Transcript Exponents - TeacherWeb

Exponents
𝑥2
Powers and Exponents
6.1 p. 433
What is our goal?
• We will review the vocabulary associated with
exponents and exponential notation
• We will extend this idea to evaluate exponents
correctly with the order of operations
Vocabulary
base
The number used as a factor
Exponent (indicates power)
tells how many times a base is used as a
factor
exponential form
2x2x2x2x2 =
This tells us to multiply
2 by itself five times.
25
10 x 10 =
Factors
2
10
Exponent
Base
3. In your own words, write the definition of an exponent. Compare it to
that of another person in your group.
Key words: base, multiplied, power
MP3 players come in different storage sizes, such as
2GB, 4GB, or 16 GB, where GB means gigabyte.
One gigabyte is equal to 10·10·10·10·10·10·10·10·10 bytes.
In exponential notation, this would be
109
Write Products as Powers
100 is a perfect square.
10 x 10 = 100
102 = 100
** 100 = 10
radical sign
10 x 10 x 10 = 1000
103 = 1000
3
** 1000 = 10
radical sign
Other perfect squares……..WHY??
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169……
Perfect cubes are numbers with three identical whole number
factors…..a number that has been “cubed” or multiplied by
itself three times. Because 4 x 4 x 4 = 64, 64 is a perfect cube.
Eight is a perfect cube because 2 x 2 x 2 = 8.
Examples
1.
Write
6 ∙ 6 · 6 ∙ 6 with an exponent
It is often helpful to write what the
exponential form means
6 x 6 x 6x 6
=
64
2. Write 4 x 4 x 4 using an exponent
4 is the base.
The factor _____
3 times.
The factor is multiplied (by itself) ______
3
The exponent is _____.
3
4
The exponential form is _________
7(7)7(7) = 74
9x9x9x9x9x9x9=
97
Got it????
Writing Powers as Products
Determine the base and the exponent.
102 10 is the base, 2 is the exponent. This would be
read as “10 squared.” if it were 103 we would read this
as “10 cubed.”
Write 52 as the product of the same factor.
By writing what it means, you can keep track of how many
times you have used the base.
52 = 5 x 5 = 25
Write 1.53 as the product of the same factor. 
1.5
2.25
(1.5)(1.5)(1.5) =
x 1.5
75
150
2.25
x 1.5
1125
2250
3.375
1
3 as the product of the same factor.
5. Write
2
1
1 1
1
1 1
𝑥
𝑥 =
𝑥
=
4
2
2
2
2
8
Got it????
=
105 = (10)10(10)10(10)
100,000
21
2.12 = 2.1 x 2.1
1
4
x 21
21
420
4.41
2
1
1
= 𝑥
4
4
=
1
16
WAIT!!!!!
Guided Practice……
1. 8 x 8 x 8 =
83
2. 1 x 1 x 1 x 1 x 1=
Write each power as a product, then find the value.
(Expand and evaluate)
1
7
3. ( )3
1
1 1
1 1
𝑥
𝑥 =
∙
7
7
7
49 7
1
=
343
4. 25 = 2 x 2 x 2 x 2 x 2 =
= 4 x 4 x2
= 16 x 2 = 32
14
5. 1.42 =
1.4 ( 1.4) =
x 14
56
140
1.96
15
6. Coal mines have shafts that can be as much as 73
feet deep. About how many feet deep into the Earth’s
crust are these shafts? (Look back at your work!!)
≈ 343 𝑓𝑡.
7. How is using exponents helpful?
Strange, but TRUE!
There are two interesting exponents that you will see this
year. The exponents 0 and 1 have values that are worth
talking about.
n0 =
0
10 = 1
1
0
12 = 1
n1 =
1
10 =
10
n
1
12 =
12
n0=1
Anything0 =
1
The exponent tells us how many times to write a base in the
multiplication problem. Well, if the exponent is 0, then the base is not
written at all……but it still has a 1 as a factor…
Later this year you will learn an exponent rule that “proves” this fact
about the exponent 0.
. . . . 100 = 1
120 = 1
0
0
=
1
=
1
n 1=n
101 = 10
Anything 1 = Anything
71 = 7
5271 = 527
So . . . . .
1
=
1
=
n1 = n
Let’s Try a Few More
2 2 
3
(2 3) 
3
2 + (2 x 2 x 2)
3
(5) 
125
2+8=
(1 3) 
2
(4) 
2
4 x 4 = 16
10
In Review……..
Base
3
3
Exponent
=3∙3∙3
9 ∙ 3 = 27
The number 3 was used as
a factor 3 times.
In review
Exponential
Form
103
53
34
Written as
Factors
Evaluated
10 x 10 x 10
1000
(5)(5)(5)
125
3∙3∙3∙3
9∙9
81
What did we accomplish?
• We have defined exponential notation.
• We have written values as the products of
repeated factors.
• We looked at perfect squares.
• We have evaluated numbers written in
exponential notation.