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Warm Up

This lesson presentation covers the basics of exponents and powers, including simplifying expressions and evaluating powers. It also includes an application problem involving exponential growth.

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Warm Up

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  1. Preview Warm Up California Standards Lesson Presentation

  2. 4 9 Warm Up Simplify. 1. 2(2) 2. (–2)(–2) 3. (–2)(–2)(–2) 4. 3(3)(3) 4 4 –8 27 5.

  3. California Standards 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.

  4. Vocabulary power base exponent

  5. 3 2 Apower 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 exponent, 2, tells how many times the base, 3, is used as a factor. The base, 3, is the number that is used as a factor.

  6. 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 = s3, where s is the side length. s s s When a number is raised to the second power, we usually say it is “squared.” The area of a square is ss = s2, where s is the side length.

  7. 5 5 5 Additional Example 1A: Writing Powers for Geometric Models Write the power represented by the geometric model. The figure is 5 units long, 5 units wide, and 5 units tall. 5  5  5 53 The factor 5 is used 3 times.

  8. Additional Example 1B: Writing Powers for Geometric Models Write the power represented by the geometric model. 6 The figure is 6 units long and 6 units wide. 6 x 6 6 62 The factor 6 is used 2 times.

  9. x x x Check It Out! Example 1 Write the power represented by each geometric model. a. The figure is 2 units long and 2 units wide. 2  2 22 The factor 2 is used 2 times. b. The figure is x units long, x units wide, and x units tall. x  x  x The factor x is used 3 times. x3

  10. 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. Reading Exponents Words Power Value Multiplication 3 to the first power 3 31 3 3 to the second power, or 3 squared 32 3  3 9 3 to the third power, or 3 cubed 33 3  3  3 27 81 3  3  3  3 34 3 to the fourth power 35 243 3  3  3  3  3 3 to the fifth power

  11. 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. Additional Example 2: Evaluating Powers Simplify each expression. A. (–6)3 (–6)(–6)(–6) Use –6 as a factor 3 times. –216 B. –102 Think of a negative sign in front of a power as multiplying by a –1. –1 •10 •10 Find the product of –1 and two 10’s. –100

  13. 2 9 2 9 2 9 2 9 2 9   Use as a factor 2 times. 4 81 = Additional Example 2: Evaluating Powers Simplify the expression. C.

  14. Check It Out! Example 2 Simplify each expression. a. (–5)3 (–5)(–5)(–5) Use –5 as a factor 3 times. –125 Think of a negative sign in front of a power as multiplying by –1. b. –62 –1  6  6 Find the product of –1 and two 6’s. –36

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

  16. Additional Example 3: Writing Powers Write each number as a power of the given base. A. 64; base 8 The product of two 8’s is 64. 8 8 82 B. 81; base –3 (–3)(–3)(–3)(–3) The product of four –3’s is 81. (–3)4

  17. Check It Out! Example 3 Write each number as a power of a given base. a. 64; base 8 8  8 The product of two 8’s is 64. 8² b. –27; base –3 (–3)(–3)(–3) The product of three –3’s is –27. (–3)3

  18. 1 Understand the problem Additional Example 4:Problem-Solving Application 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. List the important information: • The PTA president calls 3 families. • Each family then calls 3 more families.

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

  20. 3 Solve Additional Example 4 Continued Notice that after each round of calls the number of families contacted is a power of 3. 1st round of calls: 1 3 = 3 or 31 families contacted 2nd round of calls: 3 3 = 9 or 32 families contacted 3rd round of calls: 9 3 = 27 or 33 families contacted So, in the 4th round of calls, 34 families will have been contacted. Multiply four 3’s. 34 = 3  3  3  3 = 81 In the fourth round of calls, 81 families will have been contacted.

  21. 4 Look Back Additional Example 4 Continued 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.

  22. Check It Out! Example 4 What if…? How many bacteria will be on the slide after 8 hours? After each hour, the number of bacteria is a power of 2. 28 2  2  2  2  2  2  2  2 Multiply eight 2’s. 256 The product of eight 2’s.

  23. Lesson Quiz n • 1. Write the power represented by the geometric model. n2 n Simplify each expression. 3. –63 −216 2. 5. (–2)6 4. 6 216 64 Write each number as a power of the given base. 6. 343; base 7 73 7. 10,000; base 10 104

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