moon. Is it more appropriate to report this time as
1.92 × 108 seconds or 6.09 years? Explain your reasoning.
The measure 6.09 years is more appropriate. The number
1.92 × 108 seconds is very large so choosing a larger unit
of measure is more meaningful.

Answer
One light year is about 9.46 × 1012 kilometers
or 9.46 × 1018 millimeters. Is it more appropriate
to report this length as 9.46 × 1012 kilometers or
9.46 × 1018 millimeters? Explain your reasoning.
9.46 × 1012 kilometers; the number is very large
so choosing a larger unit of measure is more
meaningful.

To
• add, subtract, multiply and divide
with numbers written in scientific
notation
The Number System

1
2
3
4
5
1. Evaluate (7.2 × 103)(1.6 × 104). Express the
result in scientific notation.
(7.2 × 103)(1.6 × 104) = (7.2 × 1.6)(103 × 104)
= (11.52)(103 × 104)
= 11.52 × 103 + 4
= 11.52 × 107
= 1.152 × 108
Commutative and
Associative Properties
Multiply 7.2 by 1.6.
Product of Powers
Add the exponents.
Write in scientific notation.

Answer
Evaluate (1.1 × 10–3)(2.5 × 109). Express the
2.75 × 106

1
2
3
4
5
6
2. In 2010, the world population was about 6,860,000,000. The population
of the United States was about 3 × 108. About how many times larger is
the world population than the population of the United States?
Estimate the population of the world and write in scientific notation.
6,860,000,000 ≈ 7,000,000,000 or 7 × 109
Find .
=
≈ 2.3 ×
Associative Property
Divide 7 by 3. Round to the nearest tenth.
Quotient of Powers
7
≈ 2.3 × 101
So, the population of the world is about 23 times larger
than the population of the United States.
≈ 2.3 × 109 – 8
Subtract the exponents.

Answer
The largest planet in our solar system is Jupiter
with a diameter of about 143,000 kilometers.
The smallest planet in our solar system is
Mercury with a diameter of about 5 × 103
kilometers. About how many times greater is the
diameter of Jupiter than the diameter of
Mercury?
Sample answer: 3 × 101 or 30 times greater

1
2
3
4
3. Evaluate the expression. Express the result in
scientific notation.
(6.89 × 104) + (9.24 × 105)
(6.89 × 104) + (9.24 × 105)
= (6.89 + 92.4) × 104
Write 9.24 × 105 as 92.4 × 104.
Distributive Property
Rewrite in scientific notation.= 9.929 × 105
= (6.89 × 104) + (92.4 × 104)
= 99.29 × 104 Add 6.89 and 92.4.

Answer
Evaluate (2.85 × 107) + (1.61 × 109). Express
the result in scientific notation.
1.6385 × 109

1
2
3
4
5
4. Evaluate the expression. Express the result in scientific notation.
(7.83 × 108) – 11,610,000
(7.83 × 108) – (1.161 × 107)
= (78.3 × 107) – (1.161 × 107)
Rewrite 11,610,000 in scientific
notation.
Write 7.83 × 108 as 78.3 × 107.
Subtract 1.161 from 78.3.= 77.139 × 107
= (78.3 – 1.161) × 107 Distributive Property
(7.83 × 108) – (1.161 × 107)
Rewrite in scientific notation.= 7.7139 × 108

Answer
Evaluate (8.23 × 106) – 391,000. Express the
7.839 × 106

1
2
3
4
5. 593,000 + (7.89 × 106)
= (0.593 × 106) + (7.89 × 106)
= (5.93 × 105) + (7.89 × 106) Rewrite 593,000 in scientific notation.
Write 5.93 × 105 as 0.593 × 106
Add 0.593 and 7.89.= 8.483 × 106
= (0.593 + 7.89) × 106 Distributive Property
593,000 + (7.89 × 106)

Answer
Evaluate 6,450,000,000 – (8.27 × 107).
Express the result in scientific notation.
6.3673 × 109

To
• find the square roots of perfect squares,
• solve equations with square and cube
roots
The Number System

• square root
• perfect square
• radical sign
• cube root
• perfect cube
The Number System

1
1. Find the square root.
√64
Find the positive square
root of 64; 82 = 64.
√64 = 8

Answer
15
Find √225.

