Page 1. Algebra 1 Unit 5 Working with Exponents Per

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1 Algera 1 Unit 5 Working with Exponents Nae Per Ojective 1: Siplifying Expressions with Exponents (Powers INSIDE parentheses) Exponents are a shortcut. They are a quicker way of writing repeated ultiplication. How to siplify exponential expressions 1) Expand all exponents. Since exponents tell us how any of a certain letter are eing ultiplied to itself, we expand out the prole so we can COUNT how any of each letter there are.. ) Use exponents to "snap ack". Multiply all nuers. Multiply all letters y using the exponent. Wrote the letter ONCE with the correct exponent next to it. ) Anything raised to the power of 0 will siplify to 1 An exponent on a single variale or nuer eans to write out a ultiplication prole with that variale or nuer ultiplied to itself. Write out the variale or nuer as any ties as it says to in the exponent. Note: If the variale has a negative sign in front of it, there is a HIDDEN - 1 that is ultiplied to the power. You do NOT write out the - 1. Write a negative sign as "ties negative 1" Exaples: Expand: Ex 1: x 5 Ex : x 5 Ex : (x ) y 5 Ex : -n 5 Ex 5: -n 6 Ojective 1: Quick Practice Siplify the following y first expanding and then snapping ack: 1) n i n 7 ) x ( x ) 5) ( x )( 5x 7 ) ) n(n )(n ) ) (x )(8x) 6) (x y )(5x y 6 ) Page 1

2 Ojective : Siplifying Exponential expressions: Powers OUTSIDE parentheses Siplify inside the parentheses as uch as possile efore continuing the prole. If there is a power outside parentheses, write out the ENTIRE part in the parentheses ties itself. Write it as any ties as it says to in the exponent outside the parentheses. This is a required step. Exaples: Expand and siplify: Ex 1) (-n) 6 Ex ) -(n) 6 Ex ) (-n) 5 Ex ) -(n) 5 Ex 5) (-5x y)(xy ) Ojective : Quick Practice Siplify the following: 1) ( x 5 ) = x 5 i x 5 = ) ( x y) 5) ( x) (xy ) ) ( x ) 5 ( x ) ) ( x) ( y 5 ) 6) n(n ) 5 n Page

3 Practice: Ojective 1 1. n(n 5 ) 6. n i n 6 i n n i n 6 i n 0. y(y 5 ) 7. (-x )(-x ) 1. ( 7x )(x ). n n 7 8. (a )(8a) 1. ( x y)(xy ). (-7x )(x ) 9. (x y)(xy ) 1. ( x y )(xy 6 ) 5. x(x )(x ) 10. ( 7x )(x ) 15. x 0 Practice: Ojective 1) ( x 5 ) ) ( n) ( n) 7) -(x ) (5x ) ) ( n) ( n ) 5) ( n ) ) ( n ) 5 6) ( x y ) ( y 5 ) Page

4 Ojective : Dividing with Exponents Iportant Fact: Coon letters on different sides of a fraction ar will CANCEL to 1's. (Make sure extra letters stay in the correct place y fraction ar) Nuers will also divide or coon factors will cancel to 1's (Make sure nuers stay in the correct place y fraction ar!) Ex 1) 10x 10 Ex ) 5 5x Ex ) x x Ex ) 6 x Ex 5) 6x Ex 6) x 6 Ojective Quick Practice: Expand each expression and then snap it ack together using exponents a a a a a 6a a a. 1a 18a a a 5 9. xy 6 1y x Page

5 Practice: Dividing with Exponents Page 5

6 Practice (ixed proles) Expand each expression and then snap it ack together using exponents. 1. (a )(-a ) 1a 5. 6a 9. x(x )(x 7 ). (a ) (a )(a )( ). a a 7.. 1a 6a w x 5 7w x Page 6

7 Exponent Bingo Free Space!!! Bingo Proles: 1) 6) 11) ) 7) 1) ) 8) 1) ) 9) 1) 5) 10) 15) Page 7

8 Ojective : Dividing with Exponents and powers OUTSIDE parentheses Siplify inside the parentheses as uch as possile efore continuing the prole. If there is a power outside parentheses, write out the ENTIRE part in the parentheses ties itself. Write it as any ties as it says to in the exponent outside the parentheses. This is a required step. You ay answer the rest of the prole without further expanding if you wish. Ex 1) 1x y 5 x y ex ) x6 y x 5 y Ex ) 5x5 y 15x Ex ) 6ac 0 5 Ojective Quick Practice 1) x y 8x ) 1x 0 y 6 y ) r 5 w r w ) xy 7 xy Page 8

