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Simplifying Simplifying ExponentsExponentsSimplifying Simplifying ExponentsExponents
Algebra IAlgebra I
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Contents• Multiplication Properties of Exponents ……….1 – 13• Zero Exponent and Negative Exponents……14 – 24• Division Properties of Exponents ……………….15 –
32• Simplifying Expressions using Multiplication and
Division Properties of Exponents…………………33 – 51
• Scientific Notation ………………………………………..52 - 61
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Multiplication Properties of Exponents
•Product of Powers Property•Power of a Power Property•Power of a Product Property
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Product of Powers Property
• To multiply powers that have the same base, you add the exponents.
• Example: 53232 aaaaaaaaa
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Practice Product of Powers Property:
• Try:
• Try: 325 nnn
45 xx
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Answers To Practice Problems
1. Answer:
2. Answer:
94545 xxxx
10325325 nnnnn
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Power of a Power Property
• To find a power of a power, you multiply the exponents.
• Example:
• Therefore,
622222232 )( aaaaaa
63232 )( aaa
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Practice Using the Power of a Power
Property
1. Try:
2. Try:
44 )( p
54 )(n
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Answers to Practice Problems
1. Answer:
2. Answer:
164444 )( ppp
205454 )( nnn
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Power of a Product Property
• To find a power of a product, find the power of EACH factor and multiply.
• Example: 333333 644)4( zyzyyz
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Practice Power of a Product Property
1. Try:
2. Try:
6)2( mn
4)(abc
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Answers to Practice Problems
1. Answer:
2. Answer:
666666 642)2( nmnmmn
4444)( cbaabc
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Review Multiplication Properties of Exponents
• Product of Powers Property—To multiply powers that have the same base, ADD the exponents.
• Power of a Power Property—To find a power of a power, multiply the exponents.
• Power of a Product Property—To find a power of a product, find the power of each factor and multiply.
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Zero Exponents• Any number, besides zero, to the
zero power is 1.
• Example:
• Example:
10 a
140
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Negative Exponents
• To make a negative exponent a positive exponent, write it as its reciprocal.
• In other words, when faced with a negative exponent—make it happy by “flipping” it.
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Negative Exponent Examples
• Example of Negative Exponent in the Numerator:
• The negative exponent is in the numerator—to make it positive, I “flipped” it to the denominator.
33 1
xx
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Negative Exponents Example
• Negative Exponent in the Denominator:
• The negative exponent is in the denominator, so I “flipped” it to the numerator to make the exponent positive.
44
4 1
1y
y
y
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Practice Making Negative Exponents
Positive
1. Try:
2. Try:
3d
5
1z
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Answers to Negative Exponents Practice
1. Answer:
2. Answer:
33 1
dd
55
5 1
1z
z
z
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Rewrite the Expression with Positive Exponents
• Example:
• Look at EACH factor and decide if the factor belongs in the numerator or denominator.
• All three factors are in the numerator. The 2 has a positive exponent, so it remains in the numerator, the x has a negative exponent, so we “flip” it to the denominator. The y has a negative exponent, so we “flip” it to the denominator.
232 yx
xyyx
22 23
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Rewrite the Expression with Positive Exponents
• Example:
• All the factors are in the numerator. Now look at each factor and decide if the exponent is positive or negative. If the exponent is negative, we will flip the factor to make the exponent positive.
8334 cab
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Rewriting the Expression with
Positive Exponents• Example:
• The 4 has a negative exponent so to make the exponent positive—flip it to the denominator.
• The exponent of a is 1, and the exponent of b is 3—both positive exponents, so they will remain in the numerator.
• The exponent of c is negative so we will flip c from the numerator to the denominator to make the exponent positive.
8334 cab
8
3
83
3
644 c
ab
c
ab
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Practice Rewriting the Expressions with Positive
Exponents:
1. Try:
2. Try:
zyx 3213
dcba 4324
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Answers
1. Answer
2. Answer
32321
33
yx
zzyx
42
3432 4
4ca
dbdcba
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Division Properties of Exponents
•Quotient of Powers Property
•Power of a Quotient Property
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Quotient of Powers Property
• To divide powers that have the same base, subtract the exponents.
• Example: 2
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3
5
1x
x
x
x
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Practice Quotient of Powers Property
1. Try:
2. Try:
3
9
a
a
4
3
y
y
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Answers
1. Answer:
2. Answer:
639
3
9
1a
a
a
a
yyy
y 11344
3
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Power of a Quotient Property
• To find a power of a quotient, find the power of the numerator and the power of the denominator and divide.
