1. Copyright © Cengage Learning. All rights reserved. Polynomials 4

2. Copyright © Cengage Learning. All rights reserved. Section 4.2 Zero and Negative-Integer Exponents

3. 3 Objectives Simplify an expression containing an exponent of zero. Simplify an expression containing a negative-integer exponent. Simplify an expression containing a variable exponent. 1 1 2 2 3 3

4. 4 Simplify an expression containing an exponent of zero 1.

5. 5 Simplify an expression containing an exponent of zero When we discussed the quotient rule for exponents in the previous section, the exponent in the numerator was always greater than the exponent in the denominator. We now consider what happens when the exponents are equal. If we apply the quotient rule to the fraction, where the exponents in the numerator and denominator are equal, we obtain 5 0. However, because any nonzero number divided by itself equals 1, we also obtain 1.

6. 6 Simplify an expression containing an exponent of zero For this reason, we define 5 0 to be equal to 1. In general, the following is true. Zero Exponents If x is any nonzero real number, then x 0 = 1

7. 7 Example Write each expression without exponents. a. b. (x  0) = x 0 = 1 c. 3x 0 = 3(1) = 3 d. (3x) 0 = 1

8. 8 Example e. = 6 0 = 1 f. (y  0) = y 0 = 1 Parts c and d point out that 3x 0  (3x) 0. cont’d

9. 9 Simplify an expression containing a negative-integer exponent 2.

10. 10 Simplify an expression containing a negative-integer exponent If we apply the quotient rule to, where the exponent in the numerator is less than the exponent in the denominator, we obtain 6 –3. However, by dividing out two factors of 6, we also obtain. For these reasons, we define 6 –3 to be.

11. 11 Simplify an expression containing a negative-integer exponent In general, the following is true. Negative Exponents If x is any nonzero number and n is a natural number, then and Comment A negative exponent can be viewed as “taking the reciprocal.”

12. 12 Express each quantity without negative exponents or parentheses. Assume no variables are zero. a. b. Example

13. 13 c. d. cont’d Example

14. 14 e. f. cont’d Example

15. 15 Simplify an expression containing a negative-integer exponent Because of the definitions of negative and zero exponents, the product, power, and quotient rules are true for all integer exponents. Properties of Integer Exponents If m and n are integers and no base is 0 (x  0, y  0), then x m x n = x m + n (x m ) n = x mn (xy) n = x n y n

16. 16 Simplify an expression containing a variable exponent 3.

17. 17 Simplify an expression containing a variable exponent The properties of exponents are also true when the exponents are algebraic expressions.

18. 18 Example Simplify. Assume no base is 0. a. x 2m x 3m = x 2m + 3m = x 5m b.

19. 19 Example c. a 2m – 1 a 2m = a 2m – 1 + 2m = a 4m – 1 d. (b m + 1 ) 2 = b (m + 1)2 = b 2m + 2 cont’d