1. 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!

2. 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.

3. 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 2 + 5 (4 ∙ 5)(x 2 ∙ x 1 )(y 3 ∙ y 5 ) 4 ∙ 5 ∙ x 2 + 1 ∙ y 3+5 20x 3 y 8 Use the Comm. and Assoc. Properties. Think: x = x 1.

4. 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 2 + 1 ∙ 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)

5. 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.

6. 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 5-2 15 3 5m35m3 B. Divide coefficients. Subtract exponents that have the same base. 18a 2 b 3 16ab 3 a 2-1 b 3-3 9898 a = 1 1 / 8 a 9898

7. 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

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

9. 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 2 + 4 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 4 + 3 ∙ p

10. 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 4 + 1 ∙ 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)

11. 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 7-2 18 6 3x53x5 B. Divide coefficients. Subtract exponents that have the same base. 12m 2 n 3 9mn 2 m 2-1 n 3-2 4343 mn = 1 1 / 3 mn 4343

12. Evaluating Algebraic Expressions 4-4 Multiplying and Dividing Monomials Simplify. Check It Out! Example 3 A. (4a) 4 4 4 ∙ 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.