Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

Math humor:Math humor:

Question: what has variables Question: what has variables with whole-number exponents with whole-number exponents

and a bunch of children?and a bunch of children?

Answer: a mom-nomial!Answer: a mom-nomial!

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

A monomial is a number or a product of numbers and variables with exponents that are whole numbers.

7x5, 3a2b3, n2, 8, z 4

Monomials

Not monomials

m3,4z2.5, 5 + y, , 2x8 w3

Rule: To multiply two monomials, multiply the coefficients and add the exponents that have the same base.

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

Multiply.

Example 1: Multiplying Monomials

A. (3a2)(4a5)

12a7Multiply coefficients. Addexponents that have the same base.

B. (4x2y3)(5xy5)

Multiply coefficients. Addexponents that have the same base.

Use the Comm. and Assoc. Properties. 3 ∙ 4 ∙ a2 + 5

(4 ∙ 5)(x2 ∙ x1)(y3 ∙ y5)

4 ∙ 5 ∙ x2 + 1 ∙ y3+5

20x3y8

Use the Comm. and Assoc. Properties. Think: x = x1.

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

C. (–3p2r)(6pr3s)

Multiply coefficients. Addexponents that have the same base.

–3 ∙ 6 ∙ p2 + 1 ∙ r1+3 ∙ s

–18p3r4s

Use the Comm. and Assoc. Properties. (–3 ∙ 6)(p2 ∙ p1)(r1 ∙ r3)(s)

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

Rule: To divide a monomial by a monomial, divide the coefficients and subtract the exponents of the powers in the denominator from the exponents of the powers in the numerator that have the same base.

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

Divide. Assume that no denominator equals zero.

A.

Divide coefficients. Subtractexponents that have the same base.

Example 2: Dividing Monomials

15m5 3m2

m5-215 3

5m3

B.

Divide coefficients. Subtractexponents that have the same base.

18a2b3 16ab3

a2-1 b3-39 8

a = 1 1/8 a9 8

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

To raise a monomial to a power, you must first understand how to find a power of a product. Notice what happens to the exponents when you find a power of a product.

(xy)3 = xy ∙ xy ∙ xy = x ∙ x ∙ x ∙ y ∙ y ∙ y = x3y3

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

Simplify.Example 3: Raising a Monomial to a Power

A. (3y)3

33 ∙ y3

27y3

Raise each factor to the power.

B. (2a2b6)4

24 ∙ (a2)4 ∙ (b6)4

16a8b24

Raise each factor to the power.

Multiply exponents.

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

Multiply.Check It Out! Example 1

A. (2b2)(7b4)

14b6Multiply coefficients. Addexponents that have the same base.

Use the Comm. and Assoc. Properties.2 ∙ 7 ∙ b2 + 4

B. (4n4)(5n3)(p)

20n7pMultiply coefficients. Addexponents that have the same base.

Use the Comm. and Assoc. Properties.4 ∙ 5 ∙ n4 + 3 ∙ p

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

Multiply.

Check It Out! Example 1

C. (–2a4b4)(3ab3c)

(–2 ∙ 3)(a4 ∙ a)(b4 ∙ b3)(c)

Multiply coefficients. Addexponents that have the same base.

–2 ∙ 3 ∙ a4 + 1 ∙ b4+3 ∙ c

–6a5b7c

Use the Comm. and Assoc. Properties.

(–2 ∙ 3)(a4 ∙ a1)(b4 ∙ b3)(c)

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

Divide. Assume that no denominator equals zero.

A.

Divide coefficients. Subtractexponents that have the same base.

Check It Out! Example 2

18x7 6x2

x7-218 6

3x5

B.

Divide coefficients. Subtractexponents that have the same base.

12m2n3 9mn2

m2-1 n3-24 3

mn = 1 1/3 mn4 3

Evaluating Algebraic Expressions

4-4 Multiplying and Dividing Monomials

Simplify.

Check It Out! Example 3

A. (4a)4

44 ∙ a4

256a4

Raise each factor to the power.

B. (–3x2y)2

(–3)2 ∙ (x2)2 ∙ (y)2

9x4y2

Raise each factor to the power.

Multiply exponents.