# Polynomials and Polynomial Functions

## Properties of Exponents and Scientific Notation


### 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.
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.
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.
Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.
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.
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.
In this text, we assume any variable that we raise to the zero power is not zero.

### 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.
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.
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.
Now that we have negative exponents, we will use the Product Property with expressions that have negative exponents.
Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.



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.
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.
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.
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.

### 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:
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.
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: 
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.
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.

### Key Concepts

1. 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.
2. 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.
3. Quotient Property for Exponents
If  is a real number,  and m and n are integers, then

4. Zero Exponent


5. Negative Exponent


6. Quotient to a Negative Exponent Property
If  are real numbers,  and  is an integer, then

7. Power Property for Exponents
If  is a real number and  are integers, then

To raise a power to a power, multiply the exponents.
8. 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.
9. 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.
10. Summary of Exponent Properties
If a and b are real numbers, and m and n are integers, then


11. Scientific Notation
A number is expressed in scientific notation when it is of the form

12. How to convert a decimal to scientific notation.


13. How to convert scientific notation to decimal form.



### Practice Makes Perfect

Simplify Expressions Using the Properties for Exponents
In the following exercises, simplify each expression using the properties for exponents.
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.
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.
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.
In the following exercises, convert each number to decimal form.
In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

### Writing Exercises


### 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?