7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b

Transcription

1 Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5 b) 4. Zero Expoet Rule: 0 b Examples: 0 7 b) 0 5 c) 0 5. Product Rule: m m b g b b - keep the base ad the expoets Examples: g b) x 7 g x 4. Quotiet Rule: b b m m b - keep the base ad the expoets. Examples: 4 b) x 5 x

2 m 5. Power to a Power Rule b m b - keep the base ad the expoets. Examples: 5 b) x 4 6. Product to a power: ab Note well: a b a b ad a b a b a b - raise both bases to the same power. Examples: y 4 5 b) x y 7. Quotiet to a power: a b a b - raise both bases to the same power. Examples: 5y 4 b) x y

3 ) No paretheses. ) No powers raised to a powers. ) Each based occurs oly oce. 4) No egative expoets. 5) Simplify umerical expressios. Simplifyig Expoetial Expressios Examples: Simplify the expressio. 8 4 x y 4 5 b) x y c) 7xy x y d) 6x y 5 xy e) 5xy 6 8 5xy 4 f) 00x 0x y y 6 4 g) 4x d y g 7 4 h) 5x y 8 4 5

4 Defiitio: A umber is writte i if it is of the form where a 0 ad Z. Examples : Write i stadard form (decimal form) b) c).7 0 d) Example : Write i scietific otatio. 4,970,000,000,000 b) To multiply umbers writte i scietific otatio:. Multiply the decimal parts first. (use your calculator, if ecessary). Usig the rules of expoets, multiply the powers of te. (add the expoets). Be sure your fial aswer is i scietific otatio b) c) d) e) f)

5 To divide umbers writte i scietific otatio:. Divide the decimal parts first. (use your calculator, if ecessary). Usig the rules of expoets, divide the powers of te. (subtract the expoets). Be sure your fial aswer is i scietific otatio Examples: Fid the quotiet, express aswers i scietific otatio b) c) d) e) f)

6 Guided Notes for lesso P. More with Expoets If b ad b 0, ad m, Z the the followig properties hold: 8. Fractioal Expoets: If is eve, 0 b b b If is odd, b ad If is eve, b 0 b b If is odd, b Examples: 64 b) 5 c) 6 4 d) 7 e) 64 m m 9. Fractioal Expoets: b b b m If is eve, b 0 If is odd, b ad b m m b b m If is eve, b 0 If is odd, b Examples: 7 b) 4 c) 5 8 d) 8 4 e)

7 Examples: Simplify the expressio. Express your aswer usig a radical whe your fial aswer yields a fractioal expoet. 4 5x 7x b) x y c) x 6x 7 4 d) 4 x y x y 7 4 e) x x 5 7 f) 4 x x 7

8 Examples: Factor the expressio ad simplify. This is a skill i preparatio for calculus. x x x 4 b) x 4 x 4 8x c) x8 x 8 x 8

9 Guided Notes for lesso P.7 Absolute Value Equatios Absolute Value: Defiitio: a meas the distace the umber a is from o a umber lie. Solvig Absolute Value Equatios: If X is ay algebraic expressio ad c Z, the the solutios to X care foud by solvig the equatios ad. Example Solve the equatio. x 5 b) x 7 c) x 8 4 d) x 5 e) 7 4 x f) 4 5x 7 9

10 Guided Notes for lesso P.9 Iequalities ad Absolute Value Equatios ad Iequalities To solve a iequality meas to fid all the possible values of the variable that makes the iequality true. To solve a iequality you use the same iverse operatio(s) to both sides of the iequality with oe slight glitch. Keep i mid you wat the variable all aloe o oe side of the iequality. Whe a iequality is multiplied or divided by a positive umber the iequality sig remais uchaged. Example to illustrate: 5 6 true 50 5 true 5 6 multiply by 50 5 divide by 5 0 < true < 7 true Whe a iequality is multiplied or divided by a egative umber the iequality sig is reversed. Example to illustrate: 5 6 true 50 5 true 5 6 multiply by 50 5 divide by 5 0 < false true 0 > 7 false 0 7 true Examples: Solve each of the followig ad graph your solutio o a umber lie. x 5 b) x 5 c) 4 6 d) 8 0

