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Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Find each product.
1. 4 • 4
2. 7 • 7
3. 5 • 5
Perform the indicated operations.
5. 3 + 12 – 7
6. 6 • 1 ÷ 2
7. 4 – 2 + 9
8. 10 – 5 – 4
9. 5 • 5 + 7
10. 30 ÷ 6 • 2
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4. 9 • 9
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Solutions
1. 4 • 4 = 16
2. 7 • 7 = 49
3. 5 • 5 = 25
4. 9 • 9 = 81
5. 3 + 12 – 7 = (3 + 12) – 7 = 15 – 7 = 8
6. 6 • 1 ÷ 2 = (6 • 1) ÷ 2 = 6 ÷ 2 = 3
7. 4 – 2 + 9 = (4 – 2) + 9 = 2 + 9 = 11
8. 10 – 5 – 4 = (10 – 5) – 4 = 5 – 4 = 1
9. 5 • 5 + 7 = (5 • 5) + 7 = 25 + 7 = 32
10. 30 ÷ 6 • 2 = (30 ÷ 6) • 2 = 5 • 2 = 10
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Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify 32 + 62 – 14 • 3.
32 + 62 – 14 • 3 = 32 + 36 – 14 • 3 Simplify the power: 62 = 6 • 6 = 36.
= 32 + 36 – 42
Multiply 14 and 3.
= 68 – 42
Add and subtract in order from left to right.
= 26
Subtract.
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Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Evaluate 5x = 32 ÷ p for x = 2 and p = 3.
5x + 32 ÷ p = 5 • 2 + 32 ÷ 3
Substitute 2 for x and 3 for p.
=5•2+9÷3
Simplify the power.
= 10 + 3
Multiply and divide from left to right.
= 13
Add.
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Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Find the total cost of a pair of jeans that cost $32 and have
an 8% sales tax.
total cost
original price
C
=
p
+
sales tax
r•p
sales tax rate
C=p+r•p
= 32 + 0.08 • 32
Substitute 32 for p. Change 8% to 0.08 and
substitute 0.08 for r.
= 32 + 2.56
Multiply first.
= 34.56
Then add.
The total cost of the jeans is $34.56.
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Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify 3(8 + 6) ÷ (42 – 10).
3(8 + 6) ÷ (42 – 10) = 3(8 + 6) ÷ (16 – 10)
Simplify the power.
= 3(14) ÷ 6
Simplify within parentheses.
= 42 ÷ 6
Multiply and divide from left to right.
=7
Divide.
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Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Evaluate each expression for x = 11 and z = 16.
a. (xz)2
b. xz2
(xz)2 = (11 • 16)2
Substitute 11 for x and 16 for z.
= (176)2
Simplify within parentheses. Multiply.
= 30,976
Simplify.
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xz2 = 11 • 162
= 11 • 256
= 2816
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify 4[(2 • 9) + (15 ÷ 3)2].
4[(2 • 9) + (15 ÷ 3)2] = 4[18 + (5)2]
First simplify (2 • 9) and (15 ÷ 3).
= 4[18 + 25]
Simplify the power.
= 4[43]
Add within brackets.
= 172
Multiply.
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Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
A carpenter wants to build three decks in the shape of
regular hexagons. The perimeter p of each deck will be 60 ft. The
perpendicular distance a from the center of each deck to one of the
sides will be 8.7 ft.
pa
Use the formula A = 3 ( 2
pa
A=3( 2
=3(
) to find the total area of all three decks.
)
60 • 8.7
)
2
Substitute 60 for p and 8.7 for a.
522
Simplify the numerator.
=3( 2
= 3(261)
)
Simplify the fraction.
= 783
Multiply.
The total area of all three decks is 783 ft2.
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Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify each expression.
1. 50 – 4 • 3 + 6
44
2. 3(6 + 22) – 5
25
3. 2[(1 + 5)2 – (18 ÷ 3)]
60
Evaluate each expression.
4. 4x + 3y for x = 2 and y = 4
20
5. 2 • p2 + 3s for p = 3 and s = 11
51
6. xy2 + z for x = 3, y = 6 and z = 4
112
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Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
(For help, go to skills handbook page 725.)
Write each decimal as a fraction and each fraction as a decimal.
1. 0.5
5.
2
5
2. 0.05
6.
3
8
3. 3.25
7.
