3.2 Simplifying Expressions (AM)
Essential Question: How do you simplify algebraic expressions? You can simplify
algebraic expressions by using the Associative, Commutative, Identity, and
Distributive Properties.
Properties of Real Numbers & Combining Like Terms
Identity Property for Addition
𝑎 + 0 = 𝑎 𝑎𝑛𝑑 0 + 𝑎 = 𝑎
Identity Property for Multiplication 𝑎 ∙ 1 = 𝑎 𝑎𝑛𝑑 1 ∙ 𝑎 = 𝑎
Inverse Property for Addition
𝑎 + (−𝑎) = 0 𝑎𝑛𝑑 − 𝑎 + 𝑎 = 0
Inverse Property for Multiplication
Commutative Property for Addition
1
𝑎∙𝑎 =1
𝑎𝑛𝑑
1
𝑎
∙𝑎 =1
𝑎+𝑏 =𝑏+𝑎
Commutative Property for Multiplication 𝑎 ∙ 𝑏 = 𝑏 ∙ 𝑎
Associative Property for Addition
Associative Property for Multiplication
Distributive Property
(𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐)
(𝑎 ∙ 𝑏) ∙ 𝑐 = 𝑎 ∙ (𝑏 ∙ 𝑐)
𝑎(𝑏 + 𝑐) = 𝑎 ∙ 𝑏 + 𝑎 ∙ 𝑐
and
𝑎(𝑏 − 𝑐) = 𝑎 ∙ 𝑏 − 𝑎 ∙ 𝑐
and
Example 1: Simplify
(54 + 9) + 91
2
5
2 7
∙ (3 ∙ 5)
Example 2: Rewrite using the distributive property.
3
(5 − 2𝑦)
5
(𝑏 + 𝑐)𝑎 = 𝑏 ∙ 𝑎 + 𝑐 ∙ 𝑎
(𝑏 − 𝑐)𝑎 = 𝑏 ∙ 𝑎 − 𝑐 ∙ 𝑎
Example 3: Rewrite using the distributive property.
2
5 6
(3 𝑥 + 6) 7
Like terms are terms that are exactly the same or differ only in their numerical coefficients.
Terms that are not like terms are called unlike terms.
3𝑥 2 and 𝑥 2 are like terms because they both have 𝑥 2
4𝑎𝑏 3 and −7𝑎𝑏 3 are like terms because they both have 𝑎𝑏 3
2𝑥 and 3𝑥 2 are not like terms
−5𝑐𝑑2 and 2𝑐𝑑 are not like terms
Only like terms can be added or subtracted.
Example 4: Simplify 9 + 7𝑐 − 3𝑑 + 5𝑐 − 5𝑑 − 3
Example 5: Simplify 5𝑎3 𝑏 2 + 2𝑎2 𝑏 3
Example 6: Simplify 8𝑧 + (2𝑧 − 3)3 + 15
Example 7: Simplify
3 4
[ (5𝑎 − 10)] + 2
4 5