Transcript 1-4 Powers and Exponents

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Powers and Exponents
Preview
Warm Up
California Standards
Lesson Presentation
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Powers and Exponents
Warm Up
Simplify.
1. 2(2) 4
2. (–2)(–2) 4
3. (–2)(–2)(–2) –8
4. 3(3)(3) 27
5.
4
9
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Powers and Exponents
California
Standards
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.
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Powers and Exponents
Vocabulary
power
base
exponent
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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.
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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 = s2, 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 = s3, where s is the side
length.
s
s
s
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Powers and Exponents
Additional Example 1A: Writing Powers for
Geometric Models
Write the power represented by the
geometric model.
5
5
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.
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Powers and Exponents
Additional Example 1B: 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.
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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
x3
The factor x is used 3 times.
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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
33
32
9
333
33
27
3333
34
81
33333
35
243
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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.
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Powers and Exponents
Additional Example 2: Evaluating Powers
Simplify each expression.
A. (–6)3
(–6)(–6)(–6)
Use –6 as a factor 3 times.
–216
B. –102
–1 • 10 • 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.
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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
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Powers and Exponents
Check It Out! Example 2
Simplify each expression.
a. (–5)3
(–5)(–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.
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Powers and Exponents
Check It Out! Example 2
Simplify the expression.
c.
Use 3 as a factor 3 times.
4
27
64
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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)(–3)(–3)
(–3)4
The product of four –3’s is 81.
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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)
(–3)3
The product of three –3’s is –27.
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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 4th round of calls?
1
Understand the problem
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.
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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
1st round of calls
2nd round of calls
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Additional Example 4 Continued
3
Solve
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.
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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.
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Powers and Exponents
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
28
a power of 2.
2  2  2  2  2  2  2  2 Multiply eight 2’s.
256
The product of eight 2’s.
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Powers and Exponents
Lesson Quiz
1. Write the power
represented by the
geometric model.
n
n
n2
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