11 x e) 0 f) x 08 g) 6x 5 57 h) 4x 40 7x 65 i) x 7 5x j) 5x 9 i) x 8

12 Three ways to display solutios to iequalities: Set builder Notatio (used for Alg ) Iterval Notatio (used i pre-calc o up) Graph (used for either Alg or pre-calc o up) x a x b x a x b x x a x x a x x b x x b xx

13 Coectors of sets of umbers: Itersectio (ad) (what is i both) Set builder Notatio Symbol Iterval Notatio Symbol Uio (or) (what is i either or both) Set builder Notatio Iterval Notatio Graph x x 4 x 8 x x 7 x 9

14 Solvig Absolute Value Iequalities Solutio: If X is ay algebraic expressio ad c Z, the the solutios to X care foud by solvig ad. Examples: Solve the iequality ad graph the solutio set. x 5 4 b) x c) 5x 7 4

15 Solutio: If X is ay algebraic expressio ad c Z, the the solutios to X care foud by solvig ad. Examples: Solve the iequality ad graph the solutio set. x 7 0 b) 8 4 x c) 5x 7 5

16 Guided Notes for lesso P.A Radicals Date: Defiitio: a is called the th root or. a is called the ad is called the. Note Well: If is eve, a 0. If is odd, a. If a, b ad b 0, ad m, Z the the followig properties hold:. Product Rule: ab a b g. Note Well a b a b ad a b a b. Quotiet Rule: a b a b. Power Rule : a 4. Power Rule : a m a m To simplify m a 6

17 Example: Simplify the radical. Express the aswer i simplest radical form. b) 75 c) 700 d) 48 e) 75 f) 686 g) 4 50x h) x y i) x y j) 49x 64y 5 9 k) 54x 7 8y 6 Defiitio: Like radicals are radicals with the same ad Rule: You may oly add or subtract like radicals. 7

18 Example: Simplify the expressio b) c) 8 d) 8x 4 x 4 7x Example: Multiply. Express the aswer i simplest radical form, if ecessary. 8 5 g b) 8 c) 5 5 d) Defiitio: The irratioal umbers a b ad a b are called ad whe multiplied together will always yield a umber. Why? 8

19 e) 0 f) Example: Divide. Express the aswer i simplest radical form, if ecessary b) c) d)

20 Defiitio: To meas to fid a equivalet fractio with a deomiator that is a ratioal umber. Why was/is that importat? Example: Ratioalize the deomiator. 5 b) 0 c) 4 d) e) 0 0 f) x h x x h x 0

21 Guided Notes for lesso 0.5 The Biomial Theorem Defiitio: A is a two termed algebraic expressio. Defiitio: Whe ay biomial is raised to a positive itegral power, the result is called a Illustratio: Expad x y x y x y x y x y x y x x y y x x y xy x y xy y x x y x y y A few thigs you should otice i the expasio of x y ) the x s decrease i power,,, 0 ) the y s icrease i power 0,,, x x x x term by term. y y y y term by term. : ) the expoets o x ad y always add up to for each term. 4) the umber of terms (4) is oe greater tha the expoet. 5) there are coefficiets o the two middle terms Where the coefficiets of a biomial expasio come from? Defiitio: The coefficiet of ay term of biomial expasio is called a biomial coefficiet ad is foud! by r provided r, ad r. r! r! C r is used as well to deote r. A combiatio of thigs take r at a time. Examples: Evaluate by had b) 8 5 c) 7 Example: Evaluate with your calculator 9 9 b) 7 c) 5 0

22 The Biomial Theorem: For ay moomial expressio a ay moomial expressio b, ad r, : a b a b a b a b... a b a b a b Examples: Expad the expressio. (write small) x 4 b) x remember that x x( ) c) x y 5 make sure it s the ad the x that are raised to the expoets.

23 d) x 4y e) 4 x 6y Aother way to aid you with expasio of x y Example: Expad x y 5 Cosider x y x x y xy y is to use Pascal s Triagle x x y xy y usig Pascal s Triagle.

