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418 Chapter 7 Exponents and Exponential Functions
Zero and Negative
Exponents
Objective To simplify expressions involving zero and negative exponents
Copy and complete the table. Make a conjecture about how the value of an exponential expression (an expression containing an exponent) changes when you decrease the exponent by 1. What do you think the value of 5^2^2 is? Explain your reasoning.
Look for a pattern in the values in the table.
The patterns you found in the Solve It illustrate the definitions of zero and negative exponents.
Essential Understanding You can extend the idea of exponents to include zero
and negative exponents. Consider 33 , 32 , and 31. Decreasing the exponents by 1 is the same as dividing by 3. If you continue the pattern, 30 equals 1 and 321 equals 13.
Properties Zero and Negative Exponents
Zero as an Exponent For every nonzero number a , a^0 5 1. Examples 40 5 1 ( 2 3)^0 5 1 (5.14)^0 5
Negative Exponent For every nonzero number a and integer n , a^2 n^ (^5) a^1 n. Examples 723 5 713 ( 2 5)^22 5 ( 21 5) 2
2 x^ 10 x 24 ■ 23 ■ 22 ■ 21 ■ 20 ■ 2 ^1 ■ 2 ^2 ■
10 ^1 ■
10 ^2 ■
Content Standards
Prepares for N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values... Also prepares for N.RN.
MATHEMATICAL PRACTICES
Problem 1
Got It?
Problem 2
Got It?
Lesson 7-1 Zero and Negative Exponents 419
Why can’t you use 0 as a base with zero exponents? The first property on the previous page implies the following pattern. 30 5 1 20 5 1 10 5 1 00 5 1 However, consider the following pattern. 03 5 0 02 5 0 01 5 0 00 5 0 It is not possible for 00 to equal both 1 and 0. Therefore 00 is undefined. Why can’t you use 0 as a base with a negative exponent? Using 0 as a base with a negative exponent will result in division by zero, which is undefined.
Simplifying Powers What is the simplified form of each expression? A 9^2^2
922 5 1 92
Use the definition of negative exponent.
5 811 Simplify.
B ( 2 3.6)^0 5 1 Use the definition of zero as an exponent.
1. What is the simplified form of each expression? a. 423 b. ( 2 5)^0 c. 322 d. 621 e. ( 2 4)^22
An algebraic expression is in simplest form when powers with a variable base are written with only positive exponents.
Simplifying Exponential Expressions What is the simplified form of each expression? A 5 a^3 b^2^2
5 a^3 b^22 5 5 a^3 Q b^12 R Use the definition of negative exponent.
5 5 a
3 b^2
Simplify.
B 1 x^2^5 1 x^25
5 1 4 x^25 Rewrite using a division symbol. 5 1 4 1 x^5
Use the definition of negative exponent.
5 1? x^5 Multiply by the reciprocal of (^) x^15 , which is x^5. 5 x^5 Identity Property of Multiplication
2. What is the simplified form of each expression? a. x^29 b.^1 n^23
c. 4 c^23 b d.^2 a^23
e. n
25 m^2
Can you use the definition of zero as an exponent when the base is a negative number? Yes, the definition of zero as an exponent is true for all nonzero bases.
Which part of the expression do you need to rewrite? The base b has a negative exponent, so you need to rewrite it with a positive exponent.
Lesson Check
Got It?
Lesson 7-1 Zero and Negative Exponents 421
Practice and Problem-Solving Exercises
Simplify each expression.
9. 322 10. ( 2 4.25)^0 11. ( 2 5)^22 12. 2522 13. ( 2 4)^22 14. 226 15. 230 16. 21221 17.^1 20 18. 5821 19. 1.5^22 20. ( 2 5)^23
Simplify each expression.
21. 4 ab^0 22.^1 x^27 23. 5 x^24 24.^1 c^21 25.^3
22 n 26.^ k^24 j^^0 27.^
3 x^22 y 28.^
7 ab^22 3 w
29. c^25 d^27 30. c^25 d^7 31. (^) 2 s^823 32. (^) 57 t 2 s 3 33.^6 a
(^21) c 23 d^0 34.^^2
(^23) x (^2) z (^27) 35. 120 t (^7) u (^211) 36.^7 s^^0 t^25 221 m^2
A Practice See Problem 1.
See Problem 2.
Do you know HOW?
Simplify each expression.
1. 225 2. m^0 3. 5 s^2 t^21 4. (^) x 243
Evaluate each expression for a 5 2 and b 5 2 4.
5. a^3 b^21 6. 2 a^24 b^0
Do you UNDERSTAND?
7. Vocabulary A positive exponent shows repeated multiplication. What repeated operation does a negative exponent show? 8. Error Analysis A student incorrectly simplified x
n a^2 nb^0 as shown below. Find and correct the student’s error.
= undefined
xn a-nb^0
anxn b^0 anxn 0
4. A population of insects triples every week. The number of insects is modeled by the expression 5400? 3 w , where w is the number of weeks after the population was measured. Evaluate the expression for w 5 22 , w 5 0, and w 5 1. What does each value of the expression represent in the situation?
MATHEMATICAL PRACTICES
MATHEMATICAL PRACTICES
422 Chapter 7 Exponents and Exponential Functions
Evaluate each expression for r 5 2 3 and s 5 5.
37. r^23 38. s^23 39.^3 r s^22 40. s^
0 r^22
41. 4 s^21 42. r^0 s^22 43. r^24 s^2 44. 224 r^3 s^22 45. Internet Traffic The number of visitors to a certain Web site triples every month. The number of visitors is modeled by the expression 8100? 3 m , where m is the number of months after the number of visitors was measured. Evaluate the expression m 5 24. What does the value of the expression represent in the situation? 46. Population Growth A Galápagos cactus finch population increases by half every decade. The number of finches is modeled by the expression 45? 1.5 d , where d is the number of decades after the population was measured. Evaluate the expression for d 5 22, d 5 0, and d 5 1. What does each value of the expression represent in the situation?
Mental Math Is the value of each expression positive or negative?
47. 222 48. ( 2 2) 2 49. ( 2 2) 3 50. ( 2 2)^23
Write each number as a power of 10 using negative exponents.
51. (^) 101 52. (^) 1001 53. (^) 10001 54. (^) 10,000^1 55. a. Patterns Complete the pattern using powers of 5. 1 52 5 j^
1 51 5 j^
1 50 5 j^
1 521 5 j^
1 522 5 j b. Write 5124 using a positive exponent. c. Rewrite (^) a^12 n as a power of a.
Rewrite each fraction with all the variables in the numerator.
56. a b^22
4 g h^3
58.^5 m
6 3 n 59.^
8 c^5 11 d^4 e^22
60. Think About a Plan Suppose your drama club’s budget doubles every year. This year the budget is $500. How much was the club’s budget 2 yr ago? - What expression models what the budget of the club will be in 1 yr? In 2 yr? In y years? - What value of y can you substitute into your expression to find the budget of the club 2 yr ago? 61. Copy and complete the table at the right. 62. a. Simplify an^? a^2 n. b. Reasoning What is the mathematical relationship between an^ and a^2 n? Explain.
See Problem 3.
See Problem 4.
B^ Apply
n
n ^1
5 8
Galápagos cactus finch
STEM
