evaluating algebraic expressions 4-4 multiplying and dividing monomials af2.2 multiply and divide...
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Evaluating Algebraic Expressions
4-4 Multiplying and Dividing Monomials
AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent.
Also covered: AF1.3
California Standards
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
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.
Additional Example 1: Multiplying Monomials
A. (3a2)(4a5)(3 ∙ 4)(a2 ∙ a5)
12a7
Multiply coefficients. Addexponents that have the same base.
B. (4x2y3)(5xy5)
(4 ∙ 5)(x2 ∙ x)(y3 ∙ y5)
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
Multiply.
Additional Example 1: Multiplying Monomials
C. (–3p2r)(6pr3s)
(–3 ∙ 6)(p2 ∙ p)(r ∙ r3)(s)
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
Multiply.Check It Out! Example 1
A. (2b2)(7b4)
(2 ∙ 7)(b2 ∙ b4)
14b6
Multiply coefficients. Addexponents that have the same base.
Use the Comm. and Assoc. Properties.
2 ∙ 7 ∙ b2 + 4
B. (4n4)(5n3)(p)(4 ∙ 5)(n4 ∙ n3)(p)
20n7p
Multiply 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
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.
Additional 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
a9 8
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
mn4 3
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.Additional 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
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.