Exponents and Rules
Exponent Rules
Standard: 8.EE.1
Today
•
What are Exponents?
•
How do you write an expression
using exponents?
•
How do you evaluate an expression
that contains exponents?
•
What RULES do we have for dealing
with exponents?
What is an Exponent?
Exponents are a short-cut way of
writing
repeated multiplication
.
How do you Write an
expression using exponents?
Expressions can be written in two
different forms.
Factored Form
Exponential Form
8 • 8 • 2 • 2 • 2
8² • 2³
5 • 9 • 5 • 5 • 5
5⁴ • 9
How do you Evaluate an expression
that contains exponents?
To find 5
4
on my calculator I type in:
= 625
How do you Evaluate an expression
that contains exponents?
To find 3
4
• 2⁵ on my calculator I type in:
= 2,592
Write the expression in Exponential Form.
6 • 6 • 3 • 4 • 6 • 3
Now, evaluate your expression using the
^
button on your calculator.
= 6³ • 3² • 4
7,776
When
evaluating exponents
you must
watch the sign
and the
parenthesis
!
5²
=
5 • 5
= 25
(–5)² =
(
–
5) • (
–
5)
=
25
–5²
=
–
(5 • 5)
=
–
25
Use your calculator.
Be sure to enter the ( )
where indicated.
Evaluate.
1.
(–4)²
2.
–4²
3.
–(4)²
4.
4²
Multiplying Powers
with the Same Base
RULE 1
8
-
3
7
.
C
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.
Expand each expression into
factored form
and then rewrite it in a simplified
exponential form
as shown in the example.
8
-
3
7
a
.
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.
5
2
• 5⁵
=
5⁷
2² • 2⁴ = 2
6
3⁷ • 3²
= 3⁹
The base is the same in
both forms.
The simplified exponent can
be determined by adding (+)
the original exponents
together.
Original Form
Simplified Form
RULE for Multiplying Powers
with the Same Base
5
2
• 5⁵
=
5²⁺⁵ =
5⁷
or
78,125
2² • 2⁴ =
2²⁺⁴ =
2
6
or
64
3⁷ • 3²
=
3⁷⁺² =
3⁹
or
19,683
(20¹²
)
(20⁵¹
) =
8
-
3
7
b
.
W
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!
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.
20¹²⁺⁵¹ =
20⁶³
8
-
3
7
c
.
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1
0
³
•
5
⁴
a
s
5
0
⁷
.
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g
.
Find
SET 1
on your practice sheet.
You will have a set amount of time
to complete these problems. Then
we will review them.
Dividing Powers
with the Same Base
RULE 2
If you ADD (+) the exponents when
multiplying
powers with the same base….
What do you think you are supposed to
do when
dividing
powers with the same
base?
RULE for Dividing Powers
with the Same Base
Why does subtracting the
exponents work?
= 7³⁻² =
7
= 9⁵⁻¹ =
9⁴
= 2⁹⁻⁴
=
2⁵
Write the expression in simplified exponential
form. Use your new exponent rule! You do not
have to evaluate them.
Find
SET 2
on your practice sheet.
You will have a set amount of time
to complete these problems. Then
we will review them.
Power of a Power
RULE 3
8
-
6
1
.
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.
Complete the table below. Expand each expression into
factored form and then rewrite it with new exponents as
shown in the example.
8
-
6
1
a
.
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.
(5
2
)⁵
=
5¹⁰
(2²)⁴ = 2⁸
(3⁷)²
= 3¹⁴
The exponents in the
simplified form are the
product
of the exponent
inside the parentheses and
the exponent outside the
parentheses.
Original Form
Simplified Form
RULE for a Power of a Power
8
-
6
1
b
.
V
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l
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e
(
2
0
³
)
⁸
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?
Now, write the express (20³)⁸ in simplified
exponential form. You do not have to evaluate
it. Use your new rule!
the factored form would look like
8 sets of 20 · 20 · 20. There would
be 24 of the 20’s.
Find
SET 3
on your practice sheet.
You will have a set amount of time
to complete these problems. Then
we will review them.
Look at this example of how to find a
Power of a Product.
Write this in simplified Exponential form.
(7⁴ • 2³
)²
=
7⁸
•
2⁶
Find
SET 4
on your practice sheet.
You will have a set amount of time
to complete these problems. Then
we will review them.
Exponents are a shortcut for repeated multiplication. Learn how to write expressions using exponents, evaluate expressions containing exponents, and apply rules for dealing with exponents. Explore examples and see how to write expressions in exponential form, evaluate them using calculators, and simplify them. Understand the importance of signs and parentheses when evaluating exponents. Master multiplying powers with the same base and practice expanding expressions into factored and simplified exponential forms.