1
2.
Find both square roots of 1.21;
1.12 = 1.21.
± √1.21 = ± 1.1
Find the square root.
± √1.21

Answer
±1.2
Find ±√1.44.

1
3. Find the square root.
Find the negative square root of

1
4.
There is no real square root because no number
times itself is equal to –16.
Find the square root.
√ –16

Answer
no real square root
Find √ –81.

1
2
3
5. Solve t2 = 169. Check your solution(s).
t2 = 169 Write the equation.
t = 13 and –13
Definition of square root
Check 13 · 13 = 169 and (–13)(–13) = 169
t = ±√169

Answer
x2 = 225
±15

The Number System
Words A square root of a number is one of its two equal factors.
Symbols
Example
2If = ,then is a square root of .x y x y
25 =25 so 5 is a square root of 25.

1
6. Find the cube root.
√125
53 = 5 · 5 · 5 or 125
3
√125 = 5
3

Answer
3
Find √27.
3

1
7.
(–3)3 = (–3) · (–3) · (–3) or –27
Find the cube root.
√ –27
3
√–27 = –3
3

Answer
–10
Find √ –1,000.
3

1
2
3
4
5
8. Dylan has a planter in the shape of a cube that
holds 8 cubic feet of potting soil. Solve the equation
8 = s3 to find the side length s of the container.
Write the equation.8 = s3
Take the cube root of each side.
So, each side of the container is 2 feet.
Definition of cube root2 = s
Check (2)3 = 8
√8 = s
3

Answer
A desktop organizer that is shaped like a cube
has a volume of 125 cubic inches. What is
the length of one side of the organizer?
5 in.

The Number System
Words A cube root of a number is one of its three equal factors.
Symbols 3If = ,then is the cube root of .x y x y

To
• estimate square roots and cube
roots to the nearest integer
The Number System

1
2
3
4
5
6
1. Estimate √83 to the nearest integer.
7
• The largest perfect square less than 83 is 81.
• The smallest perfect square greater than 83 is 100.
√81 = 9
√100 = 10
Plot each square root on a number line. Then estimate √83.
81 < 83 < 100 Write an inequality.
92 < 83 < 102 81 = 92 and 100 = 102
√92 < √83 < √102 Find the square root of each number.
9 < √83 < 10 Simplify.
So, √83 is between 9 and 10. Since √83 is closer to √81
than √100, the best integer estimate for √83 is 9.

Answer
Estimate √54 to the nearest integer.
7

1
2
3
4
5
6
2. Estimate √320 to the nearest integer.
• The largest perfect cube less than 320 is 216.
• The smallest perfect cube greater than 320 is 343.
216 < 320 < 343
Find the cube root of each number.
Simplify.
So, √320 is between 6 and 7. Since
320 is closer to 343 than 216, the
best integer estimate for √320 is 7.
Write an inequality.
63 < 320 < 73 216 = 63 and 343 = 73
3
√63 < √320 < √73
6 < √320 < 7
√216 = 6.
√343 = 7.
3
3
3 3 3
3
3
3

Answer
Estimate √100 to the nearest integer.
5
3

1
2
3
4
5
6
3. Wyatt wants to fence in a square portion of the yard to make
a play area for his new puppy. The area covered is 2 square
meters. How much fencing should Wyatt buy?
√2 is between 1.41 and 1.42.
Wyatt will need 4 • √2 meters of fencing. The square root of 2 is between
1 and 2 so 4 • √2 is between 4 and 8. Is this the best approximation?
You can truncate the decimal expansion of √2 to find better approximations.
Truncate, or drop, the digits after the first decimal
place. √2 is between 1.4 and 1.5.
√2 ≈ 1.414213562 Use a calculator.
5.6 < 4√2 < 6.0
To find a closer approximation, expand √2 then truncate the
decimal expansion after the first two decimal places.
7
Estimate √2 by truncating, or dropping, the digits after the first decimal place, then after
the second decimal place, and so on until an appropriate approximation is reached.
4 • 1.4 = 5.6 and 4 • 1.5 = 6.0
4 • 1.41 = 5.64 and 4 • 1.42 = 5.685.64 < 4√2 < 5.68
The approximations indicate that Wyatt should buy
6 yards of fencing.
√2 ≈ 1.414213562
√2 ≈ 1.414213562

