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**Simplifying Exponents**

Algebra 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 ………………………………………

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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:

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**Practice Product of Powers Property:**

Try:

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**Answers To Practice Problems**

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**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**

Try:

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**Answers to Practice Problems**

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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:

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**Answers to Practice Problems**

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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:

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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.

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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:

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**Answers to Negative Exponents Practice**

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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 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.

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**Practice Rewriting the Expressions with Positive Exponents:**

Try:

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Answers 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:

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Answers 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**

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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**

Power of a Quotient Property along with Power of a Power Property to remove parenthesis “Flip” negative exponents to make them positive exponents Use Product of Powers Property 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**

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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:

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Answers 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!

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