Properties of Exponents - WordPress.com · integer exponents. 4. 4x2y 2xy2 22x y xy2 2xy 2y 2x y...
Transcript of Properties of Exponents - WordPress.com · integer exponents. 4. 4x2y 2xy2 22x y xy2 2xy 2y 2x y...
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Part 1: Multiplication Properties
Part 2: Division Properties
Part 3: Integers & Exponents
Part 4: Simplify Expressions with Exponents
Properties of Exponents
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Part 1
Multiplication Properties
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Exponent Review
An exponent is a number placed above and to the right of
another number to show that it has been raised to a power.
This is 2 to the fifth power. Which means:
2 • 2 • 2 • 2 • 2
25 = 32
25 Base
Exponent
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Multiplying Rule #1
35 = 3 • 3 • 3 • 3 • 3
38 = 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3
How many 3s all together? 13
Words Numbers Algebra
To multiply powers with the same base, keep the base and add the exponents.
MULTIPLYING POWERS WITH THE SAME BASE
35 • 38 = 35 + 8 = 313
bm • bn = bm + n
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Practice: Multiply. Write the product as one power.
1. 66 • 63
66+3
69
2. n5 • n7
n5+7
n12
3. 25 • 2
25+1
26
4. 244 • 244
244+4
248
5. 810 • 83 • 87
810+3+7
820
6. h4 • m4
Don’t have the
same base.
h4 m4
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Multiplying Rule #2
Power of powers with the same base:
For example: (43)5
43 = 4 • 4 • 4
Now do that 5 times.
(4 • 4 • 4) (4 • 4 • 4) (4 • 4 • 4) (4 • 4 • 4) (4 • 4 • 4)
How many 4s do you have?
15
(43)5 = 415
The rule is that if you are taking a power to another power,
you multiply the exponents.
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Multiplying Rule #3
The Power of a Product Property is used when you have
different bases being raised to another power.
In this case, distribute the POWER to each base by
multiplication.
For example:
(23m5)4
You are going to take (23)4 and (m5)4
(23m5)4 = 212m20
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Practice:
1. (32y3)2
3(2•2)y(3•2)
34y6
81y6
2. (5a2b4)5
5(1•5)a(2•5) b(4•5)
55a10 b20
55 is way more than 1,000
so leave it as 55.
3. (-4xy3)3
(-4)(1•3)x(1•3) y(3•3)
(-4)3x3y9
-64x3y9
4. (24h3k2)2
2(4•2)h(3•2) k(2•2)
28h6 k4
256h6 k4
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Part 2
Division Properties
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Dividing Rule #1
69 = 6 • 6 • 6 • 6 • 6 • 6 • 6 • 6 • 6
64 = 6 • 6 • 6 • 6
How many 6s are left?
This is why the answer is 65
DIVIDING POWERS WITH THE SAME BASE
Words Numbers Algebra
To divide powers with the same base, keep the base and subtract the exponents.
69
64 = 69-4 = 65 = bm-n bm
bn
5
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Practice: Divide. Write the quotient as one power.
1. 75
73
75-3
72
2. x10
x9
x10-9
x
3. 99
92
99-2
97
4. 43
43
43-3
40 = 1
4 • 4 • 4
4 • 4 • 4
= 1
1
Any number to the
zero power is 1.
5. e10
e5
e10-5
e5
6. 7017
705
7017-5
7012
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Division Rule #2
The Power of Quotient Property has you distribute the
power to every number inside the parentheses – both the
numerator and the denominator get the power.
For example
3
2 2
= 22
32
= 4
9
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Practice:
6
x 3
= x3
63
= x3
216
4
2m 4
= 24m4
44
= 16m4
256
1.
2.
Reduce the
numbers of
the fraction
part of the
answer: 16/256
16/256 reduces
to 1/16.
= m4
16
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Practice: 3.
4.
= (-2)3
33
= -8
27
2
8b 3
= 83b3
23
= 512b3
8
= 64b3
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Part 3
Integers & Exponents
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NEGATIVE EXPONENTS
Words Numbers Algebra
A power with a negative exponent equals 1 divided by that power with it’s opposite exponent.
b–n = 1 bn 5–3 = =
1
125
1 53
You can NEVER leave an answer with a negative exponent.
