Polynomials and Polynomial Functions

# Properties of Exponents and Scientific Notation

### Learning Objectives

By the end of this section, you will be able to:

• Simplify expressions using the properties for exponents
• Use the definition of a negative exponent
• Use scientific notation

Before you get started, take this readiness quiz.

1. Simplify:

If you missed this problem, review (Figure).

2. Simplify:

If you missed this problem, review (Figure).

3. Name the decimal

If you missed this problem, review (Figure).

### Simplify Expressions Using the Properties for Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression the exponent m tells us how many times we use the base a as a factor.

Let’s review the vocabulary for expressions with exponents.

Exponential Notation

This is read a to the power.

In the expression the exponent m tells us how many times we use the base a as a factor.

When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

First, we will look at an example that leads to the Product Property.

 What does this mean?

Notice that 5 is the sum of the exponents, 2 and 3. We see is or

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product Property for Exponents

If a is a real number and m and n are integers, then

To multiply with like bases, add the exponents.

Simplify each expression:

 Use the Product Property, Simplify.

 Use the Product Property, Simplify.

 Rewrite, Use the Commutative Property and use the Product Property, Simplify.

 Add the exponents, since bases are the same. Simplify.

Simplify each expression:

Simplify each expression:

Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.

 Consider and What do they mean? Use the Equivalent Fractions Property. Simplify.

Notice, in each case the bases were the same and we subtracted exponents. We see is or . We see is or When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator–notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, and so we need a 1 in the numerator. . This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If a is a real number, and m and n are integers, then

Simplify each expression:

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

 Since there are more factors of in the numerator. Use Quotient Property, Simplify.

 Since there are more factors of in the numerator. Use Quotient Property, Simplify.

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

 Since there are more factors of in the denominator. Use Quotient Property, Simplify.

 Since there are more factors of in the denominator. Use Quotient Property, Simplify. Simplify.

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.

Simplify each expression:

Simplify each expression:

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like We know for any since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify when and when by subtracting exponents. What if We will simplify in two ways to lead us to the definition of the Zero Exponent Property. In general, for

We see simplifies to and to 1. So Any non-zero base raised to the power of zero equals 1.

Zero Exponent Property

If a is a non-zero number, then

If a is a non-zero number, then a to the power of zero equals 1.

Any non-zero number raised to the zero power is 1.

In this text, we assume any variable that we raise to the zero power is not zero.

Simplify each expression:

The definition says any non-zero number raised to the zero power is 1.

To simplify the expression n raised to the zero power we just use the definition of the zero exponent. The result is 1.

Simplify each expression:

1 1

Simplify each expression:

1 1

### Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents has two forms depending on whether the exponent is larger in the numerator or the denominator. What if we just subtract exponents regardless of which is larger?

Let’s consider We subtract the exponent in the denominator from the exponent in the numerator. We see is or

We can also simplify by dividing out common factors:

This implies that and it leads us to the definition of a negative exponent. If n is an integer and then

Let’s now look at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

This implies and is another form of the definition of Properties of Negative Exponents.

Properties of Negative Exponents

If n is an integer and then or

The negative exponent tells us we can rewrite the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.

For example, if after simplifying an expression we end up with the expression we will take one more step and write The answer is considered to be in simplest form when it has only positive exponents.

Simplify each expression:

Simplify each expression:

Simplify each expression:

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Property.

Quotient to a Negative Power Property

If a and b are real numbers, and n is an integer, then

Simplify each expression:

Simplify each expression:

Simplify each expression:

Now that we have negative exponents, we will use the Product Property with expressions that have negative exponents.

Simplify each expression:

Simplify each expression:

Simplify each expression:

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

 How many factors altogether? So we have

Notice the 6 is the product of the exponents, 2 and 3. We see that is or

We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property for Exponents

If a is a real number and m and n are integers, then

To raise a power to a power, multiply the exponents.

Simplify each expression:

 Use the Power Property, Simplify.

 Use the Power Property. Simplify.

Simplify each expression:

Simplify each expression:

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

Notice that each factor was raised to the power and is

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

Product to a Power Property for Exponents

If a and b are real numbers and m is a whole number, then

To raise a product to a power, raise each factor to that power.

Simplify each expression:

 Use Power of a Product Property, Simplify.

Simplify each expression:

1

Simplify each expression:

1

Now we will look at an example that will lead us to the Quotient to a Power Property.

Notice that the exponent applies to both the numerator and the denominator.

We see that is

This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property for Exponents

If and are real numbers, and is an integer, then

To raise a fraction to a power, raise the numerator and denominator to that power.

Simplify each expression:

 Use Quotient to a Power Property, Simplify.

 Raise the numerator and denominator to the power. Use the definition of negative exponent. Multiply.

Simplify each expression:

Simplify each expression:

We now have several properties for exponents. Let’s summarize them and then we’ll do some more examples that use more than one of the properties.

Summary of Exponent Properties

If a and b are real numbers, and m and n are integers, then

Property Description
Product Property
Power Property
Product to a Power
Quotient Property
Zero Exponent Property
Quotient to a Power Property
Properties of Negative Exponents and
Quotient to a Negative Exponent

Simplify each expression by applying several properties:

Simplify each expression:

Simplify each expression:

### Use Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.

Using place value, we can rewrite the numbers 4,000 and 0.004. We know that 4,000 means and 0.004 means

If we write the 1,000 as a power of ten in exponential form, we can rewrite these numbers in this way:

 4,000 0.004

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than ten, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

Scientific Notation

A number is expressed in scientific notation when it is of the form

It is customary in scientific notation to use as the multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

The power of 10 is positive when the number is larger than 1:

The power of 10 is negative when the number is between 0 and 1:

To convert a decimal to scientific notation.
1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, n, that the decimal point was moved.
3. Write the number as a product with a power of 10. If the original number is.
• greater than 1, the power of 10 will be
• between 0 and 1, the power of 10 will be
4. Check.