1-3
2
3
4. 0.325
8. 3 5
9
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Solutions
1. 0.5 = 5 = 5 • 1 = 1
10
5•2
2
2. 0.05 = 5 = 5 • 1 = 1
100 5 • 20
20
3. 3.25 = 3 25 = 3 25 • 1 = 3 1 or 13
25 • 4
4
100
4
4. 0.325 = 325 = 25 • 13 = 13
1000
5.
25 • 40
40
2 = 2 ÷ 5 = 0.4
5
3 = 3 ÷ 8 = 0.375
8
7. 2 = 2 ÷ 3 = 0.6
3
8. 3 5 = 3 + (5 ÷ 9) = 3.5
9
6.
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Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Name the set(s) of numbers to which each number belongs.
a. –13
b. 3.28
integers
rational numbers
rational numbers
1-3
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Which set of numbers is most reasonable for displaying
outdoor temperatures?
integers
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Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Determine whether the statement is true or false. If it is false,
give a counterexample.
All negative numbers are integers.
A negative number can be a fraction, such as –
The statement is false.
1-3
2
. This is not an integer.
3
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Write – 3 , – 7 , and – 5 , in order from least to greatest.
4
12
8
– 3 = –0.75
Write each fraction as a decimal.
–0.75 < –0.625 < –0.583
Order the decimals from least to greatest.
4
– 7 = –0.583
12
– 5 = –0.625
8
From least to greatest, the fractions are – 3 , – 5 , and – 7 .
4
1-3
8
12
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Find each absolute value.
a. |–2.5|
b. |7|
–2.5 is 2.5 units from 0
on a number line.
7 is 7 units from 0
on a number line.
|–2.5| = 2.5
|7| = 7
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Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Name the set(s) of numbers to which each given number belongs.
1. –2.7
2.
rational numbers
11
3. 16
irrational numbers
Use <, =, or > to compare.
4. 3
4
>
5.
5
8
–3 < – 5
4
6. Find |– 7 |.
12
7
12
1-3
8
natural numbers,
whole numbers
integers,
rational numbers
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
x
Evaluate – y – 4z2 for x = 4, y = –2, and z = –4.
–4
– x – 4z2 = –2 – 4(–4)2
y
–4
Substitute 4 for x, –2 for y, and –4 for z.
= –2 – 4(16)
Simplify the power.
= 2 – 64
Divide and multiply.
= –62
Subtract.
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Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Evaluate p for p = 3 and r = – 3 .
r
p
=p÷r
r
2
4
Rewrite the equation.
3
(– 34 )
Substitute 2 for p and – 4 for r.
3
4
Multiply by – 3 , the reciprocal of – 4 .
= 2 ÷
= 2 (– 3
= –2
)
3
3
4
Simplify.
1-6
3
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Simplify.
1. –8(–7)
2. –6(–7 + 10) – 4
– 22
56
Evaluate each expression for m = –3, n = 4, and p = –1.
3. 8m + p
n
–7
4. (mp)3
5. mnp
27
12
1
2
6. Evaluate 2a ÷ 4b – c for a = –2, b = – 1 , and c = – .
3
1
32
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The Distributive Property
ALGEBRA 1 LESSON 1-7
(For help, go to Lessons 1-2 and 1-6.)
Use the order of operations to simplify each expression.
1. 3(4 + 7)
2. –2(5 + 6)
4. –0.5(8 – 6)
5.
1
t(10 – 4)
2
1-7
3. –1(–9 + 8)
6. m(–3 – 1)
The Distributive Property
ALGEBRA 1 LESSON 1-7
Solutions
1. 3(4 + 7) = 3(11) = 33
2. –2(5 + 6) = –2(11) = –22
3. –1(–9 + 8) = –1(–1) = 1
4. –0.5(8 – 6) = –0.5(2) = –1
5.
1
1
1
1
t(10 – 4) = t(6) = (6)t = ( • 6) t = 3t
2
2
2
2
6. m(–3 – 1) = m(–4) = –4m
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The Distributive Property
ALGEBRA 1 LESSON 1-7
Use the Distributive Property to simplify 26(98).
26(98) = 26(100 – 2)
Rewrite 98 as 100 – 2.
= 26(100) – 26(2)
Use the Distributive Property.
= 2600 – 52
Simplify.
= 2548
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The Distributive Property
ALGEBRA 1 LESSON 1-7
Find the total cost of 4 CDs that cost $12.99 each.
4(12.99) = 4(13 – 0.01)
Rewrite 12.99 as 13 – 0.01.