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Presentation Transcript
Exponent Rules Standard: 8.EE.1
Today What are Exponents? How do you write an expression using exponents? How do you evaluate an expression that contains exponents? What RULES do we have for dealing with exponents?
What is an Exponent? Exponents are a short-cut way of writing repeated multiplication.
How do you Write an expression using exponents? Expressions can be written in two different forms. Factored Form Exponential Form 8 8 2 2 2 8 2 5 9 5 5 5 5 9
How do you Evaluate an expression that contains exponents? To find 54 on my calculator I type in: = 625
How do you Evaluate an expression that contains exponents? To find 34 2 on my calculator I type in: = 2,592
Write the expression in Exponential Form. 6 6 3 4 6 3 = 6 3 4 Now, evaluate your expression using the ^ button on your calculator. 7,776
When evaluating exponents you must watch the sign and the parenthesis! 5 = (5 5) 5 = 5 5 = 25 = 25 ( 5) = ( 5) ( 5) = 25
Use your calculator. Be sure to enter the ( ) where indicated. Evaluate. 16 1. ( 4) -16 2. 4 -16 3. (4) 16 4. 4
Multiplying Powers with the Same Base RULE 1
8 8- -37. 37. Complete the table below. Expand each expression into factored form and then rewrite it in a simplified exponential form as shown in the example.
8 8- -37a. 37a. Work with your team to compare the bases and exponents of the original form to the base and exponent of the simplified exponent form. Write a statement to describe the relationships you see. Original Form Simplified Form The base is the same in both forms. 52 5 =5 The simplified exponent can be determined by adding (+) the original exponents together. 2 2 = 26 3 3 = 3
RULE for Multiplying Powers with the Same Base Step 1 Step 2 ADD the Exponents Step 3 Evaluate if needed Keep the Base 52 5 = 5 = 5 or 78,125 2 2 =2 = 26or 64 3 3 = 3 = 3 or 19,683
Step 1 Step 2 ADD the Exponents Step 3 Evaluate if needed Keep the Base 8 8- -37b. 37b. Write the expression in simplified exponential form. Use your new exponent rule! You do not have to evaluate it. 20 = 20 (20 )(20 ) =
8 8- -37c. 37c. One study team rewrote the expression 10 5 as 50 . Is their simplification correct? Explain your reasoning. NO it is not correct. 103 54= 625,000 , while 507= 781,250,000,000 The BASES have to be the SAME! On this one One of the bases is 10 and the other base is 5.
Find SET 1 on your practice sheet. You will have a set amount of time to complete these problems. Then we will review them.
Dividing Powers with the Same Base RULE 2
If you ADD (+) the exponents when multiplyingpowers with the same base . What do you think you are supposed to do when dividing powers with the same base?
RULE for Dividing Powers with the Same Base Step 1 Step 2 Subtract the Exponents Step 3 Evaluate if needed Keep the Base 65 6 = 6 = 6 or 216
Why does subtracting the exponents work? 6 6 6 6 6 6 6 65 62 = = 63
Step 1 Step 2 Subtract the Exponents Step 3 Evaluate if necessary Keep the Base Write the expression in simplified exponential form. Use your new exponent rule! You do not have to evaluate them. 7 7 = 7 = 7 1) ?? ? 2) = 9 = 9 ?? ?? = 2 = 2 3)
Find SET 2 on your practice sheet. You will have a set amount of time to complete these problems. Then we will review them.
Power of a Power RULE 3
8 8- -61. 61. When a number is raised to a power, and then raised to a power again, the result follows a consistent pattern. Complete the table below. Expand each expression into factored form and then rewrite it with new exponents as shown in the example.
8 8- -61a. 61a. Work with your team to describe the pattern between the exponents in the original form and the exponents in the simplified exponential form. Original Form Simplified Form The exponents in the simplified form are the product of the exponent inside the parentheses and the exponent outside the parentheses. (52) =5 (2 ) = 2 (3 ) = 3
RULE for a Power of a Power Step 1 Step 2 Evaluate if needed Multiply the exponent inside the ( ) by the exponent on the outside. (52) =5 =9,765,625 (2 ) = 2 = 256 (3 ) = 3 = 4,782,969
8 8- -61b. 61b. Visualize (20 ) written in factored form. What would it look like? How many 20 s would be written down? the factored form would look like 8 sets of 20 20 20. There would be 24 of the 20 s. Now, write the express (20 ) in simplified exponential form. You do not have to evaluate it. Use your new rule! (20 ) =20
Find SET 3 on your practice sheet. You will have a set amount of time to complete these problems. Then we will review them.
Look at this example of how to find a Power of a Product. (2 3 ) =2 3 Write this in simplified Exponential form. (7 2 ) = 7 2
Find SET 4 on your practice sheet. You will have a set amount of time to complete these problems. Then we will review them.
