# Polynomials

## Integer Exponents and Scientific Notation


### Use the Definition of a Negative Exponent

The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.
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.
The negative exponent tells us to re-write 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 an expression with only positive exponents.
When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.
We must be careful to follow the order of operations. In the next example, parts ⓐ and ⓑ look similar, but we get different results.
When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.
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, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.
Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form,  where  and m and n are integers.
When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, 

### Simplify Expressions with Integer Exponents

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
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 of Exponents.
If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation.
In the next two examples, we’ll use the Power Property and the Product to a Power Property.
To simplify a fraction, we use the Quotient Property.

### 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  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  and  We know that  means  and  means  If we write the  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  and the second factor is a power of  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.
Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.
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  places to get the first factor,  by itself.
1. The power of  is positive when the number is larger than 
2. The power of  is negative when the number is between  and 

### 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.

### Multiply and Divide Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

### Key Concepts

1. Summary of Exponent Properties



2. Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation:


3. Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:


### Practice Makes Perfect

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.
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 or divide and write your answer in decimal form.

### Everyday Math


### Writing Exercises


### Self Check

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



ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?
