1. Powers and Exponents1-4 Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. Powers and Exponents1-4 Warm Up Simplify. 1. 2(2) 2. (–2)(–2) 3. (–2)(–2)(–2) 4. 3(3)(3) 4 4 –8 27 4949 5.

3. Powers and Exponents1-4 Preparation for 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. California Standards

4. Powers and Exponents1-4 power base exponent Vocabulary

5. Powers and Exponents1-4 A power is an expression written with an exponent and a base or the value of such an expression. 3² is an example of a power. The base, 3, is the number that is used as a factor. 3 2 The exponent, 2, tells how many times the base, 3, is used as a factor.

6. Powers and Exponents1-4 When a number is raised to the second power, we usually say it is “squared.” The area of a square is s  s = s 2, where s is the side length. s s When a number is raised to the third power, we usually say it is “cubed.” The volume of a cube is s  s  s = s 3, where s is the side length. s s s

7. Powers and Exponents1-4 Write the power represented by the geometric model. 5 5 5 The figure is 5 units long, 5 units wide, and 5 units tall. 5  5  5 The factor 5 is used 3 times. 5353 Additional Example 1A: Writing Powers for Geometric Models

8. Powers and Exponents1-4 Write the power represented by the geometric model. The figure is 6 units long and 6 units wide. 6 x 6 The factor 6 is used 2 times. 6262 6 6 Additional Example 1B: Writing Powers for Geometric Models

9. Powers and Exponents1-4 Write the power represented by each geometric model. a. 2 b. Check It Out! Example 1 x x x The figure is 2 units long and 2 units wide. 2  2 The factor 2 is used 2 times. The figure is x units long, x units wide, and x units tall. x  x  x The factor x is used 3 times. x3x3

10. Powers and Exponents1-4 There are no easy geometric models for numbers raised to exponents greater than 3, but you can still write them using repeated multiplication or with a base and exponent. 3 to the second power, or 3 squared 3  3  3  3  3 Multiplication Power Value Words 3  3  3  3 3  3  3 3  3 3 3 to the first power 3 to the third power, or 3 cubed 3 to the fourth power 3 to the fifth power 3 9 27 81 243 3131 3232 3 3434 3535 Reading Exponents

11. Powers and Exponents1-4 Caution! In the expression –5², 5 is the base because the negative sign is not in parentheses. In the expression (–2)³, –2 is the base because of the parentheses.

12. Powers and Exponents1-4 Simplify each expression. A. (–6) 3 (–6)(–6)(–6) –216 B. –10 2 –1 10 10 –100 Use –6 as a factor 3 times. Find the product of –1 and two 10’s. Additional Example 2: Evaluating Powers Think of a negative sign in front of a power as multiplying by a –1.

13. Powers and Exponents1-4 Use as a factor 2 times. 2929 Simplify the expression. C. 2929  2929 Additional Example 2: Evaluating Powers = 4 81 2929  2929

14. Powers and Exponents1-4 Simplify each expression. a. (–5) 3 b. –6 2 Check It Out! Example 2 (–5)(–5)(–5) Use –5 as a factor 3 times. –125 –1  6  6 –36 Think of a negative sign in front of a power as multiplying by –1. Find the product of –1 and two 6’s.

15. Powers and Exponents1-4 Check It Out! Example 2 Simplify the expression. c. Use as a factor 3 times. 3434 27 64

16. Powers and Exponents1-4 Write each number as a power of the given base. A. 64; base 8 8  8 8282 B. 81; base –3 (–3)(–3)(–3)(–3) (–3) 4 The product of two 8’s is 64. The product of four –3’s is 81. Additional Example 3: Writing Powers

17. Powers and Exponents1-4 Write each number as a power of a given base. a. 64; base 8 b. –27; base –3 Check It Out! Example 3 8  8 The product of two 8’s is 64. 8²8² (–3)(–3)(–3) (–3) 3 The product of three –3’s is –27.

18. Powers and Exponents1-4 In case of a school closing, the PTA president calls 3 families. Each of these families calls 3 other families, and so on. How many families will have been called in the 4th round of calls? The answer will be the number of families contacted in the 4th round of calls. Additional Example 4: Problem-Solving Application Understand the problem 1 List the important information: The PTA president calls 3 families. Each family then calls 3 more families.

19. Powers and Exponents1-4 Draw a diagram to show the number of families called in each round of calls. 2 Make a Plan Additional Example 4 Continued 2nd round of calls 1st round of calls PTA President

20. Powers and Exponents1-4 Notice that after each round of calls the number of families contacted is a power of 3. 1 st round of calls: 1  3 = 3 or 3 1 families contacted So, in the 4 th round of calls, 3 4 families will have been contacted. 3 4 = 3  3  3  3 = 81 Multiply four 3’s. In the fourth round of calls, 81 families will have been contacted. 2 nd round of calls: 3  3 = 9 or 3 2 families contacted 3 rd round of calls: 9  3 = 27 or 3 3 families contacted Solve 3 Additional Example 4 Continued

21. Powers and Exponents1-4 Drawing a diagram helps you visualize the problem, but the numbers become too large for a diagram. The diagram helps you recognize the pattern of multiplying by 3 so that you can write the number as a power of 3. Look Back4 Additional Example 4 Continued

22. Powers and Exponents1-4 What if…? How many bacteria will be on the slide after 8 hours? Check It Out! Example 4 2828 After each hour, the number of bacteria is a power of 2. 2  2  2  2  2  2  2  2 Multiply eight 2’s. The product of eight 2’s. 256

23. Powers and Exponents1-4 1. Write the power represented by the geometric model. n n n2n2 Simplify each expression. 2. 4. 6 3. –6 3 5. (–2) 6 −216 21664 Lesson Quiz Write each number as a power of the given base. 6. 343; base 77. 10,000; base 107373 10 4