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UNIT 1.7 COMMUTATIVE, UNIT 1.7 COMMUTATIVE, ASSOCIATIVE AND DISTRIBUTIVE ASSOCIATIVE AND DISTRIBUTIVE
PROPERTIESPROPERTIES
Use the Commutative, Associative, and Distributive Properties to simplify expressions.
Combine like terms.
Objectives
The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression to simplify it.
Example 1A: Using the Commutative and Associative Properties
Simplify.
11(5)
55
Use the Commutative Property.
Use the Associative Property to make groups of compatible numbers.
Simplify.
Example 1B: Using the Commutative and Associative Properties
45 + 16 + 55 + 4
45 + 55 + 16 + 4
(45 + 55) + (16 + 4)
(100) + (20)
120
Use the Commutative Property.
Use the Associative Property to make groups of compatible numbers.
Helpful Hint
Compatible numbers help you do math
mentally. Try to make multiples of 5
or 10. They are simpler to use when
multiplying.
Check It Out! Example 1a
Simplify.
21
Use the Commutative Property.
Use the Associative Property to make groups of compatible numbers.
Check It Out! Example 1b
Simplify.
410 + 58 + 90 + 2
410 + 90 + 58 + 2
(410 + 90) + (58 + 2)
(500) + (60)
560
Use the Commutative Property.
Use the Associative Property to make groups of compatible numbers.
Check It Out! Example 1c
Simplify.
28
Use the Commutative Property.
Use the Associative Property to make groups of compatible numbers.
12 • 7 • 8
12 • 8 • 7
( )12 • 8 7
4 • 7
The Distributive Property is used with Addition to Simplify Expressions.
The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.
Example 2A: Using the Distributive Property with Mental Math
Write the product using the Distributive Property. Then simplify.
5(59)
5(50 + 9)
5(50) + 5(9)
250 + 45
295
Rewrite 59 as 50 + 9.
Use the Distributive Property.
Multiply.
Add.
8(33)
8(30 + 3)
8(30) + 8(3)
240 + 24
264
Rewrite 33 as 30 + 3.
Use the Distributive Property.
Multiply.
Add.
Example 2B: Using the Distributive Property with Mental Math
Write the product using the Distributive Property. Then simplify.
Check It Out! Example 2a
9(52)
9(50) + 9(2)
9(50 + 2)
450 + 18
468
Rewrite 52 as 50 + 2.
Use the Distributive Property.
Multiply.
Add.
Write the product using the Distributive Property. Then simplify.
Check It Out! Example 2b
12(98)
1176
Rewrite 98 as 100 – 2.
Use the Distributive Property.
Multiply.
Subtract.
12(100 – 2)
1200 – 24
12(100) – 12(2)
Write the product using the Distributive Property. Then simplify.
Check It Out! Example 2c
7(34)
7(30 + 4)
7(30) + 7(4)
210 + 28
238
Rewrite 34 as 30 + 4.
Use the Distributive Property.
Multiply.
Add.
Write the product using the Distributive Property. Then simplify.
The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms.
4x – 3x + 2
Like terms Constant
A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1.
1x2 + 3x
Coefficients
Using the Distributive Property can help you combine like terms. You can factor out the common factors to simplify the expression.
7x2 – 4x2 = (7 – 4)x2
= (3)x2
= 3x2
Factor out x2 from both terms.
Perform operations in parenthesis.
Notice that you can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same.
Example 3A: Combining Like Terms
Simplify the expression by combining like terms.
72p – 25p
72p – 25p
47p
72p and 25p are like terms.
Subtract the coefficients.
Example 3B: Combining Like Terms
Simplify the expression by combining like terms.
A variable without a coefficient has a coefficient of 1.
Write 1 as .
Add the coefficients.
and are like terms.
Example 3C: Combining Like Terms
Simplify the expression by combining like terms.
0.5m + 2.5n
0.5m + 2.5n
0.5m + 2.5n
0.5m and 2.5n are not like terms.
Do not combine the terms.
Check It Out! Example 3
Simplify by combining like terms.
3a. 16p + 84p
16p + 84p
100p16p + 84p are like terms.
Add the coefficients.
3b. –20t – 8.5t2
–20t – 8.5t2 20t and 8.5t2 are not like terms.
–20t – 8.5t2 Do not combine the terms.
3m2 + m3 3m2 and m3 are not like terms.
3c. 3m2 + m3
Do not combine the terms.3m2 + m3
Example 4: Simplifying Algebraic ExpressionsSimplify 14x + 4(2 + x). Justify each step.
14x + 4(2) + 4(x) Distributive Property
Multiply.Commutative Property
Associative Property
Combine like terms.
14x + 8 + 4x
(14x + 4x) + 8
14x + 4x + 8
18x + 8
14x + 4(2 + x)1. 2. 3. 4. 5. 6.
Procedure Justification
6(x) – 6(4) + 9 Distributive Property
Multiply.Combine like terms.
6x – 24 + 9
6x – 15
6(x – 4) + 91. 2. 3. 4.
Procedure Justification
Check It Out! Example 4a
Simplify 6(x – 4) + 9. Justify each step.
–12x – 5x + x + 3a Commutative Property
Combine like terms.–16x + 3a
–12x – 5x + 3a + x1. 2. 3.
Procedure Justification
Check It Out! Example 4b
Simplify −12x – 5x + 3a + x. Justify each step.
Lesson Quiz: Part I
Simplify each expression.
1. 165 +27 + 3 + 5
2.
Write each product using the Distributive Property. Then simplify.
3. 5($1.99)
4. 6(13)
200
8
5($2) – 5($0.01) = $9.95
6(10) + 6(3) = 78
Lesson Quiz: Part II
Simplify each expression by combining like terms. Justify each step with an operation or property.
7. 301x – x
8. 24a + b2 + 3a + 2b2
5.
300x
27a + 3b2
6. 14c2 – 9c 14c2 – 9c
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