9 Ojective 5: Negative Exponents Iportant fact: Negative Exponents are "position upers" of the entire power relative to fraction ar. To siplify expressions with negative exponents: 1) Make sure there is a fraction ar. (if there is no fraction ar, create one under the prole ) Move all powers with negative exponents to their correct position relative to the fraction ) FULLY siplify expressions INSIDE the ( ) if possile. ) ALWAYS EXPAND if there is a power outside the parentheses. Rewrite the following without negative exponents Ex 1: 5a - Ex ) -x - Ex ) Ojective 5: Quick Practice 1) ) p 8 p ) ) ( a ) ( a) 5) ( x 1 y) 0 w 1 y 6) x x 5 7) x y 0 x 8) n 8 1 n 0 Page 9

10 Practicing all types Hidden Message Card 1 s Letter Copy and solve elow Card 5 s Letter Copy and solve elow Card s Letter Copy and solve elow Card 6 s Letter Copy and solve elow Card s Letter Copy and solve elow Card 7 s Letter Copy and solve elow Card s Letter Copy and solve elow Card 8 s Letter Copy and solve elow Page 10

11 Card 9 s Letter Copy and solve elow Card 1 s Letter Copy and solve elow Card 10 s Letter Copy and solve elow Card 1 s Letter Copy and solve elow Card 11 s Letter Copy and solve elow Card 15 s Letter Copy and solve elow Card 1 s Letter Copy and solve elow Card 16 s Letter Hidden Message: Hidden Message Page 11

12 More Practice Proles Set #1: Siplify the following copletely. All final expressions should only have positive exponents. 1) (x)( xy ) ) ( x) ( x) 7) ( xy ) 8x 5 y ) x(xy)( x5 y ) 5) (x)( xy ) 8) ( 5x 0 y ) ) xy 7 6x y 5 6) 9x 18y 9) 1x y 6x 7 y 5 Set #: Expand each expression and then snap it ack together using exponents x y x 9. 10a 5a x x y a 5a. 8 p p 7. 5x x y a 5a a 5a Page 1

13 Ojective 6: Scientific Notation Scientific notation is a forat used to write very large and very sall nuers. It saves space and akes coparing nuers easier. It is often used in science for long distances or sall sizes (atos) There are parts to a nuer in scientific notation. Part 1: A nuer in scientific notation starts with a nuer etween 1 and 10 (can e 1, can t e 10). Part : A ultiplication syol is next. Usually is used ut soeties or. Part : The nuer 10 to soe power is last. The power can e any integer ( -,-1, 0, 1,, ) n Forat: c 10 Notice the old ultiplication syol is used here. This is very coon although n n n again, a dot or asterisk ay also e used. c 10 or c 10 or c 10 Nuers in scientific notation Nuers NOT in scientific notation (why NOT on the lank) n NOTE: Calculators (not people) often write nuers in scientific notation like this: ce with a lot of decial nuers in c. 5.E eans Put in your calculator and press ENTER. POSITIVE POWERS OF 10 If the exponent on the 10 is positive, the nuer represented in scientific notation is very LARGE. How do I know this? The first nuer is ultiplied over and over y 10, oving the decial point one place to the RIGHT for every exponent value on Exaple 1: Change.5 10 fro scientific notation to decial for. This is a large nuer. Answer: The nuer.5 will e ultiplied y 10 five ties, ecoing 5,000,000,000. Rewrite the following nuers fro scientific notation to decial (standard) for If a nuer is very large, it can e put in scientific notation, Exaple : Put 5,19 into scientific notation Answer: The exponent on 10 will e positive since 5,19 is a large nuer. Find the spot where the decial point should e so the first nuer will e etween 1 and 10. The spot is etween 5 and. Rewrite Notice the exponent is positive to indicate a large nuer. Put the following nuers into scientific notation. 1) 6.1 ),56. ) 718,569 ) 9. Page 1

