If m and n are positive integers and a and b are real numbers (b0 when it is a denominator), then: Let’s review some properties of exponents where the exponents were always positive integers. Examples: Properties: In this example, the expression cannot be simplified because the bases are different. Next Slide If the bases are the same, subtract the smaller exponent from the larger exponent. Keep the variable where the exponent is larger.

We will now introduce how to work with negative exponents We will now introduce how to work with negative exponents. Before we give a definition, let’s experiment with our calculators. Directions: Type 2 then the exponent button. The exponent button will either be xy or . Then type in –3. Use the (+-) button to make the 3 negative. You should see 0.125 on the calculator. Most calculators can convert from decimal to fraction. Some calculators, you can just press the fraction button then enter. On others, the DF button will convert the decimal to a fraction. Notice the result of a positive number with a negative exponent. The result is not a negative number. You may notice, the result is the base to the positive exponent under one. Definition:

Definition: If b is any nonzero real number, b0=1. Example: 70=1 Try using a scientific calculator to verify this is true. If you don’t have one, then get one. I’m sure your instructor would be more than happy to recommend an appropriate calculator. More Examples: , This definition is consistent with property 5i) from the previous slide. If we subtract the exponents, the exponent equals zero. Example: Next Slide

Example 1. Evaluate the following: Solutions: Your Turn Problem #1 Evaluate the following: Answers:

We’ll now simplify a quotient of two variables with negative exponents We’ll now simplify a quotient of two variables with negative exponents. An expression is considered simplified if it does not contain negative exponents and like variables are combined using properties of exponents. Simplify: (Shortcuts given after example.) 2. Simplify the complex fraction. Shortcut: If the exponent is negative, move it from denominator to numerator or numerator to denominator, then change the exponent to positive. If the exponent is positive, leave it where it is. Next Slide

Example 2. Simplify the following. Use the shortcut: Solutions: Shortcut Procedure: If the exponent is negative, move it from denominator to numerator or numerator to denominator, then change the exponent to positive. If the exponent is positive, leave it where it is. Your Turn Problem #2 Simplify the following: Answers:

Now that we know how to deal with negative integer exponents, we will be able to simplify more types of problems. Please note: There will usually be more than one method of simplifying these expressions. Recommendations will be given on the “easiest” method, however, it certainly will not be the only method. Recall the properties of exponents and the definition of a negative exponent: We can rewrite #5 simply as: Next Slide Then use the definition of a negative exponent if necessary:

Example 3. Simplify the following: Solutions: Best to 1st multiply the exponent outside the parentheses with the exponents inside the parentheses. Then use the definition of a negative exponent. Your Turn Problem #3 Simplify the following: Answers:

Solutions: Example 4. Simplify the following: These examples involve quotients. We have several options. Some prefer to make the exponents positive first. Others may prefer to use the property: Answers: Your Turn Problem #4 Simplify the following:

Example 5. Simplify the following: Solutions: Your Turn Problem #5 Simplify the following: Answers:

Solutions: Example 6. Simplify the following: These examples involve quotients with exponents on the outside of the parentheses. We again have several options. Let’s first use the property Your Turn Problem #6 Simplify the following: Answers:

Example 7. Simplify the following. Solutions: More examples of quotients with exponents on the outside of the parentheses. Let’s first use the property Your Turn Problem #7 Simplify the following: Answers:

Example 8. Simplify the following: Solutions: It is definitely a good idea to apply the exponent outside the parentheses to all of the exponents inside. Be careful, if a variable doesn’t have an exponent written, then the exponent is 1. The exponent on the coefficients is also a 1. It is a good idea to just write the exponent of 1 to avoid mistakes. Your Turn Problem #8 Simplify the following: Answers:

Example 9. Simplify the following. Solutions: Again, there are many methods to simplifying. Just don’t be a rule-breaker and it will work out. Since there are no exponents on the outside of parentheses, reduce the coefficients and make the exponents positive. Then use the appropriate properties. Your Turn Problem #9 Simplify the following: Answers:

Example 10. Simplify the following. Solutions: It is still a good idea to apply the exponent outside the parentheses to all of the exponents inside. If the coefficients can be reduced (b), do so first to make the operations more manageable. Your Turn Problem #10 Simplify the following: Answers:

We’re almost done. We just need to cover addition and subtraction of expressions with negative exponents. Make the definition of a negative exponent to make the exponents positive. Then to combine fractions using the LCD. Example 11. Simplify the following: Solutions: (Do inside parentheses 1st) Your Turn Problem #11 Simplify the following: The End B.R. 5-18-07 Answers: