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Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 1 Chapter 5 Exponents and Polynomials Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 2 5.5 Integer Exponents and the Quotient Rule Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 3 5.5 Integer Exponents and the Quotient Rule Objectives 1. 2. 3. 4. Use 0 as an exponent. Use negative numbers as exponents. Use the quotient rule for exponents. Use combinations of rules. Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 4 5.5 Integer Exponents and the Quotient Rule Using 0 as an Exponent Zero Exponent For any nonzero real number a, a0 = 1. Example: 170 = 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 5 5.5 Integer Exponents and the Quotient Rule Using 0 as an Exponent Example 1 Evaluate. (a) 380 = 1 (b) (–9)0 = 1 (c) –90 = –1(9)0 = –1(1) = –1 (d) x0 = 1 (e) 5x0 = 5·1 = 5 (f) (5x)0 = 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 6 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents Negative Exponents For any nonzero real number a and any integer n, 1 n a n. a 1 1 Example: 32 2 . 3 9 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 7 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents Example 2 Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. 1 1 –3 (a) 9 3 9 729 3 3 5 5 1 4 (b) 64 4 1 Notice that we can change the base to its reciprocal if we also change the sign of the exponent. 2 3 243 (c) 32 3 2 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 8 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents Example 2 (concluded) Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. 1 1 (d) 6 3 6 3 1 2 6 6 1 6 1 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. 3 3 x (e) 4 1 x 4 x x4 3x 4 4 =1 5.5 – Slide 9 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents CAUTION A negative exponent does not indicate a negative number. Negative exponents lead to reciprocals. Expression Example a–n 1 1 2 3 2 8 –a–n 23 3 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1 1 23 8 Not negative Negative 5.5 – Slide 10 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents Changing from Negative to Positive Exponents For any nonzero numbers a and b and any integers m and n, m n a b m n b a 5 Examples: 4 3 2 5 4 2 3 a and b 3 m m b . a 3 4 5 and . 5 4 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 11 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents CAUTION a m bn Be careful. We cannot use the rule n m to b a change negative exponents to positive exponents if the exponents occur in a sum or difference of terms. For example, 52 31 1 1 7 23 2 5 3. would be written with positive exponents as 1 7 3 2 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 12 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents Quotient Rule for Exponents For any nonzero number a and any integers m and n, am m n a . n a (Keep the same base and subtract the exponents.) 58 Example: 6 586 =52 =25. 5 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 13 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents CAUTION 58 A common error is to write 6 186 12. This is 5 incorrect. By the quotient rule, the quotient must have the same base, 5, so 58 86 2 5 =5 . 6 5 We can confirm this by using the definition of exponents to write out the factors: 58 55555555 . 6 555555 5 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 14 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents Example 3 Simplify. Assume that all variables represent nonzero real numbers. 1 34 4 6 2 (a) 6 3 3 32 3 y 4 (b) 9 y 4( 9) y 49 y 5 y Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 15 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents Example 3 (continued) Simplify. Assume that all variables represent nonzero real numbers. 24 ( z a ) 7 (c) 5 2 ( z a)6 24 ( z a ) 7 5 2 ( z a)6 24( 5) ( z a)76 29 ( z a) Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 16 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents Example 3 (concluded) Simplify. Assume that all variables represent nonzero real numbers. 5x 3 y8 5 x 3 y 8 (d) 2 4 6 2 4 6 3 x y 3 x y 5 32 x 34 y86 5 9x 7 y 2 45y 2 7 x Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 17 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents Definitions and Rules for Exponents For any integers m and n: Product rule am · an = am+n Zero exponent a0 = 1 (a ≠ 0) Negative exponent Quotient rule Copyright © 2010 Pearson Education, Inc. All rights reserved. a n 1 n a am m n a an (a 0) 5.5 – Slide 18 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents Definitions and Rules for Exponents (concluded) For any integers m and n: Power rules (a) (am)n = amn (b) (ab)m = ambm m a am (c) b b m Negative-to-Positive Rules a m bn m n b a a b Copyright © 2010 Pearson Education, Inc. All rights reserved. m (b 0) (a, b 0) b a m 5.5 – Slide 19 5.5 Integer Exponents and the Quotient Rule Using Combinations of Rules Example 4 Simplify each expression. Assume all variables represent nonzero real numbers. 6 (23 ) 2 2 (a) 6 26 2 266 20 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 20 5.5 Integer Exponents and the Quotient Rule Using Combinations of Rules Example 4 (continued) Simplify each expression. Assume all variables represent nonzero real numbers. (3y ) 4 (3y ) 2 (3 y ) 42 (b) 1 (3y ) (3 y ) 1 (3 y)6( 1) (3 y)7 37 y 7 2187 y 7 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 21 5.5 Integer Exponents and the Quotient Rule Using Combinations of Rules Example 4 (concluded) Simplify each expression. Assume all variables represent nonzero real numbers. 2 2 5a 2 b (c) 1 4 3 2 b 5a 2 a3 b4 1 2 5 3 1 4 a 6 b8 (10) 2 a 6 b8 100 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 22 5.5 Integer Exponents and the Quotient Rule Using Combinations of Rules Note Since the steps can be done in several different orders, there are many equally correct ways to simplify expressions like those in Example 4. Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 23