Write in scientific notation: 37,000

 The original number, 37,000, is greater than 1 so we will have a positive power of 10. 37,000 Move the decimal point to get 3.7, a number between 1 and 10. Count the number of decimal places the point was moved. Write as a product with a power of 10. Check:

 The original number, 0.0052, is between 0 and 1 so we will have a negative power of 10. 0.0052 Move the decimal point to get 5.2, a number between 1 and 10. Count the number of decimal places the point was moved. Write as a product with a power of 10.

Write in scientific notation: 96,000 0.0078.

Write in scientific notation: 48,300 0.0129.

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

Convert scientific notation to decimal form.
1. Determine the exponent, n, on the factor 10.
2. Move the decimal n places, adding zeros if needed.
• If the exponent is positive, move the decimal point n places to the right.
• If the exponent is negative, move the decimal point places to the left.
3. Check.

Convert to decimal form:

 Determine the exponent, n, on the factor 10. The exponent is 3. Since the exponent is positive, move the decimal point 3 places to the right. Add zeros as needed for placeholders.

 Determine the exponent, n, on the factor 10. The exponent is Since the exponent is negative, move the decimal point 2 places to the left. Add zeros as needed for placeholders.

Convert to decimal form:

1,300

Convert to decimal form:

0.075

When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

Multiply or divide as indicated. Write answers in decimal form:

Multiply or divide as indicated. Write answers in decimal form:

20,000

Multiply or divide as indicated. Write answers in decimal form:

;

400,000

Access these online resources for additional instruction and practice with using multiplication properties of exponents.

### Key Concepts

• Exponential Notation

This is read a to the power.

In the expression , the exponent m tells us how many times we use the base a as a factor.

• Product Property for Exponents

If a is a real number and m and n are integers, then

To multiply with like bases, add the exponents.

• Quotient Property for Exponents

If is a real number, and m and n are integers, then

• Zero Exponent
• If a is a non-zero number, then
• If a is a non-zero number, then a to the power of zero equals 1.
• Any non-zero number raised to the zero power is 1.
• Negative Exponent
• If n is an integer and then or
• Quotient to a Negative Exponent Property

If are real numbers, and is an integer, then

• Power Property for Exponents

If is a real number and are integers, then

To raise a power to a power, multiply the exponents.

• Product to a Power Property for Exponents

If a and b are real numbers and m is a whole number, then

To raise a product to a power, raise each factor to that power.

• Quotient to a Power Property for Exponents

If and are real numbers, and is an integer, then

To raise a fraction to a power, raise the numerator and denominator to that power.

• Summary of Exponent Properties

If a and b are real numbers, and m and n are integers, then

Property Description
Product Property
Power Property
Product to a Power
Quotient Property
Zero Exponent Property
Quotient to a Power Property:
Properties of Negative Exponents and
Quotient to a Negative Exponent
• Scientific Notation

A number is expressed in scientific notation when it is of the form

• How to convert a decimal to scientific notation.
1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, that the decimal point was moved.
3. Write the number as a product with a power of 10. If the original number is.
• greater than 1, the power of 10 will be
• between 0 and 1, the power of 10 will be
4. Check.
• How to convert scientific notation to decimal form.
1. Determine the exponent, on the factor 10.
2. Move the decimal places, adding zeros if needed.
• If the exponent is positive, move the decimal point places to the right.
• If the exponent is negative, move the decimal point places to the left.
3. Check.

#### Practice Makes Perfect

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

1 1

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

In the following exercises, simplify each expression using the Product Property.

1

In the following exercises, simplify each expression using the Power Property.

In the following exercises, simplify each expression using the Product to a Power Property.

1

1

In the following exercises, simplify each expression using the Quotient to a Power Property.

In the following exercises, simplify each expression by applying several properties.

#### Mixed Practice

In the following exercises, simplify each expression.

Use Scientific Notation

In the following exercises, write each number in scientific notation.

57,000 0.026

340,000 0.041

8,750,000 0.00000871

1,290,000 0.00000103

In the following exercises, convert each number to decimal form.

0.038

16,000,000,000

0.00000843

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

0.02 500,000,000

0.0000056 20,000,000

#### Writing Exercises

Use the Product Property for Exponents to explain why

Jennifer thinks the quotient simplifies to What is wrong with her reasoning?

Explain why but

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After reviewing this checklist, what will you do to become confident for all goals?

### Glossary

Product Property
According to the Product Property, a to the m times a to the n equals a to the m plus n.
Power Property
According to the Power Property, a to the m to the n equals a to the m times n.
Product to a Power
According to the Product to a Power Property, a times b in parentheses to the m equals a to the m times b to the m.
Quotient Property
According to the Quotient Property, a to the m divided by a to the n equals a to the m minus n as long as a is not zero.
Zero Exponent Property
According to the Zero Exponent Property, a to the zero is 1 as long as a is not zero.
Quotient to a Power Property
According to the Quotient to a Power Property, a divided by b in parentheses to the power of m is equal to a to the m divided by b to the m as long as b is not zero.
Properties of Negative Exponents
According to the Properties of Negative Exponents, a to the negative n equals 1 divided by a to the n and 1 divided by a to the negative n equals a to the n.
Quotient to a Negative Exponent
Raising a quotient to a negative exponent occurs when a divided by b in parentheses to the power of negative n equals b divided by a in parentheses to the power of n.