24 Cosider the expasio of x y x x y xy y. If we just wated the secod term of its expasio without expadig it, how could we fid it? It would be x y x y For ay moomial expressio a ay moomial expressio b, ad r, The rth term of a Biomial Expasio without expadig is: r r a b r Examples: Fid the give term of the epasio with out expadig it. x 5 (third term) b) 6 8 x (fourth term) c) x 5 6 (sixth term) d) x 7 9 (secod term) 5 e) 4 x y (third) 4

25 Good luck to: GMP 4: Chapter PA Test: Sectios: 0.5, P., P., ad P.9 Multiple Choice Questios: For -8, circle the best aswer to the questio. Whe you see the saw graphic, SHOW ALL WORK i the space provided. Correct aswers with o work will receive miimal credit. Icorrect aswers with work will receive partial credit. Icorrect aswers with o work will receive o credit. You may use a calculator. Each questio is worth 9 poits. ) What is the solutio of the iequality x 5? x 8 x b) x x 8 c) x x 8or x d) x x or x 8 ) What is the solutio of the iequality x 4? x 7 x b) 7 x x c) 7 x x or x d) 7 x x or x ) Which graph represets the solutio to the iequality x 7? b) c) d) 5

26 4) Which graph represets the solutio to the iequality x 7? b) c) d) 5) Evaluate the biomial coefficiet: b) 5 c) 0,40 d),68,800 6) Evaluate the biomial coefficiet: 4. 4 b) 4 c) d) 7) Expad 5x usig the biomial theorem. c) 5x 8 b) 5x 50x 60x 8 d) 6 5x 50x 6x 8 5x 0x 0x 8

27 8) Expad 4 x y usig the biomial theorem. c) x 8x y 4x y 8x y 6y b) 4 4 x 8x y 4x y xy 6y d) x x y 4x y 6x y 6y x 8x y 4x y x y 6y ) The fifth term i the expasio of 6 x y is 4 40x y b) 4 40x y c) 4 60x y d) 4 60x y 9) The last term i the expasio of x y 4 is 08xy b) 4 8y c) 4 y d) 5 4y 0) The middle term i the expasio of x y 4 is 4x y b) 4xy c) 6x y d) 6x y 7

28 ) Simplify b) 0 5 c) 5 0 d) 0 50 ) Simplify 0g 9 0 b) 0 c) 6 0 d) 40 ) Simplify b) 6 c) d) 6 4 4) Simplify a b c 4 5 5a bc 5a c b) 4 a bc 5ac c) 4 5a bc ac d) 7 5a c ac 8

29 5) Simplify b) 8 c) 4 0 d) 4 4 6) Simplify b) c) d) 5 7) Simplify b) c) 4 d) ) A Space statio rotates i order to simulate gravity, such that N, where N is the 7r umber of rotatios per miute required to simulate earth's gravity, ad r is the radius of the space statio. If the umber of rotatios per miute is 0, which of the followig is a reasoable estimate for the radius of the space statio? 5m b) 5.4m c) 0m d) 44m 9

30 Free Respose Questios: For 9-4. Whe you see the saw graphic, SHOW ALL WORK i the space provided. Correct aswers with o work will receive miimal credit. Icorrect aswers with work will receive partial credit. Icorrect aswers with o work will receive o credit. You may use a calculator. Each questio is worth 7.5 poits. For 9-, simplify the expressio completely with oly positive expoets. 9) x 6 y 6x y 5 0) 0x y z x y z 6 9) 0) 8 9 ) 4 4x y z ) xy 6x y 5 6 ) ) ) Write i stadard form without the use of expoets. ) 4) Write 6,400,000 i scietific otatio. 4) 0

31 ) Evaluate (express aswer i scietific otatio) 5) For 6-8, simplify ad express aswer i radical form, if ecessary. 6) 0x 5x 7) 7x 8x 4 5 6) 7) 8) 4 0 8x y 8)

32 For 9-4, solve the iequality. Graph your solutio o a umber lie. 9) 4 9 0) x 4 54 ).6 y. ) 8 k 4 ) 4 p 9 4) d d

33 For 5-6, solve algebraically for x. 5) x 6 8 6) 5x 0 5) 6) For 7-8, solve algebraically for x. Express your aswer i iterval otatio. 7) x 7 4 8) 7x 9 7) 8)

34 For 9-40, expad the expressio completely. 9) 4 x 5y 9) 40) x y 40) 4

35 4) Express i simplest radical form x y b) x y b) 4) Simplify the expressio b) 6 9 b) c) 5 d) 5 5 c) d) 5

36 Bous) Fid the exact value of: 7 (+5) Bous) 6