Polynomials
53 Integer Exponents and Scientific Notation
Learning Objectives
By the end of this section, you will be able to:
 Use the definition of a negative exponent
 Simplify expressions with integer exponents
 Convert from decimal notation to scientific notation
 Convert scientific notation to decimal form
 Multiply and divide using scientific notation
Before you get started, take this readiness quiz.
Use the Definition of a Negative Exponent
We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator.
If is a real number, , and are whole numbers, then
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 can also simplify by dividing out common factors:
This implies that and it leads us to the definition of a negative exponent.
If is an integer and , then .
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: ⓐ ⓑ
ⓐ  
Use the definition of a negative exponent, .  
Simplify.  
ⓑ  
Use the definition of a negative exponent, .  
Simplify. 
Simplify: ⓐ ⓑ
ⓐⓑ
Simplify: ⓐ ⓑ
ⓐⓑ
In (Figure) we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.
Use the definition of a negative exponent, .  
Simplify the complex fraction.  
Multiply. 
This leads to the Property of Negative Exponents.
If is an integer and , then .
Simplify: ⓐ ⓑ
ⓐ  
Use the property of a negative exponent, .  
ⓑ  
Use the property of a negative exponent, .  
Simplify. 
Simplify: ⓐ ⓑ
ⓐⓑ
Simplify: ⓐ ⓑ
ⓐⓑ
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.
Use the definition of a negative exponent, .  
Simplify the denominator.  
Simplify the complex fraction.  
But we know that is .  
This tells us that: 
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.
If are real numbers, and is an integer, then .
Simplify: ⓐ ⓑ
ⓐ  
Use the Quotient to a Negative Exponent Property, .  
Take the reciprocal of the fraction and change the sign of the exponent.  
Simplify.  
ⓑ  
Use the Quotient to a Negative Exponent Property, .  
Take the reciprocal of the fraction and change the sign of the exponent.  
Simplify. 
Simplify: ⓐ ⓑ
ⓐⓑ
Simplify: ⓐ ⓑ
ⓐⓑ
When simplifying an expression with exponents, we must be careful to correctly identify the base.
Simplify: ⓐ ⓑ ⓒ ⓓ
ⓐ Here the exponent applies to the base .  
Take the reciprocal of the base and change the sign of the exponent.  
Simplify.  
ⓑ The expression means “find the opposite of .” Here the exponent applies to the base .  
Rewrite as a product with .  
Take the reciprocal of the base and change the sign of the exponent.  
Simplify.  
ⓒ Here the exponent applies to the base .  
Take the reciprocal of the base and change the sign of the exponent.  
Simplify.  
ⓓ The expression means “find the opposite of .” Here the exponent applies to the base .  
Rewrite as a product with .  
Take the reciprocal of the base and change the sign of the exponent.  
Simplify. 
Simplify: ⓐ ⓑ ⓒ ⓓ
ⓐⓑⓒ 25 ⓓ
Simplify: ⓐ ⓑ , ⓒ ⓓ
ⓐⓑⓒ 49 ⓓ
We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.
Simplify: ⓐ ⓑ
ⓐ Do exponents before multiplication. 

Use .  
Simplify.  
ⓑ  
Simplify inside the parentheses first.  
Use .  
Simplify. 
Simplify: ⓐ ⓑ
ⓐⓑ
Simplify: ⓐ ⓑ
ⓐ 2 ⓑ
When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are nonzero.
Simplify: ⓐ ⓑ
 ⓐ
 ⓑ
Simplify: ⓐ ⓑ
ⓐⓑ
Simplify: ⓐ ⓑ
ⓐⓑ
When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example.
Simplify: ⓐ ⓑ ⓒ
 ⓐ
 ⓑ
 ⓒ
Simplify: ⓐ ⓑ ⓒ
ⓐⓑⓒ
Simplify: ⓐ ⓑ ⓒ
ⓐⓑⓒ
With negative exponents, the Quotient Rule needs only one form , for . When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.
Simplify Expressions with Integer Exponents
All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
If are real numbers, and are integers, then
Simplify: ⓐ ⓑ ⓒ
 ⓐ
 ⓑ
 ⓒ
Simplify: ⓐ ⓑ ⓒ
ⓐⓑⓒ
Simplify: ⓐ ⓑ ⓒ
ⓐⓑⓒ
In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.
