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Exponents Bell Ringers
In your journal find the following solutions.
1. 32
7. (-3)2
2. 45
8. -32
3. 510
4. 63
5.
6.
1 2
( )^
2
1 3
( )^
4
8.EE.1 Exponents
Created By Melissa Forsyth
Exponents
Exponents are shorthand for repeated
multiplication of the same thing by itself. For
instance, the shorthand for multiplying three
copies of the number 5 is shown on the righthand side of the "equals" sign in (5)(5)(5) = 53.
Base: The thing that's being
multiplied, being 5 in this
example.
3
5
Exponent: stands for
however many times the
value is being multiplied. The
3 is the exponent in this
example.
• This process of using exponents is called
"raising to a power", where the exponent is
the "power".
• The expression "53" is pronounced as "five,
raised to the third power" or "five to the
third".
• There are two specially-named powers: "to
the second power" is generally pronounced as
"squared", and "to the third power" is
generally pronounced as "cubed".
Negative Numbers with Exponents
(-3)2
(–3)2 = (–3)(–3) = (+3)(+3) = 9
-32
–32 = –(3)(3) = (–1)(9) = –9
Fractions with exponents
When dealing with fractions raised to a power
you will apply the exponent to the denominator.
2 2
^
6
=
2
6^2
=
2
6x6
=
2
36
=
1
18
Exponential vs Expanded Form
Exponential Form: 35
Expanded Form: 3 x 3 x 3 x 3 x3
How can we multiply or divide two powers with
the same base without expanding the
expression?
What is the expanded form of this
expression?
4
53 x 5
53 x 54 = (5 x 5 x 5) x (5 x 5 x 5 x 5)
So, 53 x 54 = 57
57
• What do you notice about multiplying powers with the
same base?
• Is there a shortcut that we can use, without having to
write the exponents in expanded form?
• Does the base change?
• Does the exponent change? How?
• What is the rule?
Rule for Multiplying Powers with the
same base
When multiplying powers with the same base,
add the exponents:
m
a
x
n
a
=
m
+
n
a
Solve the Following Problems
1. 67 x 625 =
2. 31 x 31 =
3. 72 x ____ = 76
4.
2
^2
3
x
2
^8
3
=
5. k4 x k10 =
6. x7 · x4 =
7. Find two powers that will make the equation
true: _______ x _______ = 513
What is the expanded form of this
expression?
2^6
2^2
=
3^8
=
3^3
𝑥^10
=
𝑥^8
2x2x2x2x2x2 2x2x2x2x2x2
=
=
2x2
2x2
24
What is the expanded form of this
expression?
• Is there a shortcut that we can use, without
having to write the exponents in expanded
form?
• Does the base change?
• Does the exponent change? How?
• What is the rule?
Rule for Dividing Powers with the
same base
When dividing powers with the same base,
subtract the exponents:
𝑎^𝑚
𝑎^𝑛
= a m-n
Solve the Following Problems
1.
2.
3.
4.
3^8
3^7
5^10
=
?
𝑥^20
𝑥^3
(−4)^2
(−4)^1
52
5. Find two powers that will make the equation
?
true = 95
?
How can we raise a power to a power without
expanding the expression?
Expand and simplify
4
^3
(5 ) =
4
5 x
4
5 x
4
5
=
(5 x 5 x 5 x 5) x (5 x 5 x 5 x 5) x (5 x 5 x 5 x 5)
512
Expand and simplify
1. (46)^2
2. (42)^6
3. (x4)^1
4.
1 ^3 ^3
(( ) )
4
• Is there a shortcut that we can use, without
having to write the exponents in expanded
form?
• Does the base change?
• Does the exponent change? How?
• What is the rule?
Rule for raising a Power to a power
When raising a power to a power, multiply the
exponents.
(am)^n = am x n
Solve the Following Problems
1. (57)^7
2. (-73)^9
3. (46)^? = 412
4. (x12)^5
5. (46)^7
6. (a?)^3 = a18
7. Find two exponents that will make the
equation true:(x?)^? = x24
Exponents of Variables
(-x2y5)^3
• First, it has a term with two variables, and as you can see the
exponent from outside the parentheses must multiply EACH
of them.
• Second, there is a negative sign inside the parentheses. Since
the exponent on the parentheses is 3, the negative sign is
written in front of the term three times. Then the multiple
signs are simplified
- - -x2 x 3 y5 x 3 = + + -x2 x 3 y5 x 3 = -x6 y15
Exponents of Variables
-(x2y5)^3
• This problem has a negative sign outisde the parentheses.
Again, because of the Order of Operations, the exponent must
be simplified before you do anything with the negative sign.
Look at the work below
-(x2x 3 y5x 3) = -(x6y15) = -x6y15
What does an exponent of 0 mean?
Use the rules and simplify
3^4
3^4
= 34 – 4 = 3 0
Or
3^4
3^4
=
3x3x3x3
3x3x3x3
=1
When dealing with fractions what does it mean
when the numerator and the denominator are the
same?
*it equals 1
The Power of 0
For every nonzero number, a
0
a =1
Anything to the power zero is
just "1".
Simplify
1. (57)^0
2.
3^8
3^8
3. 5-4 x 54
4. 2.[(3x4y7z12)5 (–5x9y3z4)2]0
Negative Exponents
What are negative exponents?
Let’s use what we know to look at negative
exponents
23 x 2-3
We know that 23 x 2-3 = 23 + -3 = 20
23 = 8
2-3 = ?
20 = 1
8x?=1
What times 8 equals 1?
1
8
1
8
8
1
x =1
So,
2-3
=
1
8
Negative Exponents
For every nonzero number, a and integer n
a-n
=
1
𝑎^𝑛
Simplify
1.
2.
3.
4.
4-6
10-20
(-3)-3
7-5
Distributive property and Polynomial exponents
(x3 + y4)^2
Because the two terms inside parentheses are not being multiplied or
divided, the exponent outside the parentheses can not just be
"distributed in". Instead, a 1 must be multiplied by the entire
polynomial the number of times indicated by the exponent. In this
problem the exponent is 2, so it is multiplied two times:
1(x3 + y4) (x3 + y4)
Use the FOIL Method to simplify the multiplication above, then
combine like terms
(x3 + y4) (x3 + y4) =( x3 · x3 ) + (x3 · y4) + (y4 · x3) + (y4 · y4)
( x3 · x 3 ) = x 6
(x3 · y4) + (y4 · x3) = 2(x3 · y4)
(y4 · y4) = y8
x6 + 2(x3 · y4) + y8
Simplify
1.
2.
3.
4.
(x + y)^2
(x + 3)^2
(x + 7)^2
(3a + 3b)^2
Distributive property and Polynomial exponents
When we distributed (x3 + y4)^2 we got
x6 + 2(x3 · y4) + y8
Can we reverse the distribution process?
1. We have to identify the greatest common factor for each
variable.
For the variable x the largest exponent is x3 so we will first take
out x3
For the variable y the largest exponent is y4 so we will first take
out y4
2. Since these terms are being added together we can say
x3 + y4
3. Finally we need to determine what we need to raise these to
the power of. What do we need to multiply 3 by to get to 6 and
what do we need to multiply 4 by to get 8.
We have to raise these terms to the power of 2.
So (x3 + y4)^2
Reverse Distribute
x2 + 2xy + y2