1. Powers and Exponents
CONFIDENTIAL

2. Multiply or divide if possible:
Warm Up
Multiply or divide if possible: 2) 3) 4)
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3. 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.
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4. 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.
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5. 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.
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6. Write the power represented by each geometric model.
Now you try!
Write the power represented by each geometric model.
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7. 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.
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8. 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
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9. Simplify each expression.
Now you try!
Simplify each expression.
2a) ( -5)3
2b) – 62
2c) (3)2
(4)2
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10. Write each number as a power of the given base.
Writing Powers
Write each number as a power of the given base.
A) ; base 2
= 2 · 2 · 2
= 23
The product of three 2’s is 8.
B) ; base -5
= (-5) (-5) (-5)
= (-5)3
The product of three -5’s is -125.
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11. 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
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12. 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.
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13. 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.
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14. 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?
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15. Write the power represented by each geometric model.
Assessment
Write the power represented by each geometric model. 1) 2) 3)
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16. Simplify each expression:
4) 72
5) (-2)4
6) (-2)5
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17. Write each number as a power of the given base.
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18. 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?
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19. 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

20. 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

21. 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

22. 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
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23. 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.
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24. 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

25. You did a great job today!
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