1-1 Lesson 1 Exponents We'll begin this lesson with a review of what we know about exponents. If any of this is new, spend what time you need to learn the new material and do all of the practice problems. Once you feel comfortable with what you have learned, move on to the next topic. Even though we'll review this material again in the worksheets, it won't be taught again. So get it down now, for it will be assumed from this point on that you understand these concepts. If this is all pure review, do a few problems in each sub-section, then proceed to the next lesson. Negative Exponents There are two options for placing a number, or variable, when writing a fraction; either put it in the numerator, or in the denominator. Similarly, there are two signs to use when describing a number, positive or negative. When you put these two concepts together you have everything you need to understand negative exponents. The key phrase is: when you change the place for a number or variable, you change the sign at the same time. Another way to state this is: opposite place, opposite sign. Closely observe the following examples, do the practice problems, and compare your work with the solutions. Make the exponent positive, then simplify. Example 1 9- 2 = _1_=_1 9 2 81 Example 2 Move the term with the exponent to the numerator, then simplify. Ex I _ 1 A-4 ampe3 A4 1 3 --=10 =1000 10- 3 Example 4 Practice Problems 1) X-5 = y-4 = 4) 3- 3 = 3) 2) 1 "0 Solutions 1 2-3 = 7 2 = 8) 5) 6) 7) 1) -2 1 1 5 =-=- 52 25 2) X-5 =_1_ 3) y-4 =_1_ 4) -3 1 1 3 =-=- X 5 y4 3 3 27 1 3 7 2 =49 1 6) -=2 =8 7) 8) ---=4= 10,000 2- 3 10 5) Multiplying Numbers with the Same Base If a number is multiplied by another number with the same base, you may add the exponents. The same holds true for variables with exponents. Study the examples and observe this relationship being worked out. Example 1 23.24=(21.21.21)(21.21.21.21)=27 =128 or 2 3 .2 4 =(8)(16)=128 Example 2 4 2 .4 3 = (4 1 A 1)(4 1 .4 1 A 1)=4 5 or 4 2 .4 3 = (16)(64)=1,024 Practice Problems 1) X2 .X 3 .X 4 = y-1.y5. y2 = 4) X 8 .X 3 ·X A = 8) 7 2 X .7 3 .7 X = 4 2 .4 1 .4- 1 = 5- 2 .5 6 ·5° = 2) 3) 5) 6) 7) .-------------------------------- ------------- --------- -- , • •