40 CHAPTER R Review of Basic Concepts R.4 Integer and Rational Exponents ■ Product Rule for Exponents ■ Power Rules for Exponents ■ Zero as an Exponent ■ Negative Exponents and the Quotient Rule ■ Rational Exponents Product Rule for Exponents Recall that the notation am (where m is a positive integer and a is a real number) means that a appears as a factor m times. In the same way, an (where n is a positive integer) means that a appears as a factor n times. Consider the product 25 # 23, which can be simplified as follows. 5 + 3 = 8 25 # 23 = 12 # 2 # 2 # 2 # 2212 # 2 # 22 = 28 This result suggests the product rule for exponents. Product Rule for Exponents Let m and n be integers and a be a real number. am # an =am+n That is, when multiplying powers of like bases, keep the same base and add the exponents. EXAMPLE 1 Using the Product Rule Simplify each expression. (a) y4 # y7 (b) 16z5219z3212z2 SOLUTION (a) y4 # y7 = y4+7 = y11 Product rule: Keep the base and add the exponents. (b) 16z5219z3212z2 = 16 # 9 # 22 # 1z5z3z12 Commutative and associative properties; z = z1 = 108z5+3+1 Multiply. Apply the product rule. = 108z9 Add. S Now Try Exercises 13 and 17. Power Rules for Exponents We can simplify 13422 as follows. 13422 = 34 # 34 = 34+4 = 38 Notice that 4 # 2 = 8. This example suggests power rule 1 for exponents. Power rules 2 and 3 can be demonstrated similarly. 175. If Tom Brady, of the New England Patriots, during the 2018 regular season had 375 completions, 29 touchdowns, 11 interceptions, and 4355 yards, what was his rating in 2018 for 570 attempts? 176. Refer to the passer rating formula and determine the highest rating possible (considered a “perfect” passer rating).
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