1-4 Powers and Exponents

Transcription

1 Warm Up Lesson Presentation Lesson Quiz

2 Warm Up Simplify. 1. 2(2) 4 2. ( 2)( 2) 4 3. ( 2)( 2)( 2) (3)(3)

3 Objective Evaluate expressions containing exponents.

4 power base exponent Vocabulary

5 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 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 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, is the side length. S S When a number is raised to the third power, we usually say its cubed. The of volume of a cube is s s s = s 3 is the side length. S S S

7 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 The factor 5 is used 3 times.

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

9 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 b. x 2 2 x x x 3 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.

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 a base and exponent. Words Reading Exponents Multiplication Power Value 3 to the first power 3 to the second power, or 3 squared 3 to the third power, or 3 cubed 3 to the fourth power to the fifth power

11 Caution! In the expression 5 2, 5 is the base because the negative sign is not in parentheses. In the expression ( 2), 2 is the base because of the parentheses.

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

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

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

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

16 Example 3: Writing Powers Write each number as a power of the given base. A. 64; base The product of two 8 s is 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 The product of three 4 s is b. 27; base 3 ( 3)( 3)( 3) 3 3 The product of three ( 3) s is 27.

18 1 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 4 th round of calls? Understand the problem The answer will be the number of families contacted in the 4 th round of calls. List the important information: The PTA president calls 3 families. Each family then calls 3 more families.

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

20 3 Example 4 Continued Solve 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 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 So, in the 4 th round of calls, 3 4 families will have been contacted. 3 4 = = 81 Multiply four 3 s. In the fourth round of calls, 81 families will have been contacted.

21 4 Look Back 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? 2 8 After each hour, the number of bacteria is a power of Multiply eight 2 s. 256 The product of eight 2 s.

23 1. Write the power represented by the geometric model. Lesson Quiz Simplify each expression. n n n ( 2) 6 64 Write each number as a power of the given base ; base ,000; base