• Example: 3
33
b
a
b
a
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Simplifying Expressions
• Simplify
343
3
2
mn
nm
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Simplifying Expressions
• First use the Power of a Quotient Property along with the Power of a Power Property
333
1293
333
34333343
3
2
3
2
3
2
nm
nm
nm
nm
mn
nm
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Simplify Expressions
• Now use the Quotient of Power Property
27
8
27
8
3
2 9631239
333
1293 nmnm
nm
nm
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Simplify Expressions
• Simplify 3
24
243
3
3
2
zyx
zyx
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Steps to Simplifying Expressions
1. Power of a Quotient Property along with Power of a Power Property to remove parenthesis
2. “Flip” negative exponents to make them positive exponents
3. Use Product of Powers Property4. Use the Quotient of Powers Property
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Power of a Quotient Property and Power of a
Power Property• Use the power of a quotient property to remove
parenthesis and since the expression has a power to a power, use the power of a power property.
3332343
32343333
24
243
3
2
3
23
zyx
zyx
zyx
zyx
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Continued
• Simplify powers
96123
61293
3332343
3234333
3
2
3
2
zyx
zyx
zyx
zyx
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“Flip” Negative Exponents to make Positive Exponents
• Now make all of the exponents positive by looking at each factor and deciding if they belong in the numerator or denominator.
123
9612693
96123
61293
2
3
3
2
y
zyxzx
zyx
zyx
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Product of Powers Property
• Now use the product of powers property to simplify the variables.
12
15621
12
966129
123
9612693
6
27
6
27
2
3
y
zyx
y
zyx
y
zyxzx
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Quotient of Powers Property
• Now use the Quotient of Powers Property to simplify.
6
1521
612
1521
12
15621
6
27
6
27
6
27
y
zx
y
zx
y
zyx
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Simplify the Expression
• Simplify:
4
432
523
2
5
zyx
zyx
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Step 1: Power of a Quotient Property and
Power of a Power Property
161284
208124
2
5
zyx
zyx
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Step 2: “Flip” Negative Exponents
1282084
16124
5
2
yxzy
zx
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Step 3: Product of Powers Property
202084
16124
5
2
zyx
zx
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Step 4: Quotient of Powers Property
420
4
625
16
zy
x
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Simplifying Expressions
• Given
• Step 1: Power of a Quotient Property
22
31 3
2
2
4
xy
xy
yx
xy
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Power of Quotient Property
• Result after Step 1:
• Step 2: Flip Negative Exponents
222
422
31 3
2
2
4
yx
yx
yx
xy
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“Flip” Negative Exponents
• Step 3: Make one large Fraction by using the product of Powers Property
422
2223
2
3
2
4
yx
yxxyxy
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Make one Fraction by Using Product of Powers
Property
423
642
2
34
yx
yx
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Use Quotient of Powers Property
2
9 22 yx
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Simplify the Expressions
1. Try:
2. Try:
1
2
33
1
2
42
3
a
x
x
a
253
4
2 22
y
x
y
x
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Answers
1. Answer:
2. Answer:
2
27
42
3 641
2
33
1
2 xa
a
x
x
a
104
253
4
2 222
yxy
x
y
x
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Scientific Notation• Scientific Notation uses powers of ten to
express decimal numbers.
• For example:
• The positive exponent means that you move the decimal to the right 5 times.
• So,
51039.2
000,2391039.2 5
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Scientific Notation
• If the exponent of 10 is negative, you move the decimal to the left the amount of the exponent.
• Example: 0000000265.01065.2 8
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Practice Scientific Notation
Write the number in decimal form:
1.
2.
6109.4 31023.1
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Answers
1.
2.
000,900,4109.4 6
00123.01023.1 3
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Write a Number in Scientific Notation
• To write a number in scientific notation, move the decimal to make a number between 1 and 9. Multiply by 10 and write the exponent as the number of places you moved the decimal.
• A positive exponent represents a number larger than 1 and a negative exponent represents a number smaller than 1.
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Example of Writing a Number in Scientific
Notation1. Write 88,000,000 in scientific notation
• First place the decimal to make a number between 1 and 9.
• Count the number of places you moved the decimal.
• Write the number as a product of the decimal and 10 with an exponent that represents the number of decimal places you moved.
• Positive exponent represents a number larger than 1. 7108.8
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Write 0.0422 in Scientific Notation
• Move the decimal to make a number between 1 and 9 – between the 4 and 2
• Write the number as a product of the number you made and 10 to a power 4.2 X 10
• Now the exponent represents the number of places you moved the decimal, we moved the decimal 2 times. Since the number is less than 1 the exponent is negative.
2102.4
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Operations with Scientific Notation
• For example:• Multiply 2.3 and
1.8 = 4.14• Use the product
of powers property
• Write in scientific notation
)108.1)(103.2( 53
531014.4
21014.4
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Try These:• Write in scientific notation
1.
2.
)103)(101.4( 62
)105.2)(106( 15
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Answers
1.
2.
962 1023.1)103)(101.4(
515 105.1)105.2)(106(
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The End• We have completed all the
concepts of simplifying exponents. Now we just need to practice the concepts!