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California Standards 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

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

Monomials Not monomials 7x5, -3a2b3, n2, 8, z 4 m-3,4z2.5, 5 + y, , 2x 8 w3 To multiply two monomials, multiply the coefficients and add the exponents that have the same base.

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**Additional Example 1: Multiplying Monomials**

A. (3a2)(4a5) Use the Comm. and Assoc. Properties. (3 ∙ 4)(a2 ∙ a5) 3 ∙ 4 ∙ a2 + 5 Multiply coefficients. Add exponents that have the same base. 12a7 B. (4x2y3)(5xy5) Use the Comm. and Assoc. Properties. Think: x = x1. (4 ∙ 5)(x2 ∙ x)(y3 ∙ y5) (4 ∙ 5)(x2 ∙ x1)(y3 ∙ y5) Multiply coefficients. Add exponents that have the same base. 4 ∙ 5 ∙ x2 + 1 ∙ y3+5 20x3y8

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**Additional Example 1: Multiplying Monomials**

C. (–3p2r)(6pr3s) Use the Comm. and Assoc. Properties. (–3 ∙ 6)(p2 ∙ p)(r ∙ r3)(s) (–3 ∙ 6)(p2 ∙ p1)(r1 ∙ r3)(s) Multiply coefficients. Add exponents that have the same base. –3 ∙ 6 ∙ p2 + 1 ∙ r1+3 ∙ s –18p3r4s

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Check It Out! Example 1 Multiply. A. (2b2)(7b4) Use the Comm. and Assoc. Properties. (2 ∙ 7)(b2 ∙ b4) 2 ∙ 7 ∙ b2 + 4 Multiply coefficients. Add exponents that have the same base. 14b6 B. (4n4)(5n3)(p) Use the Comm. and Assoc. Properties. (4 ∙ 5)(n4 ∙ n3)(p) Multiply coefficients. Add exponents that have the same base. 4 ∙ 5 ∙ n4 + 3 ∙ p 20n7p

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Check It Out! Example 1 Multiply. C. (–2a4b4)(3ab3c) Use the Comm. and Assoc. Properties. (–2 ∙ 3)(a4 ∙ a)(b4 ∙ b3)(c) (–2 ∙ 3)(a4 ∙ a1)(b4 ∙ b3)(c) Multiply coefficients. Add exponents that have the same base. –2 ∙ 3 ∙ a4 + 1 ∙ b4+3 ∙ c –6a5b7c

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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.

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**Additional Example 2: Dividing Monomials**

Divide. Assume that no denominator equals zero. 15m5 3m2 A. m5-2 15 3 Divide coefficients. Subtract exponents that have the same base. 5m3 18a2b3 16ab3 B. a2-1 b3-3 9 8 Divide coefficients. Subtract exponents that have the same base. a 9 8

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Check It Out! Example 2 Divide. Assume that no denominator equals zero. 18x7 6x2 A. x7-2 18 6 Divide coefficients. Subtract exponents that have the same base. 3x5 12m2n3 9mn2 B. m2-1 n3-2 4 3 Divide coefficients. Subtract exponents that have the same base. mn 4 3

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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

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**Additional Example 3: Raising a Monomial to a Power**

Simplify. A. (3y)3 33 ∙ y3 Raise each factor to the power. 27y3 B. (2a2b6)4 24 ∙ (a2)4 ∙ (b6)4 Raise each factor to the power. 16a8b24 Multiply exponents.

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Check It Out! Example 3 Simplify. A. (4a)4 44 ∙ a4 Raise each factor to the power. 256a4 B. (–3x2y)2 (–3)2 ∙ (x2)2 ∙ (y)2 Raise each factor to the power. 9x4y2 Multiply exponents.

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