14 Scientific Notation and NEGATIVE powers of 10 If the exponent on the 10 is negative, the nuer represented in scientific notation is very SMALL. How do I know this? The first nuer is divided over and over y 10, oving the decial point one place to the LEFT for every exponent value on Exaple 1: Change.6 10 fro scientific notation to decial for. This is a sall nuer. Answer: The nuer.6 will e divided y 10 seven ties, ecoing Rewrite the following nuers fro scientific notation to decial (standard) for If a nuer is very sall, it can e put in scientific notation, Exaple : Put into scientific notation. Answer: The exponent on 10 will e negative since is a sall nuer. Find the spot where the decial point should e so the first nuer will e etween 1 and 10. The spot is etween 8 and 5. Rewrite Notice the exponent is negative to indicate a large nuer. Put the following nuers into scientific notation. 1) ).5 ) ) Page 1

15 Practice Ojective 6 Express each nuer in standard for Express each nuer in scientific notation. 9. 5,100, ,00,000, ,070,000,000, , ,050,000, ,000,000 Page 15

16 Ojective 7: Applying Exponents: Exponential Growth and Decay y = the final aount Exponential Growth a = the initial aount r = rate of growth/decay as a Money in anks and populations often increase decial y a percentage rate over tie. If anything grows at a t = tie passed percentage rate, r%, then for every tie period that passes, the ite is worth (100% + r%) of its previous value. The exponent stands for tie that passes. Exponential Growth: y = a(1 + r) t Exaple 1: $1500 is deposited in a savings account paying an annual yield of 6%. ) Write an exponential growth equation. c) If the account is left alone, how uch oney will e in the account at the end of 10 years? Exaple ) Jaie invests $100 at an annual yield of 5%. ) Write an exponential growth equation. c) If the account is left alone, how uch oney will e in the account at the end of 10 years? Exaple : A hoe is worth $15,000. It is appreciating at a rate of % per year. ) Write an exponential growth equation. c) How uch oney will the house e worth in 5 years? Page 16

17 y = the final aount Exponential Decay and Depreciation a = the initial aount r = rate of growth/decay Autooiles and other anufactured goods t = often tie decrease passed in value over tie. This decrease is called depreciation. If anything decreases at r%, then each tie period the ite is worth (100%-r%) of its previous value. Exponential Decay: y = a(1 r) t Exaple 1 In 010, a new Fireird Trans A cost $15,798. Suppose the car depreciates 1% each year. ) Write an exponential decay equation. c) How uch oney will the car e worth in 5 years? d) How uch oney will the car e worth in 10 years? Exaple : Suppose a oat purchased for $8,000 depreciations y 10% each year. ) Write an exponential decay equation. c) How uch oney will the oat e worth in years? d) How uch oney will the car e worth in 5 years? Page 17

18 y = the final aount a = the initial aount r = rate of growth/decay t = tie passed Practice Growth and Decay You ay use a calculator to solve the proles elow. Write directly on this page. Exponential Decay y = a(1 r) t Exponential Growth y = a(1 + r) t 1) There were 0 sports radio stations in the year 000. The nuer of radio stations has since increased y approxiately 1.% per year. ) Write an exponential equation. c) What will e the value of t in the year 015? d) If the trend continues, how any sports radio stations will there e in 015? ) Marlene invested $500 in a savings account with an annual interest rate of.5%. ) Write an exponential equation. c) How uch oney will Marlene have after 0 years? ) A new car costs $18,000. It is expected to depreciate (decrease in value) at an average rate of 1% per year. ) Write an exponential equation. c) Find the value of the car in 8 years. Page 18

19 5) The current value of Connie's car is $500. The value of her car has een depreciating % per year. ) Write an exponential equation. c) She hopes to keep the car for ore years. What will it e worth then? 6) In 000, the population of Anytown was 50,000. It has een declining at a rate of.% each year. ) Write an exponential equation. c) What will e its population in 00? 7) In 000, Betterville had a population of 175,000 and was growing at a rate of f 5.6% each year. ) Write an exponential equation. c) What will e its population in 00? 8) In 005, Sally's car was worth $00 and has een appreciating 1% each year. ) Write an exponential equation. c) What will e its population in 010? Page 19

20 Exponent rules quick review: Siplify the following copletely 1. (10a )(a ) 5. (-xy)(x y 9. xy x y. (10a ) 6. (-xy) 10. xy x y. 10a a 7. xy x y. 10a a 8. xy x y Page 0