Preview Warm Up California Standards Lesson Presentation

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

Standards California 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.

Vocabulary power base exponent

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.

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. 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

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 5 53 The factor 5 is used 3 times.

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.

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. x 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

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

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.

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

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

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

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

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)(–3)(–3) The product of four –3’s is 81. (–3)4

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

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? Understand the problem 1 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.

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 1st round of calls 2nd round of calls

Additional Example 4 Continued Solve 3 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. 34 = 3  3  3  3 = 81 Multiply four 3’s. In the fourth round of calls, 81 families will have been contacted.

Additional Example 4 Continued Look Back 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.

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.

n n Lesson Quiz 1. Write the power represented by the geometric model. Simplify each expression. 3. –63 −216 2. 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