1 4 Powers and Exponents Preview Warm Up

1 -4 Powers and Exponents Preview Warm Up California Standards Lesson Presentation

1 -4 Powers and Exponents Warm Up Simplify. 1. 2(2) 4 2. (– 2) 4 3. (– 2)(– 2) – 8 4. 3(3)(3) 27 5. 4 9

1 -4 Powers and Exponents California Standards Preparation for 2. 0 Students understand use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand use the rules of exponents.

1 -4 Powers and Exponents Vocabulary power base exponent

1 -4 Powers and Exponents 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.

1 -4 Powers and Exponents 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

1 -4 Powers and Exponents Additional Example 1 A: Writing Powers for Geometric Models Write the power represented by the geometric model. 5 53 The figure is 5 units long, 5 units wide, and 5 units tall. 5 5 5 The factor 5 is used 3 times.

1 -4 Powers and Exponents Additional Example 1 B: Writing Powers for Geometric Models Write the power represented by the geometric model. 6 6 62 The figure is 6 units long and 6 units wide. 6 x 6 The factor 6 is used 2 times.

1 -4 Powers and Exponents Check It Out! Example 1 Write the power represented by each geometric model. The figure is 2 units long and 2 units wide. 2 2 a. b. x 22 x The factor 2 is used 2 times. The figure is x units long, x units wide, and x units tall. x x x x x 3 The factor x is used 3 times.

1 -4 Powers and Exponents 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 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 3 to the fifth power Multiplication Power Value 3 31 3 32 9 3 33 27 3 3 34 81 3 3 35 243

1 -4 Powers and Exponents 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.

1 -4 Powers and Exponents Additional Example 2: Evaluating Powers Simplify each expression. A. (– 6)3 (– 6)(– 6) Use – 6 as a factor 3 times. – 216 B. – 102 – 1 • 10 – 100 Think of a negative sign in front of a power as multiplying by a – 1. Find the product of – 1 and two 10’s.

1 -4 Powers and Exponents Additional Example 2: Evaluating Powers Simplify the expression. C. 2 2 9 9 2 2= 4 9 9 81 Use 2 as a factor 2 times. 9

1 -4 Powers and Exponents Check It Out! Example 2 Simplify each expression. a. (– 5)3 (– 5)(– 5) Use – 5 as a factor 3 times. – 125 b. – 62 – 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.

1 -4 Powers and Exponents Check It Out! Example 2 Simplify the expression. c. Use 3 as a factor 3 times. 4 27 64

1 -4 Powers and Exponents Additional Example 3: Writing Powers Write each number as a power of the given base. A. 64; base 8 8 8 The product of two 8’s is 64. 82 B. 81; base – 3 (– 3)(– 3)4 The product of four – 3’s is 81.

1 -4 Powers and Exponents 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.

1 -4 Powers and Exponents 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 4 th round of calls? 1 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.

1 -4 Powers and Exponents Additional Example 4 Continued 2 Make a Plan 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

1 -4 Powers and Exponents Additional Example 4 Continued 3 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 31 families contacted 2 nd round of calls: 3 3 = 9 or 32 families contacted 3 rd round of calls: 9 3 = 27 or 33 families contacted So, in the 4 th round of calls, 34 families will have been contacted. 34 = 3 3 = 81 Multiply four 3’s. In the fourth round of calls, 81 families will have been contacted.

1 -4 Powers and Exponents Additional Example 4 Continued 4 Look Back 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.

1 -4 Powers and Exponents Check It Out! Example 4 What if…? How many bacteria will be on the slide after 8 hours? 28 After each hour, the number of bacteria is a power of 2. 2 2 Multiply eight 2’s. 256 The product of eight 2’s.

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