Example 2: Zero and Negative Exponents

4. Notice the phrase “nonzero number” in the previous table. This is because 00 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above the table with a base of 0 instead of 5, you would get 0º = . Also 0–6 would be = . Since division by 0 is undefined, neither value exists.

5. Example 2: Zero and Negative Exponents Simplify. A. 4–3 B. 70 7º = 1 C. (–5)–4 D. –5–4

6. Caution In (–3)–4, the base is negative because the negative sign is inside the parentheses. In –3–4 the base (3) is positive.

7. Check It Out! Example 2 Simplify. a. 10–4 b. (–2)–4 c. (–2)–5 d. –2–5

8. for a = –2 and b = 6 Example 3A: Evaluating Expressions with Zero and Negative Exponents Evaluate the expressions for the given value of the variables. x–2 for x = 4 p–3 for p = 4 –2a0b-4 for a = 5 and b = –3 2

9. What if you have an expression with a negative exponent in a denominator, such as ? ***An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents.*** So if a base with a negative exponent is in a denominator, it is equivalent to the same base with the opposite (positive) exponent in the numerator.

10. [(–2)4]5 6. 5. (42)7 (n3)6 [(m + 1)5]4 7. 8. Simplify the expression. = 414 = (–2)20 = n18 = (m + 1)20

11. 3 a. = – 7 (– 7)2 49 x x2 x2 7 x3 x 2 2 y3 x y – = = = b. Example 2 Simplify the expression.

12. Find the values of x and y with the given information.

13. Solve for x and/or y 1. 2.

14. Simplify each expression. 1. 2. 3. 4. 4 1 10 –3

15. Another way to write nth roots is by using fractional exponents. For example, for b >1, suppose Square both sides. Power of a Power Property b1 = b2k 1 = 2k If bm = bn, then m = n. Divide both sides by 2. So for all b > 1,

16. 1 1 n n b b Check It Out! Example 1 Simplify each expression. a. Use the definition of . = 3 b. Use the definition of . = 11 + 4 = 15

17. = 243 = 25 Additional Example 2: Simplifying Expressions with Fractional Exponents Simplify each expression. A. B. C. D. = 1 = 8 E. = 81

18. Additional Example 4B: Properties of Exponents to Simplify Expressions Simplify. All variables represent nonnegative numbers.

20. A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents.

21. Monomials

22. The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.

23. A. 4p4q3 Example 1: Finding the Degree of a Monomial Find the degree of each monomial. The degree is 7. B. 7ed The degree is 2. C. 3 The degree is 0.

24. a. b. c. 1.5k2m 4x 2c3 Check It Out! Example 1 Find the degree of each monomial. The degree is 3. The degree is 1. The degree is 3.

25. A polynomial is a monomial or a sum or difference of monomials. Each monomial in a polynomial is called a term. The degree of a polynomial is the degree of the term with the greatest degree.

26. Polynomials Degree of polynomial Leading Coefficient Constant term

27. SpecialPolynomials • Binomial • Polynomial with two terms • Trinomial • Polynomial with three terms

28. B. Example 2: Finding the Degree of a Polynomial Find the degree of each polynomial. A. 11x7 + 3x3 The degree of the polynomial is the greatest degree, 7. The degree of the polynomial is the greatest degree, 4.

29. Check It Out! Example 2 Find the degree of each polynomial. a. 5x – 6 The degree of the polynomial is the greatest degree, 1. b. x3y2 + x2y3 – x4 + 2 The degree of the polynomial is the greatest degree, 5.

31. Terms Name 1 Monomial 0 Constant 2 Binomial 1 Linear 3 Trinomial Quadratic 2 Polynomial 4 or more Cubic 3 Quartic 4 Quintic 5 6 or more 6th,7th,degree and so on Some polynomials have special names based on their degree and the number of terms they have.

32. Example 4: Classifying Polynomials Classify each polynomial according to its degree and number of terms. A. 5n3 + 4n cubic binomial. B. 4y6 – 5y3 + 2y – 9 6th-degree polynomial. C. –2x linear monomial.

33. constant monomial. Check It Out! Example 4 Classify each polynomial according to its degree and number of terms. a. x3 + x2 – x + 2 cubic polynomial. b. 6 8th-degree trinomial. c. –3y8 + 18y5+ 14y

34. Expression Is it a polynomial? Classify by degree and number of terms a. 9 b. 2x2 + x – 5 c. 6n4 – 8n d. n– 2 – 3 e. 7bc3 + 4b4c Example 2 Tell whetheris a polynomial. If it is a polynomial, find its degree and classify it by the number of its terms. Otherwise, tell why it is not a polynomial. Yes constant monomial Yes Quadratic trinomial No; variable exponent No; negative exponent Yes Quintic binomial

35. Example 5: Application Continued A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds? After 3 seconds the lip balm will be 76 feet from the water.

36. 1606 feet Check It Out! Example 5 What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by –16t2 +400t + 6. How high will this firework be when it explodes?

37. Solve for x and/or y 1. 2.

38. b. (2x)–2y5 –4x2y2 8x3 = y3 y15 – = 16x4 Simplify the expression. Write your answer using only positive exponents. a. (2xy–5)3

39. Lesson Quiz: Part II 5. In an experiment, the approximate population P of a bacteria colony is given by , where t is the number of days since start of the experiment. Find the population of the colony on the 8th day. 480 Simplify. All variables represent nonnegative numbers. 6. 7.

40. Lesson Quiz: Part I Find the degree of each polynomial. 1. 7a3b2 – 2a4 + 4b –15 2. 25x2 – 3x4 Write each polynomial in standard form. Then give the leading coefficient. 3. 24g3 + 10 + 7g5 – g2 4. 14 – x4 + 3x2 5 4 7g5 + 24g3 – g2 + 10; 7 –x4 + 3x2 + 14; –1

41. Lesson Quiz: Part II Classify each polynomial according to its degree and number of terms. quadratic trinomial 5. 18x2 – 12x + 5 6. 2x4 – 1 quartic binomial 7. The polynomial 3.675v + 0.096v2 is used to estimate the stopping distance in feet for a car whose speed is v miles per hour on flat dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? 727.65 ft