You must have all positive exponents in your final answer.
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Practice simplifying products and quotients
that may have negative exponents.
1. 2–5 • 23
2 –5+3
2 –2
22
1
1 4
Bases are the same, so add the exponents.
6 5–8
6 –3
1 63
65 68
216
1
2. Bases are the same, so subtract the exponents.
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Practice simplifying products and quotients
that may have negative exponents.
5 2–3
5 –1
1
5 1
52 53
1 5
3. Bases are the same, so subtract the exponents.
7 –6+7
7 1
7
4. 7–6 • 77 Bases are the same, so add the exponents.
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Part 4
Simplify Expressions with Exponents
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Simplifying Expressions with Exponents means…
You need to know ALL of the properties of exponents
and
you need to know when to use them.
For example: Simplify the following expression.
4c0
4(1)
4
This is 4 times c to the zero power.
Order of operations states that you
have to solve exponents before
multiplying.
Anything to the zero power is 1.
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Practice simplifying expressions with
integer exponents.
1. (2m3 • 6m4)2
(12m7)2
122m14
144m14
Simplify what is in parenthesis.
2 • 6 = 12
m3 • m4 = m7
Distribute the Power of 2 to
everything in the parenthesis.
Solve 122.
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Practice simplifying expressions with
integer exponents.
2. 3y2(2y)3
= 3y2 23y3
= 3 23y5
= 3 8y5
= 24y5
Distribute the Power of 3 to
everything in the parenthesis.
The “y”s have the same base, add
the exponents.
Solve 23.
Multiply 3 and 8.
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Practice simplifying expressions with
integer exponents.
3. (6j2k4)-3
6-3j-6k-12
1 . 63 j6k12
1 .
216j6k12
Distribute the Power of -3 to
everything in the parenthesis.
You cannot have negative
exponents in the answer. To
make them positive you need to
move them to the denominator
(take the reciprocal).
Solve 63.
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Practice simplifying expressions with
integer exponents.
4. 4x2y
2xy2
2x2y
xy2
2xy
y2
2x
y
Simplify 4 divided by 2.
The “x”s have the same
base, subtract their
exponents: 2 – 1 = 1. Since
this is positive, x1 stays in
the numerator.
The “y”s have the same
base, subtract their
exponents: 1 – 2 = -1.
Since this is negative, y-1 it
moves to the denominator.
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Practice simplifying expressions with
integer exponents.
4 Before distributing the Power of 4 to
everything in the parentheses, first
simplify the variables.
The “x”s have the same base, subtract
their exponents. 7 – 2 = 5. Since it is
positive 5, x5 stays in the numerator. 4
The “y”s have the same base, subtract
their exponents. 3 – 1 = 2. Since it is
positive 2, y2 stays in the numerator.
4 Distribute the Power of 4 to everything
in parentheses.
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Practice simplifying expressions with
integer exponents.
-2 Before distributing the Power of -2 to
everything in parentheses, first reduce what is
in parentheses.
Reduce 27/18.
They are both divisible by 9.
Which reduces to 3/2.
The “w”s are the same base. Since they are
being divided, subtract their exponents.
-4 – 6 = - 10
Since it is a negative exponent, “w” is in the
denominator with a positive 10 exponent.
-2
-2
The “x”s are the same base. Since they are
being divided, subtract their exponents.
3 – -5 = 8
Since it is a positive exponent, “x” is in the
numerator with a positive 8 exponent.
-2
6.
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#6 Continued…
-2 Now distribute the Power of -2 to everything
in parentheses.
You cannot have negative exponents in the
answer. To make them positive you need to
take the reciprocal.
Move the terms with negative exponents from
the numerator to the denominator.
Move the terms with negative exponents from
the denominator to the numerator.
Solve 22 and 32.
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The key to these exponent problems is to
take your time.
Keep all of the rules handy so they are easy
to reference.