Simplifying Expressions

The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression to simplify it.

Example 1

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 2

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.

Example 3

Simplify.

21

Use the Commutative Property.

Use the Associative Property to make groups of compatible numbers.

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 4

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.

Example 5

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.

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

Example 6

Simplify the expression by combining like terms.

72p – 25p

72p – 25p

47p

72p and 25p are like terms.

Subtract the coefficients.

Example 7

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 8

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.

Example 9

Simplify by combining like terms.

A. 16p + 84p

100p 16p + 84p are like terms.

Add the coefficients.

B. –20t – 8.5t2

20t and 8.5t2 are not like terms.

Do not combine the terms.

3m2 and m3 are not like terms.

C. 3m2 + m3

Do not combine the terms.

Example 10

Simplify 14x + 4(2 + x).

14x + 4(2) + 4(x)

14x + 8 + 4x

(14x + 4x) + 8

18x + 8

14x + 4(2 + x)

There is a difference…

• Between (-4)2 and -42

• You HAVE to know the difference.

• (-4)2 means (-4)(-4) or 16

• -42 means - 4 · 4 or -16

• BIG difference!!

Try These…

Simplify each expression.

1. 165 +27 + 3 + 5

2.

Write each product using the Distributive Property. Then simplify.

4. 5($1.99)

5. 6(13)

200

8

5($2) – 5($0.01) = $9.95

6(10) + 6(3) = 78

3. -62 -36

Try These (cont)…

Simplify each expression by combining like terms.

8. 301x – x

9. 24a + b2 + 3a + 2b2

6.

300x

27a + 3b2

7. 14c2 – 9c 14c2 – 9c