Simplifying Exponents

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Simplifying Exponents

• Algebra I

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Contents

• Multiplication Properties of Exponents .1 13
• Zero Exponent and Negative Exponents14 24
• Division Properties of Exponents .15 32
• Simplifying Expressions using Multiplication and
  Division Properties of Exponents33 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

5
Practice Product of Powers Property

• Try
• Try

6
Answers To Practice Problems

• Answer
• Answer

7
Power of a Power Property

• To find a power of a power, you multiply the
  exponents.
• Example
• Therefore,

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Practice Using the Power of a Power Property

1. Try
2. Try

9
Answers to Practice Problems

• Answer
• Answer

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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

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Practice Power of a Product Property

• Try
• Try

12
Answers to Practice Problems

• Answer
• Answer

13
Review Multiplication Properties of Exponents

• Product of Powers PropertyTo multiply powers
  that have the same base, ADD the exponents.
• Power of a Power PropertyTo find a power of a
  power, multiply the exponents.
• Power of a Product PropertyTo 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

15
Negative Exponents

• To make a negative exponent a positive exponent,
  write it as its reciprocal.
• In other words, when faced with a negative
  exponentmake 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 numeratorto make
  it positive, I flipped it to the denominator.

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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.

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Practice Making Negative Exponents Positive

• Try
• Try

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Answers to Negative Exponents Practice

• Answer
• Answer

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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.

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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.

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Rewriting the Expression with Positive Exponents

• Example
• The 4 has a negative exponent so to make the
  exponent positiveflip it to the denominator.
• The exponent of a is 1, and the exponent of b is
  3both 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.

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Practice Rewriting the Expressions with Positive
Exponents

1. Try
2. Try

24
Answers

1. Answer
2. Answer

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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

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Practice Quotient of Powers Property

• Try
• Try

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Answers

• Answer
• Answer

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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

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Simplifying Expressions

• Simplify

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Simplifying Expressions

• First use the Power of a Quotient Property along
  with the Power of a Power Property

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Simplify Expressions

• Now use the Quotient of Power Property

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Simplify Expressions

• Simplify

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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 Property
4. 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.

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Continued

• Simplify powers

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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.

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Product of Powers Property

• Now use the product of powers property to
  simplify the variables.

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Quotient of Powers Property

• Now use the Quotient of Powers Property to
  simplify.

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Simplify the Expression

• Simplify

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Step 1 Power of a Quotient Property and Power
of a Power Property
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Step 2 Flip Negative Exponents
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Step 3 Product of Powers Property
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Step 4 Quotient of Powers Property
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Simplifying Expressions

• Given
• Step 1 Power of a Quotient Property

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Power of Quotient Property

• Result after Step 1
• Step 2 Flip Negative Exponents

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Flip Negative Exponents

• Step 3 Make one large Fraction by using the
  product of Powers Property

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Make one Fraction by Using Product of Powers
Property
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Use Quotient of Powers Property
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Simplify the Expressions

• Try
• Try

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Answers

• Answer
• Answer

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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,

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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

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Practice Scientific Notation

• Write the number in decimal form
• 1.
• 2.

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Answers

• 1.
• 2.

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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 Notation

• 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.

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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.

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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

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Try These

• Write in scientific notation
• 1.
• 2.

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Answers

• 1.
• 2.

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The End

• We have completed all the concepts of simplifying
  exponents. Now we just need to practice the
  concepts!