Simplify:
Simplify:
Simplify:
If the monomials have numerical coefficients, we multiply the coefficients, just like we did earlier.
Simplify:
Simplify:
Simplify:
In the next two examples, we’ll use the Power Property and the Product to a Power Property.
Simplify:
Simplify:
Simplify:
Simplify:
Simplify:
Simplify:
To simplify a fraction, we use the Quotient Property and subtract the exponents.
Simplify:
Simplify:
Simplify:
Convert from Decimal Notation to Scientific Notation
Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and . We know that 4,000 means and 0.004 means .
If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:
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 10, and the second factor is a power of 10 written in exponential form, it is said to be in 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.
Write in scientific notation: 37,000.
Write in scientific notation:
Write in scientific notation:
 Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
 Count the number of decimal places, n, that the decimal point was moved.
 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 10^{n}.
 between 0 and 1, the power of 10 will be 10^{−n}.
 Check.
Write in scientific notation:
The original number, , is between 0 and 1 so we will have a negative power of 10.
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.  
Check.  
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Write in scientific notation:
Write in scientific notation:
Convert Scientific Notation to Decimal Form
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 to decimal form:
Convert to decimal form:
1,300
Convert to decimal form:
92,500
The steps are summarized below.
To convert scientific notation to decimal form:
 Determine the exponent, , on the factor 10.
 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.
 Check.
Convert to decimal form:
Determine the exponent, n, on the factor 10.  
Since the exponent is negative, move the decimal point 2 places to the left.  
Add zeros as needed for placeholders. 
Convert to decimal form:
0.00012
Convert to decimal form:
0.075
Multiply and Divide Using Scientific Notation
Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. 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. Write answers in decimal form:
Multiply . Write answers in decimal form.
0.06
Multiply . Write answers in decimal form.
0.009
Divide. Write answers in decimal form:
Divide . Write answers in decimal form.
400,000
Divide . Write answers in decimal form.
20,000
Access these online resources for additional instruction and practice with integer exponents and scientific notation:
Key Concepts
 Property of Negative Exponents
 If is a positive integer and , then
 Quotient to a Negative Exponent
 If are real numbers, and is an integer , then
 To convert a decimal to scientific notation:
 Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
 Count the number of decimal places, , that the decimal point was moved.
 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
 Check.
 To convert scientific notation to decimal form:
 Determine the exponent, , on the factor 10.
 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.
 Check.
Section Exercises
Practice Makes Perfect
Use the Definition of a Negative Exponent
In the following exercises, simplify.
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ⓐⓑ 25
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ⓐⓑ 10000
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ⓐⓑⓒ 49 ⓓ
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ⓐⓑⓒⓓ
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ⓐⓑⓒ
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ⓐⓑⓒ
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
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ⓑ
ⓒ
ⓐ
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ⓒ
ⓐⓑⓒ
ⓐ
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ⓐ
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ⓐ 1 ⓑ ⓒ
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
57,000
340,000
8,750,000
1,290,000
0.026
0.041
0.00000871
0.00000103
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
830
16,000,000,000
0.038
0.0000193
Multiply and Divide Using Scientific Notation
In the following exercises, multiply. Write your answer in decimal form.
0.02
In the following exercises, divide. Write your answer in decimal form.
500,000,000
20,000,000
Everyday Math
The population of the United States on July 4, 2010 was almost 310,000,000. Write the number in scientific notation.
The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation
.
The average width of a human hair is 0.0018 centimeters. Write the number in scientific notation.
The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation.
In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was . Convert this number to decimal form.
At the start of 2012, the US federal budget had a deficit of more than . Convert this number to decimal form.
15,000,000,000,000
The concentration of carbon dioxide in the atmosphere is . Convert this number to decimal form.
The width of a proton is of the width of an atom. Convert this number to decimal form.
0.00001
Health care costs The Centers for Medicare and Medicaid projects that consumers will spend more than ?4 trillion on health care by 2017.
 ⓐ Write 4 trillion in decimal notation.
 ⓑ Write 4 trillion in scientific notation.
Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation.
