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Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 4 Objectives 1.Use 0 as an exponent. 2.Use negative numbers as exponents. 3.Use the quotient rule for exponents. 4.Use combinations of rules. 5.5 Integer Exponents and the Quotient Rule

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 5 Zero Exponent For any nonzero real number a, a 0 = 1. Example: 17 0 = 1 5.5 Integer Exponents and the Quotient Rule Using 0 as an Exponent

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 6 (a) 38 0 Example 1 Evaluate. 5.5 Integer Exponents and the Quotient Rule Using 0 as an Exponent (b) (–9) 0 (c) –9 0 = –1(9) 0 = –1(1) = –1 (d) x 0 = 1 (e) 5x 0 = 5·1= 5 (f) (5x) 0 = 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 7 Negative Exponents For any nonzero real number a and any integer n, Example: 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 8 Example 2 Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents (a) 9 –3 Notice that we can change the base to its reciprocal if we also change the sign of the exponent.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 9 Example 2 (concluded) Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents = 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 10 CAUTION A negative exponent does not indicate a negative number. Negative exponents lead to reciprocals. 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents ExpressionExample a–na–n Not negative –a –n Negative

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 Changing from Negative to Positive Exponents For any nonzero numbers a and b and any integers m and n, Examples:

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 12 CAUTION Be careful. We cannot use the rule to change negative exponents to positive exponents if the exponents occur in a sum or difference of terms. For example, 5.5 Integer Exponents and the Quotient Rule Using Negative Numbers as Exponents would be written with positive exponents as

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 Quotient Rule for Exponents For any nonzero number a and any integers m and n, Example: (Keep the same base and subtract the exponents.)

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 14 CAUTION A common error is to write This is incorrect. By the quotient rule, the quotient must have the same base, 5, so 5.5 Integer Exponents and the Quotient Rule We can confirm this by using the definition of exponents to write out the factors: Using the Quotient Rule for Exponents

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 15 Example 3 Simplify. Assume that all variables represent nonzero real numbers. 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 16 Example 3 (continued) Simplify. Assume that all variables represent nonzero real numbers. 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 17 Example 3 (concluded) Simplify. Assume that all variables represent nonzero real numbers. 5.5 Integer Exponents and the Quotient Rule Using the Quotient Rule for Exponents

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 18 5.5 Integer Exponents and the Quotient Rule Definitions and Rules for Exponents For any integers m and n: Product rule a m · a n = a m+n Zero exponent a 0 = 1 (a ≠ 0) Negative exponent Quotient rule Using the Quotient Rule for Exponents

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 19 5.5 Integer Exponents and the Quotient Rule Definitions and Rules for Exponents (concluded) For any integers m and n: Power rules (a) (a m ) n = a mn (b) (ab) m = a m b m (c) Negative-to-Positive Rules Using the Quotient Rule for Exponents

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 20 Example 4 Simplify each expression. Assume all variables represent nonzero real numbers. 5.5 Integer Exponents and the Quotient Rule Using Combinations of Rules

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 21 Example 4 (continued) Simplify each expression. Assume all variables represent nonzero real numbers. 5.5 Integer Exponents and the Quotient Rule Using Combinations of Rules

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 22 Example 4 (concluded) Simplify each expression. Assume all variables represent nonzero real numbers. 5.5 Integer Exponents and the Quotient Rule Using Combinations of Rules

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.5 – Slide 23 5.5 Integer Exponents and the Quotient Rule 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. Using Combinations of Rules