1. Preview
Warm Up
California Standards
Lesson Presentation

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

3. 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.

4. Vocabulary
power
base
exponent

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, 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.

6. 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

7. 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.

8. 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.

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 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

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 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

11. 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.

12. 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

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

14. 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

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

16. 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

17. 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

18. 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.

19. Additional Example 4 Continued2
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

20. 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.

21. 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.

22. 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.

23. 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.
; base 7 73
7. 10,000; base 10
104