 # California Standards AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results.

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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

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.

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

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

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

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

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.

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

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

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

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.

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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