Distance The distance between Earth and one of the brightest stars in the night star is 33.7 light years. One light year is about 6,000,000,000,000 (6 trillion), miles.
 ⓐ Write the number of miles in one light year in scientific notation.
 ⓑUse scientific notation to find the distance between Earth and the star in miles. Write the answer in scientific notation.
Debt At the end of fiscal year 2015 the gross United States federal government debt was estimated to be approximately ?18,600,000,000,000 (?18.6 trillion), according to the Federal Budget. The population of the United States was approximately 300,000,000 people at the end of fiscal year 2015.
 ⓐ Write the debt in scientific notation.
 ⓑ Write the population in scientific notation.
 ⓒ Find the amount of debt per person by using scientific notation to divide the debt by the population. Write the answer in scientific notation.
ⓐⓑⓒ
Writing Exercises
 ⓐ Explain the meaning of the exponent in the expression .
 ⓑ Explain the meaning of the exponent in the expression .
When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?
answers will vary
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are wellprepared for the next section? Why or why not?
Chapter 6 Review Exercises
Add and Subtract Polynomials
Identify Polynomials, Monomials, Binomials and Trinomials
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.
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ⓓ 10
ⓔ
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ⓐ binomial ⓑ monomial ⓒ trinomial ⓓ trinomial ⓔ other polynomial
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
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 ⓔ 100
ⓐ 3 ⓑ 4 ⓒ 2 ⓓ 4 ⓔ 0
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
Subtract
Find the sum of
Evaluate a Polynomial for a Given Value of the Variable
In the following exercises, evaluate each polynomial for the given value.
Evaluate when:
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Evaluate when:
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ⓐⓑ 10 ⓒ 22
Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial gives the height of a stone seconds after it is dropped from the cliff. Find the height after seconds.
A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial Find the revenue received when dollars.
12,000
Use Multiplication Properties of Exponents
Simplify Expressions with Exponents
In the following exercises, simplify.
17
0.125
Simplify Expressions Using the Product Property for Exponents
In the following exercises, simplify each expression.
Simplify Expressions Using the Power Property for Exponents
In the following exercises, simplify each expression.
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
Multiply Monomials
In the following exercises 8, multiply the monomials.
Multiply Polynomials
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
3
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using: ⓐ the Distributive Property, ⓑ the FOIL method, ⓒ the Vertical Method.
ⓐⓑⓒ
In the following exercises, multiply the binomials. Use any method.
Multiply a Trinomial by a Binomial
In the following exercises, multiply using ⓐ the Distributive Property, ⓑ the Vertical Method.
ⓐⓑ
In the following exercises, multiply. Use either method.
Special Products
Square a Binomial Using the Binomial Squares Pattern
In the following exercises, square each binomial using the Binomial Squares Pattern.
Multiply Conjugates Using the Product of Conjugates Pattern
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
Recognize and Use the Appropriate Special Product Pattern
In the following exercises, find each product.
Divide Monomials
Simplify Expressions Using the Quotient Property for Exponents
In the following exercises, simplify.
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
1
1
1
0
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
1
Divide Monomials
In the following exercises, divide the monomials.
Divide Polynomials
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial.
Divide a Polynomial by a Binomial
In the following exercises, divide each polynomial by the binomial.
Integer Exponents and Scientific Notation
Use the Definition of a Negative Exponent
In the following exercises, simplify.
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
8,500,000
0.00429
The thickness of a dime is about 0.053 inches.
In 2015, the population of the world was about 7,200,000,000 people.
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
Multiply and Divide Using Scientific Notation
In the following exercises, multiply and write your answer in decimal form.
In the following exercises, divide and write your answer in decimal form.
Chapter Practice Test
For the polynomial
ⓐ Is it a monomial, binomial, or trinomial?
ⓑ What is its degree?
In the following exercises, simplify each expression.
Convert 83,000,000 to scientific notation.
Convert to decimal form.
In the following exercises, simplify, and write your answer in decimal form.
74,800
A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial gives the height of the package seconds a after it was dropped. Find the height when seconds.
424 feet
Glossary
 negative exponent
 If is a positive integer and , then .
 scientific notation
 A number is expressed in scientific notation when it is of the form where and is an integer.