13 Exponents and Polynomials

13.5 Integer Exponents and the Quotient Rule 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.

Use 0 as an Exponent Zero Exponent For any nonzero real number a, a0 = 1. Example: 170 = 1

Use 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

Use Negative Numbers as Exponents Negative Exponents For any nonzero real number a and any integer n, Example:

Use Negative Numbers as Exponents Example 2 Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. (a) 9–3 Notice that we can change the base to its reciprocal if we also change the sign of the exponent.

Use Negative Numbers as Exponents Example 2 (concluded) Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. 7

Use Negative Numbers as Exponents CAUTION A negative exponent does not indicate a negative number. Negative exponents lead to reciprocals. Expression Example a–n Not negative –a–n Negative

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

Use Negative Numbers as Exponents 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, would be written with positive exponents as

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

Use the Quotient Rule for Exponents CAUTION A common error is to write This is incorrect. By the quotient rule, the quotient must have the same base, 5, so We can confirm this by using the definition of exponents to write out the factors:

Use the Quotient Rule for Exponents Example 3 Simplify. Assume that all variables represent nonzero real numbers.

Use the Quotient Rule for Exponents Example 4 Simplify. Assume that all variables represent nonzero real numbers. 14

Use the Quotient Rule for Exponents Example 4 (concluded) Simplify. Assume that all variables represent nonzero real numbers. 15

Use 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

Use 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 (c) Negative-to-Positive Rules 17

Use Combinations of Rules Example 5 Simplify each expression. Assume all variables represent nonzero real numbers.

Use Combinations of Rules Example 5 (continued) Simplify each expression. Assume all variables represent nonzero real numbers. 19

Use Combinations of Rules Example 5 (concluded) Simplify each expression. Assume all variables represent nonzero real numbers. 20

Use 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 5.