Powers and Exponents CONFIDENTIAL

Multiply or divide if possible: Warm Up Multiply or divide if possible: 2) 3) 4) CONFIDENTIAL

Powers and Exponents When bacteria divide, their number increases exponentially. This means that the number of bacteria is multiplied by the same factor each time the bacteria divide. Instead of writing repeated multiplication to express a product, you can use a power. 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. CONFIDENTIAL

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. 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. CONFIDENTIAL

Writing Powers for Geometric Models Write the power represented by each geometric model. There are 3 rows of 3 dots. 3 × 3 The factor 3 is used 2 times. The figure is 4 cubes long, 4 cubes wide, and 4 cubes tall. 4 × 4 × 4 The factor 4 is used 3 times. CONFIDENTIAL

Write the power represented by each geometric model. Now you try! Write the power represented by each geometric model. CONFIDENTIAL

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. CONFIDENTIAL

Simplify each expression. Evaluating Powers Simplify each expression. A) (-2)3 =(-2) (-2) (-2) = -8 Use -2 as a factor 3 times. B) -52 = -1 · 5 · 5 = -1 · 25 = -25 Think of a negative sign in front of a power as multiplying by -1. Find the product of -1 and two 5’s. C) (2)2 (3)2 = (2) . (2) (3) (3) = 4 9 Use 2 as a factor 2 times. 3 CONFIDENTIAL

Simplify each expression. Now you try! Simplify each expression. 2a) ( -5)3 2b) – 62 2c) (3)2 (4)2 CONFIDENTIAL

Write each number as a power of the given base. Writing Powers Write each number as a power of the given base. A) 8; base 2 = 2 · 2 · 2 = 23 The product of three 2’s is 8. B) -125; base -5 = (-5) (-5) (-5) = (-5)3 The product of three -5’s is -125. CONFIDENTIAL

Write each number as a power of the given base. Now you try! Write each number as a power of the given base. 3a) 64; base 8 3b) -27; base -3 CONFIDENTIAL

Problem-Solving Application A certain bacterium splits into 2 bacteria every hour. There is 1 bacterium on a slide. How many bacteria will be on the slide after 6 hours? • There is 1 bacterium on a slide that divides into 2 bacteria. • Each bacterium then divides into 2 more bacteria. The diagram shows the number of bacteria after each hour. Notice that after each hour, the number of bacteria is a power of 2. CONFIDENTIAL

After 1 hour: 1 · 2 = 2 or 21 bacteria on the slide After 2 hours: 2 · 2 = 4 or 22 bacteria on the slide After 3 hours: 4 · 2 = 8 or 23 bacteria on the slide So, after the 6th hour, there will be 2 6 bacteria. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64 After 6 hours, there will be 64 bacteria on the slide. Multiply six 2’s. The numbers become too large for a diagram quickly, but a diagram helps you recognize a pattern. Then you can write the numbers as powers of 2. CONFIDENTIAL

How many bacteria will be on the slide after 8 hours? Now you try! 4) A certain bacterium splits into 2 bacteria every hour. There is 1 bacterium on a slide. How many bacteria will be on the slide after 8 hours? CONFIDENTIAL

Write the power represented by each geometric model. Assessment Write the power represented by each geometric model. 1) 2) 3) CONFIDENTIAL

Simplify each expression: 4) 72 5) (-2)4 6) (-2)5 CONFIDENTIAL

Write each number as a power of the given base. CONFIDENTIAL

10) Jan wants to predict the number of hits she will get on her Web page. Her Web page received 3 hits during the first week it was posted. If the number of hits triples every week, how many hits will the Web page receive during the 5th week? CONFIDENTIAL

Powers and Exponents Let’s review When bacteria divide, their number increases exponentially. This means that the number of bacteria is multiplied by the same factor each time the bacteria divide. Instead of writing repeated multiplication to express a product, you can use a power. 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. CONFIDENTIAL

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. 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. CONFIDENTIAL

Writing Powers for Geometric Models Write the power represented by each geometric model. There are 3 rows of 3 dots. 3 × 3 The factor 3 is used 2 times. The figure is 4 cubes long, 4 cubes wide, and 4 cubes tall. 4 × 4 × 4 The factor 4 is used 3 times. CONFIDENTIAL

Simplify each expression. Evaluating Powers Simplify each expression. A) (-2)3 =(-2) (-2) (-2) = -8 Use -2 as a factor 3 times. B) -52 = -1 · 5 · 5 = -1 · 25 = -25 Think of a negative sign in front of a power as multiplying by -1. Find the product of -1 and two 5’s. C) (2)2 (3)2 = (2) . (2) (3) (3) = 4 9 Use 2 as a factor 2 times. 3 CONFIDENTIAL

Problem-Solving Application A certain bacterium splits into 2 bacteria every hour. There is 1 bacterium on a slide. How many bacteria will be on the slide after 6 hours? • There is 1 bacterium on a slide that divides into 2 bacteria. • Each bacterium then divides into 2 more bacteria. The diagram shows the number of bacteria after each hour. Notice that after each hour, the number of bacteria is a power of 2. CONFIDENTIAL

After 1 hour: 1 · 2 = 2 or 21 bacteria on the slide After 2 hours: 2 · 2 = 4 or 22 bacteria on the slide After 3 hours: 4 · 2 = 8 or 23 bacteria on the slide So, after the 6th hour, there will be 2 6 bacteria. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64 After 6 hours, there will be 64 bacteria on the slide. Multiply six 2’s. The numbers become too large for a diagram quickly, but a diagram helps you recognize a pattern. Then you can write the numbers as powers of 2. CONFIDENTIAL

You did a great job today! CONFIDENTIAL