Evaluating Algebraic Expressions 4-4 Multiplying and Dividing Monomials Math humor: Question: what has variables with whole-number exponents and a bunch of children? 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. 7x 5, 3a 2 b 3, n 2, 8, z4z4 Monomials Not monomials m 3,4z 2.5, 5 + y,, 2 x 8w38w3 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. (3a 2 )(4a 5 ) 12a 7 Multiply coefficients. Add exponents that have the same base. B. (4x 2 y 3 )(5xy 5 ) Multiply coefficients. Add exponents that have the same base. Use the Comm. and Assoc. Properties. 3 ∙ 4 ∙ a (4 ∙ 5)(x 2 ∙ x 1 )(y 3 ∙ y 5 ) 4 ∙ 5 ∙ x ∙ y x 3 y 8 Use the Comm. and Assoc. Properties. Think: x = x 1.

Evaluating Algebraic Expressions 4-4 Multiplying and Dividing Monomials C. (–3p 2 r)(6pr 3 s) Multiply coefficients. Add exponents that have the same base. –3 ∙ 6 ∙ p ∙ r 1+3 ∙ s –18p 3 r 4 s Use the Comm. and Assoc. Properties. (–3 ∙ 6)(p 2 ∙ p 1 )(r 1 ∙ r 3 )(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. Subtract exponents that have the same base. Example 2: Dividing Monomials 15m 5 3m 2 m m35m3 B. Divide coefficients. Subtract exponents that have the same base. 18a 2 b 3 16ab 3 a 2-1 b a = 1 1 / 8 a 9898

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 = x 3 y 3

Evaluating Algebraic Expressions 4-4 Multiplying and Dividing Monomials Simplify. Example 3: Raising a Monomial to a Power A. (3y) ∙ y 3 27y 3 Raise each factor to the power. B. (2a 2 b 6 ) ∙ (a 2 ) 4 ∙ (b 6 ) 4 16a 8 b 24 Raise each factor to the power. Multiply exponents.

Evaluating Algebraic Expressions 4-4 Multiplying and Dividing Monomials Multiply. Check It Out! Example 1 A. (2b 2 )(7b 4 ) 14b 6 Multiply coefficients. Add exponents that have the same base. Use the Comm. and Assoc. Properties. 2 ∙ 7 ∙ b B. (4n 4 )(5n 3 )(p) 20n 7 p Multiply coefficients. Add exponents that have the same base. Use the Comm. and Assoc. Properties. 4 ∙ 5 ∙ n ∙ p

Evaluating Algebraic Expressions 4-4 Multiplying and Dividing Monomials Multiply. Check It Out! Example 1 C. (–2a 4 b 4 )(3ab 3 c) (–2 ∙ 3)(a 4 ∙ a)(b 4 ∙ b 3 )(c) Multiply coefficients. Add exponents that have the same base. –2 ∙ 3 ∙ a ∙ b 4+3 ∙ c –6a 5 b 7 c Use the Comm. and Assoc. Properties. (–2 ∙ 3)(a 4 ∙ a 1 )(b 4 ∙ b 3 )(c)

Evaluating Algebraic Expressions 4-4 Multiplying and Dividing Monomials Divide. Assume that no denominator equals zero. A. Divide coefficients. Subtract exponents that have the same base. Check It Out! Example 2 18x 7 6x 2 x x53x5 B. Divide coefficients. Subtract exponents that have the same base. 12m 2 n 3 9mn 2 m 2-1 n mn = 1 1 / 3 mn 4343

Evaluating Algebraic Expressions 4-4 Multiplying and Dividing Monomials Simplify. Check It Out! Example 3 A. (4a) ∙ a 4 256a 4 Raise each factor to the power. B. (–3x 2 y) 2 (–3) 2 ∙ (x 2 ) 2 ∙ (y) 2 9x4y29x4y2 Raise each factor to the power. Multiply exponents.