Answer
A square flower garden has an area of
250 square feet. A stone path runs along the
outermost edge of the flower garden. Find
three sets of approximations for the length of
the path. Then determine the length of the path.
Sample answer: 60 ft and 64 ft; 63.2 ft and
63.6 ft; 63.24 ft and 63.28 ft; 63.2 ft

1
2
3
4
5
6
4. The golden rectangle is found frequently in the nautilus shell. The
length of the longer side divided by the length of the shorter side is
equal to
. Estimate this value.
4 = 22 and 9 = 32
4 < 5 < 9 4 and 9 are the closest perfect squares.
√22 < √5 < √32
First estimate the value of √5.
Find the square root of each number.
Since 5 is closer to 4 than 9, the best integer estimate
for √5 is 2. Use this value to evaluate the expression.
7
22 < 5 < 32
2 < √5 < 3 Simplify.

Answer
To estimate the time in seconds it will take an
object to fall h feet, you can use the
expression . About how long will it take an
object to fall from a height of 38 feet?
1.5 s

To
• classify numbers,
• compare and order real numbers
The Number System

• irrational number
• real number
The Number System

The Number System
Words
Examples
Rational Number
A rational number is a
number that can be
expressed as the ratio ,
where a and b are integers
and b ≠ 0.
Irrational Number
An irrational number is a
number that cannot be
expressed as the ratio,
where a and b are integers
and b ≠ 0.
7
, -12
8
76-2,5,3. 1.414213562....2

1
1. Name all sets of numbers to which the real
number belongs. 0.2525...
The decimal ends in a repeating pattern. It is a
rational number because it is equivalent to .

Answer
Name all sets of numbers to which
0.090909… belongs.
rational

1
number belongs. √36
Since √36 = 6, it is a natural number, a whole
number, an integer, and a rational number.

Answer
Name all sets of numbers to which √25
belongs.
natural, whole, integer, rational

1
number belongs. –√7
–√7 ≈ –2.645751311… The decimal does not
terminate nor repeat, so it is an irrational number.

Answer
Name all sets of numbers to which −√12
belongs.
irrational

1
2
3
4
4. Fill in the with <, >, or = to make a true statement.
√7 ≈ 2.645751311…
2 = 2.666666666…
Since 2.645751311… is less than 2.66666666…,
√7 < 2 .

Answer
Replace the in √15 3 with <, >, or =
to make a true statement.
<

1
2
3
4
5. Fill in the with <, >, or = to make a true statement.
15.7% √0.02
15.7% = 0.157
√0.02 ≈ 0.141
Since 0.157 is greater than 0.141, 15.7% > √0.02.

Answer
Replace the in 12.3% √0.01 with <, >, or = to
make a true statement.
>

1
2
3
4
5
6
6. Order the set √30 , 6, 5 , 5.36 from least to
greatest. Verify your answer by graphing on a number line.
Write each number as a decimal. Then order the decimals.
√30 ≈ 5.48
5.36 ≈ 5.37
7
6 = 6.00
5 = 5.80
From least to greatest, the order is 5.36, √30, 5 , and 6.

Answer
3, √15, 4.21, 4
Order the set √15, 3, 4 , 4.21 from least to
greatest. Verify your answer by graphing on a
number line.

1
2
3
4
7. On a clear day, the number of miles a person can see to the horizon is
about 1.23 times the square root of his or her distance from the ground in
feet. Suppose Frida is at the Empire Building observation deck at 1,250 feet
and Kia is at the Freedom Tower observation deck at 1,362 feet. How much
farther can Kia see than Frida?
Use a calculator to approximate the distance each person can see.
Frida: 1.23 • √1,250 ≈ 43.49
Kia can see 45.39 – 43.49 or 1.90 miles farther than Frida.
Kia: 1.23 • √1,362 ≈ 45.39

Answer
The time in seconds that it takes an object
to fall d feet is . About how many seconds
would it take for a volleyball thrown 32 feet
up in the air to fall from its highest point to
the sand?
1.4 s

Chapter 1 Review

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Chapter 1 Review