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Notes 7-1 Multiplying Monomials I. What is a monomial? Discuss. A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. Monomials that are real numbers are called constants. A. Identifying monomials Expression Monomial? Reason -5 Yes p+q No x Yes -5 is a real number and an example of a constant The expression involves the addition, not the product, of two variables Single variables are monomials B. Examples: Identifying monomials a. mn2 Yes b. 3x2 + 5x + 7 No c. 0.05ab Yes d. -19x +5 No e. -19x Yes Multiplying monomials is often used when comparing a characteristic of several items, such as acidity of different fruits. It is also used when determining the probability of something, like guessing the correct answer on a test or winning the lottery (Chapter 12). Today, you will learn three new properties that will help you multiply monomials. II. Products of Powers Products of powers with the same base can be found by writing each power as a repeated multiplication. am an = (a a … a) (a a … a) m factors n factors = a a … a = am+n m + n factors KEY CONCEPT Product of Powers Words: To multiply two powers that have the same base, add their exponents. Symbols: For any number a and all integers m and n, am • an = am + n Example: a4 • a12 = a4 + 12 or a16 Remember! A number or variable written without an exponent actually has an exponent of 1. 10 = 101 y = y1 Simplify. A. Since the powers have the same base, keep the base and add the exponents. Your turn! Simplify. a. Since the powers have the same base, keep the base and add the exponents. Simplify. B. Group powers with the same base together. Add the exponents of powers with the same base. Your turn! b. a2b6a4b9 a2a4b6b9 a2 + 4b 6 + 9 a6b15 Group powers with the same base together. Add the exponents of powers with the same base. Multiply. C. (6y3)(3y5) (6y3)(3y5) (6 3)(y3 y5) Group factors with like bases together. 18y8 Multiply. D. (3mn2) (9m2n) (3mn2)(9m2n) (3 9)(m m2)(n2 n) 27m3n3 Group factors with like bases together. Multiply. Your turn! Multiply. c. (3x3)(6x2) (3x3)(6x2) (3 6)(x3 x2) 18x5 Group factors with like bases together. Multiply. d. (2r2t)(5t3) (2r2t)(5t3) (2 5)(r2)(t3 t) 10r2t4 Group factors with like bases together. Multiply. Again… When multiplying powers with the same base, keep the base and add the exponents. x2 x3 = x2+3 = x5 III. Power of a Power To find a power of a power, you can use the meaning of exponents. = am am … am n factors = a a … a a a … a … a a … a = amn m factors m factors m factors n groups of m factors KEY CONCEPT Power of a Power Words: To find the power of a power, multiply the exponents. Symbols: For any number a and all integers m and n, (am)n = am • n Example: (k5)9 = k5 • 9 or k45 Simplify. Use the Power of a Power Property. Simplify. Your turn! Simplify. Use the Power of a Power Property. Simplify. Simplify B. [(32)3]2 (32•3)2 (36)2 36•2 312 Power of a Power Simplify Power of a Power Simplify You Try! Simplify B. [(22)2]4 (22•2)4 (24)4 24•4 216 Power of a Power Simplify Power of a Power Simplify Powers of products can be found by using the meaning of an exponent. (ab)n = ab ab … ab n factors = a a … a b b … b = anbn n factors n factors KEY CONCEPT Power of a Product Words: To find the power of a product, find the power of each factor and multiply. Symbols: For all numbers a and b and any integer m, (ab)m = ambm Example: (-2xy)3 = (-2)3x3y3 or -8x3y3 Simplify. A. Use the Power of a Product Property. Simplify. B. Use the Power of a Product Property. Simplify. Caution! In Example 4A, the negative sign is not part of the base. –(2y)2 = –1(2y)2 Classwork Workbook Section 7-1 Page 87