= 4(13) – 4(0.01)
Use the Distributive Property.
= 52 – 0.04
Simplify.
= 51.96
The total cost of 4 CDs is $51.96.
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The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify 3(4m – 7).
3(4m – 7) = 3(4m) – 3(7)
= 12m – 21
Use the Distributive Property.
Simplify.
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The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify –(5q – 6).
–(5q – 6) = –1(5q – 6)
Rewrite the expression using –1.
= –1(5q) – 1(–6)
Use the Distributive Property.
= –5q + 6
Simplify.
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The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify –2w2 + w2.
–2w2 + w2 = (–2 + 1)w2
= –w2
Use the Distributive Property.
Simplify.
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The Distributive Property
ALGEBRA 1 LESSON 1-7
Write an expression for the product of –6 and the quantity 7
minus m.
Relate: –6 times the quantity 7 minus m
Write:
–6
•
(7 – m)
–6(7 – m)
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The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify each expression.
3. – 3(2y – 7)
– 6y + 21
1. 11(299)
3289
2. 4(x + 8)
4x + 32
4. –(6 + p)
5. 1.3a + 2b – 4c + 3.1b – 4a
–6–p
–2.7a + 5.1b – 4c
6. Write an expression for the product of 4 and the quantity b minus 3 .
7
4
3
b
–
(
)
7
5
1-7
5
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
(For help, go to Lessons 1-4 and 1-6.)
Simplify each expression.
1. 8 + (9 + 2)
2. 3 • (–2 • 5)
3. 7 + 16 + 3
4. –4(7)(–5)
5. –6 + 9 + (–4)
6. 0.25 • 3 • 4
7. 3 + x – 2
8. 2t – 8 + 3t
9. –5m + 2m – 4m
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Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Solutions
1. 8 + (9 + 2) = 8 + (2 + 9) = (8 + 2) + 9 = 10 + 9 = 19
2. 3 • (–2 • 5) = 3 • (–10) = –30
3. 7 + 16 + 3 = 7 + 3 + 16 = 10 + 16 = 26
4. –4(7)(–5) = –4(–5)(7) = 20(7) = 140
5. –6 + 9 + (–4) = –6 + (–4) + 9 = –10 + 9 = –1
6. 0.25 • 3 • 4 = 0.25 • 4 • 3 = 1 • 3 = 3
7. 3 + x – 2 = 3 + (–2) + x = 1 + x
8. 2t – 8 + 3t = 2t + 3t – 8 = (2 + 3)t – 8 = 5t – 8
9. –5m + 2m – 4m = (–5 + 2 – 4)m = –7m
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Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Name the property each equation illustrates.
a. 3 • a = a • 3
Commutative Property of Multiplication,
because the order of the factors changes
b. p • 0 = 0
Multiplication Property of Zero, because a
factor multiplied by zero is zero
c. 6 + (–6) = 0
Inverse Property of Addition, because the sum of a
number and its inverse is zero
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Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Suppose you buy a shirt for $14.85, a pair of pants for
$21.95, and a pair of shoes for $25.15. Find the total amount you
spent.
14.85 + 21.95 + 25.15 = 14.85 + 25.15 + 21.95
Commutative Property of Addition
= (14.85 + 25.15) + 21.95 Associative Property of Addition
= 40.00 + 21.95
Add within parentheses first.
= 61.95
Simplify.
The total amount spent was $61.95.
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Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Simplify 3x – 4(x – 8). Justify each step.
3x – 4(x – 8) = 3x – 4x + 32
Distributive Property
= (3 – 4)x + 32
Distributive Property
= –1x + 32
Subtraction
= –x + 32
Identity Property of Multiplication
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Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Name the property that each equation illustrates.
1. 1m = m
2. (– 3 + 4) + 5 = – 3 + (4 + 5)
Iden. Prop. Of Mult.
3. –14 • 0 = 0
Assoc. Prop. Of Add.
Mult. Prop. Of Zero
4. Give a reason to justify each step.
a. 3x – 2(x + 5) = 3x – 2x – 10
Distributive Property
b.
= 3x + (– 2x) + (– 10)
Definition of Subtraction
c.
= [3 + (– 2)]x + (– 10)
Distributive Property
d.
= 1x + (– 10)
Addition
e.
= 1x – 10
Definition of Subtraction
f.
= x – 10
Identity Property of Multiplication
1-8