# Prerequisites

## Exponents and Scientific Notation


### Learning Objective:

1. Plan your weekly academic schedule for the term.

### Objective 1: Plan your weekly academic schedule for the term.

1. 
Most college instructors advocate studying at least 2 hours for each hour in class. With this recommendation in mind, complete the following table showing credit hours enrolled in, the study time required, and total time to be devoted to college work. Assume 2 hours of study time for each hour in class to complete this table, and after your first exam you can fine tune this estimate based on your performance.


Consider spending at least 2 hours of your study time each week at your campus (or virtual) math tutoring center or with a study group, the time will be well spent!


2. 
Another way to optimize your class and study time is to have a plan for efficiency, meaning make every minute count. Below is a list of good practices, check off those you feel you could utilize this term.


3. 
Creating your Semester Calendar- complete the following weekly schedule being sure to labelOptional: also include if you want a more comprehensive view of your time commitments




Term: ________________________________
Name: ________________________________________
Date: ________________________________________
Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per pixel, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number.
Using a calculator, we enter 
  and press ENTER. The calculator displays 1.304596316E13. What does this mean? The “E13” portion of the result represents the exponent 13 of ten, so there are a maximum of approximately 
  bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers.

### Using the Product Rule of Exponents

Consider the product 
       Both terms have the same base, x, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.
The result is that 
      
    
Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.
Now consider an example with real numbers.
We can always check that this is true by simplifying each exponential expression. We find that 
       is 8, 
       is 16, and 
       is 128. The product 
       equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

### Using the Quotient Rule of Exponents

The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as 
       where 
       Consider the example 
       Perform the division by canceling common factors.
Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.
In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.
For the time being, we must be aware of the condition 
       Otherwise, the difference 
       could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.

### Using the Power Rule of Exponents

Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents. Consider the expression 
       The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.
The exponent of the answer is the product of the exponents: 
       In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.
Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.

### Using the Zero Exponent Rule of Exponents

Return to the quotient rule. We made the condition that 
       so that the difference 
       would never be zero or negative. What would happen if 
       In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example.
If we were to simplify the original expression using the quotient rule, we would have
If we equate the two answers, the result is 
       This is true for any nonzero real number, or any variable representing a real number.
The sole exception is the expression 
       This appears later in more advanced courses, but for now, we will consider the value to be undefined.

### Using the Negative Rule of Exponents

Another useful result occurs if we relax the condition that 
       in the quotient rule even further. For example, can we simplify 
       When 
       —that is, where the difference 
       is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal.
Divide one exponential expression by another with a larger exponent. Use our example, 
       
If we were to simplify the original expression using the quotient rule, we would have
Putting the answers together, we have 
       This is true for any nonzero real number, or any variable representing a nonzero real number.
A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.
We have shown that the exponential expression 
       is defined when 
       is a natural number, 0, or the negative of a natural number. That means that 
       is defined for any integer 
       Also, the product and quotient rules and all of the rules we will look at soon hold for any integer 
      
    

### Finding the Power of a Product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider 
       We begin by using the associative and commutative properties of multiplication to regroup the factors.
In other words, 
      
    

### Finding the Power of a Quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.
Let’s rewrite the original problem differently and look at the result.
It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

### Simplifying Exponential Expressions

Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

### Using Scientific Notation

Recall at the beginning of the section that we found the number 
  when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these?
A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n. If you moved the decimal left as in a very large number, 
     is positive. If you moved the decimal right as in a very small number, 
     is negative.
For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.
We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.
Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.
Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

### Converting from Scientific to Standard Notation 

To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal 
     places to the right if 
     is positive or 
     places to the left if 
     is negative and add zeros as needed. Remember, if 
     is positive, the absolute value of the number is greater than 1, and if 
     is negative, the absolute value of the number is less than one.

### Using Scientific Notation in Applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around 
  molecules of water and 1 L of water holds about 
  average drops. Therefore, there are approximately 
  atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!
When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product 
  The answer is not in proper scientific notation because 35 is greater than 10. Consider 35 as 
  That adds a ten to the exponent of the answer.

### Key Equations


### Key Concepts

1. Products of exponential expressions with the same base can be simplified by adding exponents. See .
2. Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See .
3. Powers of exponential expressions with the same base can be simplified by multiplying exponents. See .
4. An expression with exponent zero is defined as 1. See .
5. An expression with a negative exponent is defined as a reciprocal. See  and .
6. The power of a product of factors is the same as the product of the powers of the same factors. See .
7. The power of a quotient of factors is the same as the quotient of the powers of the same factors. See .
8. The rules for exponential expressions can be combined to simplify more complicated expressions. See .
9. Scientific notation uses powers of 10 to simplify very large or very small numbers. See  and .
10. Scientific notation may be used to simplify calculations with very large or very small numbers. See  and .

### Section Exercises


### Verbal


### Numeric

For the following exercises, simplify the given expression. Write answers with positive exponents.
For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.
For the following exercises, express the decimal in scientific notation.
For the following exercises, convert each number in scientific notation to standard notation.

### Algebraic

For the following exercises, simplify the given expression. Write answers with positive exponents.

### Real-World Applications


### Technology

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

### Extensions

For the following exercises, simplify the given expression. Write answers with positive exponents.