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Simplifying Expressions using Properties

Learn how to simplify expressions using the Commutative, Associative, and Distributive Properties. Practice combining like terms and using mental math shortcuts.

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Simplifying Expressions using Properties

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  1. Purpose Students will be able to use the Commutative, Associative, and Distributive Properties to simplify expressions and combine like terms.

  2. Example 1: Using the Commutative and Associative Properties Simplify. Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. 11(5) 55

  3. Example 2: Using the Commutative and Associative Properties Simplify. 45 + 16 + 55 + 4 45 + 55 + 16 + 4 Use the Commutative Property. (45 + 55) + (16 + 4) Use the Associative Property to make groups of compatible numbers. (100) + (20) 120

  4. Helpful Hint Compatible numbers help you do math mentally. Try to make multiples of 5 or 10. They are simpler to use when multiplying. Example: 15 + 18 + 5 easier(15 + 5) + 18 = 20 + 18 harder (15 + 18) +5 = 33 + 5

  5. You Try! Simplify. Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. 21

  6. ( ) 7 1 2 1 2 7 • 8 8•7 • • 1 2 8 • 4 7 • You Try! Simplify. Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. 28

  7. Note: The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

  8. Example 3: Using the Distributive Property with Mental Math Write the product using the Distributive Property. Then simplify. 5(59) Rewrite 59 as 50 + 9. 5(50 + 9) Use the Distributive Property. 5(50) + 5(9) 250 + 45 Multiply. 295 Add.

  9. You Try! Write the product using the Distributive Property. Then simplify. 9(52) 9(50 + 2) Rewrite 52 as 50 + 2. 9(50) + 9(2) Use the Distributive Property. 450 + 18 Multiply. 468 Add.

  10. You Try! Write the product using the Distributive Property. Then simplify. 12(98) 12(100 – 2) Rewrite 98 as 100 – 2. 12(100) –12(2) Use the Distributive Property. 1200 – 24 Multiply. 1176 Subtract.

  11. 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. Like terms Constant 4x –3x + 2

  12. A coefficientis a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1. Coefficients 1x2 + 3x

  13. Using the Distributive Property can help you combine like terms. 7x2 – 4x2 = (7 – 4)x2 Factor out x2 from both terms. = (3)x2 Perform operations in parenthesis. = 3x2

  14. 3m2 + m3 You Try! Simplify by combining like terms. 3a. 16p + 84p 16p + 84p 16p + 84p are like terms. 100p 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. 3c. 3m2 + m3 3m2 + m3 3m2 and m3are not like terms. Do not combine the terms.

  15. Example 4: Simplifying Algebraic Expressions Simplify 14x + 4(2 + x). Justify each step. Procedure Justification 1. 14x + 4(2 + x) 2. 14x + 4(2) + 4(x) Distributive Property 3. 14x + 8 + 4x Multiply. Commutative Property 4. 14x + 4x + 8 5. (14x + 4x) + 8 Associative Property 6. 18x + 8 Combine like terms.

  16. You Try! Simplify 6(x –4) + 9.Justify each step. Procedure Justification 1. 6(x –4) + 9 2. Distributive Property 6(x)–6(4) + 9 3. 6x – 24 + 9 Multiply. Combine like terms. 4. 6x – 15

  17. 2. Exit Task Simplify each expression. 1. 165 +27 + 3 + 5 200 8 Write each product using the Distributive Property. Then simplify. 3. 5($1.99) 5($2) – 5($0.01) = $9.95 4. 6(13) 6(10) + 6(3) = 78

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