simplifying exponents

Simplifying Exponents

Simplifying Exponents

Algebra I

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

Multiplication Properties of Exponents

Product of Powers Property

Power of a Power Property

Power of a Product Property


To multiply powers that have the same base, you add the exponents.

Example:

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Practice Product of Powers Property:

Try:

Try:

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Answers To Practice Problems

Answer:

Answer:

Power of a Power Property

To find a power of a power, you multiply the exponents.

Example:

Therefore,

Practice Using the Power of a Power Property

Try:

Try:

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Answers to Practice Problems

Answer:

Answer:

Power of a Product Property

To find a power of a product, find the power of EACH factor and multiply.

Example:

Practice Power of a Product Property

Try:

Try:

Answers to Practice Problems

Answer:

Answer:

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.

Zero Exponents

Any number, besides zero, to the zero power is 1.

Example:

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 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.

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.

Practice Making Negative Exponents Positive

Try:

Try:

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Answers to Negative Exponents Practice

Answer:

Answer:

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.

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.

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.

Practice Rewriting the Expressions with Positive Exponents:

Try:

Try:

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Answers

Answer

Answer

Division Properties of Exponents

Quotient of Powers Property

Power of a Quotient 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:

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


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

Simplify Expressions

Now use the Quotient of Power Property

Simplify Expressions

Simplify

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

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.

Continued

Simplify powers

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.


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


Now use the Quotient of Powers Property to simplify.

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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Given

Step 1: Power of a Quotient Property

Power of Quotient Property

Result after Step 1:

Step 2: Flip Negative Exponents

Flip Negative Exponents

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

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:

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,

Scientific Notation

If the exponent of 10 is negative, you move the decimal to the left the amount of the exponent.

Example:

Practice Scientific Notation

Write the number in decimal form:

1.

2.

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Answers

1.

2.

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.

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.

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.

Operations with Scientific Notation

For example:

Multiply 2.3 and 1.8 = 4.14

Use the product of powers property

Write in scientific notation

Try These:

Write in scientific notation

1.

2.

Answers

1.

2.

The End

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

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