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Please close your laptops and turn off and put away your cell phones, and get out your note-taking materials. Section 5.5A Negative Exponents When might you have to deal with a negative exponent? Example: Use the quotient rule to simplify (Assume that x ≠ 0.) 4 x 46 2 Solution: x x 6 x But what does x -2 mean? x x x x x 1 1 2 6 x x x x x x x x x x 4 So: x 2 1 2 x In order to extend the quotient rule to cases where the difference of the exponents would give us a negative number we define negative exponents as follows: If a 0, and n is an integer, then a n 1 n a Example Simplify by writing each of the following expressions with positive exponents. (Calculate out any number terms.) 1 1 2 2 1) 3 3 9 1 7 2) x 7 x 3) 2x 4 2 4 x Remember that without parentheses, only the x is the base for the exponent –4, not the entire expression 2x. Note that in the previous problem, 2x-4 gave us 2 on top and x4 on the bottom, and only the x term, not the 2, was raised to the -4th power. Here’s a question for you: What would (2x)-4 look like when simplified? Would it look different than 2x-4? (Good question for a quiz!) Watch out for negatives that are NOT in the exponent! For example: What’s the difference between 3-1 and -3? More examples with negative signs in exponents and other negative signs that are not in the exponent: Simplify by writing each of the following expressions with positive exponents or calculating: 1) 2) 3) 1 x 3 x 3 3 2 (3) 1 1 2 3 9 2 1 1 2 9 (3) Notice the difference in results when parentheses enclose the -3 Problem from today’s homework: Solution: 1/6 + 1/25 = 31/150 Example Simplify by writing each of the following expressions with positive exponents. 1) 2) 1 x3 1 3 x 3 1 x 1 x3 1 (Note that to convert a power with a negative 2 4 2 x y exponent to one with a positive exponent, you x 2 simply switch the power from a numerator to 4 1 y x a denominator, or vice versa, and switch the exponent to its positive value.) 4 y Problem from today’s homework: Answer: 5 x9z8 . Example from today’s homework: Answer: 7 . 9xy13 Example from today’s homework: (do this in your notebook) Answer: 16 . x12y40 Summary of exponent rules If m and n are integers and a and b are real numbers, then: Product Rule for exponents am • an = am+n Power Rule for exponents (am)n = amn Power of a Product (ab)n = an • bn n an a Power of a Quotient n , b 0 b b am mn Quotient Rule for exponents a , a0 n a Zero exponent a0 = 1, a 0 1 n Negative exponent a n , a 0 a Example Simplify the following expression, using only positive exponents in your answer. Note: Problems like this are much easier to solve if you start by simplifying the part inside the parentheses by combining the exponents of identical bases using the quotient rule, and then apply the power rule using the exponent outside of the parentheses . 3 ab 4 7 3 3 a b 2 3 2 = (3-2- -4 a3-7 b1- - 3)-2 = (32 a -4 b 4)-2 = 32*-2 a-4*-2 b4*-2 = 3-4 a8 b-8 = a8 34b8 = a8 81b8 REMINDER: The assignment on today’s material (HW 5.5A) is due at the start of the next class session. Homework Questions? Use the Open Lab! Please remember to sign in! You may now OPEN your LAPTOPS and begin working on the homework assignment. We expect all students to stay in the classroom to work on your homework till the end of the 55minute class period. If you have already finished the homework assignment for today’s section, you should work ahead on the next one or work on the next practice quiz/test.