Chapter 01 – Section 06

Commutative and Associative Properties

© William James Calhoun

To recognize and use the commutative and associative properties and to simplify expressions.

This section is about two more properties.

You do need to memorize their names, but

Focus on understanding how these properties work with problems.

Remember that SIMPLIFY means:

1) Get rid of all parenthesis (distribution)

2) Combine Like Terms

© William James Calhoun

Question: What is 2 + 5? How about 5 + 2? Does it matter in what order you add two numbers? How about three numbers?

Question: What about multiplying numbers?

• commutative property - the order in which you add or multiply two numbers does not change their sum or product.

For any numbers a and b, a + b = b + a and a * b = b * a.

COMMUTATIVE PROPERTY

You can add/multiply two numbers in any order.

77

nono

Same as adding.

© William James Calhoun

Question: Is there any mathematical difference to these two statements?Two plus five is seven. Seven plus four is eleven.Five plus four is nine. Nine plus two is eleven.

No – both give you eleven in the end.

• associate property - the way you group three or more numbers when adding or multiplying does NOT change their sum or product.

For any numbers a, b, and c, (a + b) + c = a + (b + c) andab(c) = a(bc).

ASSOCIATIVE PROPERTY

When adding and multiplying - group any way you want to.2 + 3 + 5 = (2 + 3) + 5 = 2 + (3 + 5) = 10 and2 * 3 * 5 = (2 * 3) * 5 = 2 * (3 * 5) = 30.

© William James Calhoun

NOTE: Associative and Commutative Properties do NOT work for subtraction and division!

Here is a chart of the properties mainly used to simplify expressions.

Addition MultiplicationCommutative a + b = b + a ab = baAssociative (a + b) + c = a + (b + c) (ab)c = a(bc)

Identity0 is the identity.a + 0 = 0 + a = a

1 is the identity.a * 1 = 1 * a = a

Zero a * 0 = 0 * a = 0DistributiveSubstitution If a = b, then a may be substituted for b.

The following properties are true for any numbers a, b, and c.

a(b + c) = ab + ac and (b + c)a = ba + ca

© William James Calhoun

EXAMPLE 1α: Simplify each expression.a. 6(a + b) – a + 3b b.

EX1EX1β

2 1 4x 10

3 2 3

Distribute.

Reorder (if you want.)Signs in front stay the same.

CLT

Clean up, if necessary.

6a + 6b – a + 3b

6a – a + 6b + 3b

5a + 9b

5a + 9b

2 x 4

3 2 35

4

2 3

x 2

35

x

27

x7

2

© William James Calhoun

EXAMPLE 1β: Simplify each expression.a. 6(2x + 4y) + 2(x + 9) b. 5(0.3x + 0.1y) + 0.2x

© William James Calhoun

EXAMPLE 2α: a. Write an algebraic expression for the verbal expression

the sum of two and the square of t increased by the sum of t squared and 3.

b. Then simplify the algebraic expression.

EX2EX2β

Remember: Your main goal should be to employ these properties. There is no need to memorize their proper names.

32t ++t 22 +

2 + t2 + t2 + 3

t2 + t2 + 2 + 3

2t2 + 5

© William James Calhoun

EXAMPLE 2β: a. Write an algebraic expression for the verbal expression

the difference of x cubed and three increased by the difference of 5 and x cubed.

b. Then simplify the algebraic expression.