I do slope intercept form With my shades on Martin-Gay, Developmental Mathematics

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1 AAT-A Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #17-45 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped HW Weeks 11-1 Announcements: Tutoring: Tues. nd Thurs. -4 I do slope intercept form With my shdes on Mrtin-Gy, Developmentl Mthemtics 1

2 Squre Roots Opposite of squring number is tking the squre root of number. A number b is squre root of number if b =. In order to find squre root of, you need # tht, when squred, equls. Mrtin-Gy, Developmentl Mthemtics

3 Principl Squre Roots The principl (positive) squre root is noted s The negtive squre root is noted s Mrtin-Gy, Developmentl Mthemtics

4 Rdicnds Rdicl expression is n expression contining rdicl sign. Rdicnd is the expression under rdicl sign. Note tht if the rdicnd of squre root is negtive number, the rdicl is NOT rel number. Mrtin-Gy, Developmentl Mthemtics 4

5 Finding the n th root of number Finding the squre root of number involves finding number tht, when squred, equls the given number. b Some vocbulry involved with n th roots: In other words, finding such tht b =. n is the index of the expression. The index tells us wht mount of fctors we should look for in order to simplify quntity. Exmples: If n =, we re looking for some vlue r such tht r = s. If n = 4, we re looking for some vlue r such tht r 4 = s. n s This is clled rdicl symbol. s is clled the rdicnd of the rdicl expression. If the index n is even, then s must be positive. This is becuse there is no vlue of r such tht r = -s. Mrtin-Gy, Developmentl Mthemtics 5

6 nth Roots The nth root of is defined s n b only if If the index, n, is even, the root is NOT rel number when is negtive. If the index is odd, the root will be rel number. b n * Mrtin-Gy, Developmentl Mthemtics 6

7 Rdicnds Exmple Mrtin-Gy, Developmentl Mthemtics 7

8 Perfect Squres Squre roots of perfect squre rdicnds simplify to rtionl numbers (numbers tht cn be written s quotient of integers). Squre roots of numbers tht re not perfect squres (like 7, 10, etc.) re irrtionl numbers. IF REQUESTED, you cn find deciml pproximtion for these irrtionl numbers. Otherwise, leve them in rdicl form. Mrtin-Gy, Developmentl Mthemtics 8

9 Perfect Squre Roots Rdicnds might lso contin vribles nd powers of vribles. To void negtive rdicnds, ssume for this chpter tht if vrible ppers in the rdicnd, it represents positive numbers only. Exmple 10 64x 5 8x Mrtin-Gy, Developmentl Mthemtics 9

10 nth Roots Exmple Simplify the following. 0 5 b 10 5b 64 b 9 4 b Mrtin-Gy, Developmentl Mthemtics 10

11 Cube Roots The cube root of rel number b only Note: is not restricted to non-negtive numbers for cubes. if b Mrtin-Gy, Developmentl Mthemtics 11

12 Cube Roots Exmple 7 6 8x x Mrtin-Gy, Developmentl Mthemtics 1

13 Chpter Sections 15.1 Introduction to Rdicls 15. Simplifying Rdicls 15. Adding nd Subtrcting Rdicls 15.4 Multiplying nd Dividing Rdicls 15.5 Solving Equtions Contining Rdicls 15.6 Rdicl Equtions nd Problem Solving Mrtin-Gy, Developmentl Mthemtics 1

14 15. Simplifying Rdicls

15 Product Rule for Rdicls If nd b re rel numbers, b b b b if b 0 Mrtin-Gy, Developmentl Mthemtics 15

16 Simplifying Rdicls Exmple Simplify the following rdicl expressions No perfect squre fctor, so the rdicl is lredy simplified. Mrtin-Gy, Developmentl Mthemtics 16

17 Simplifying Rdicls Exmple Simplify the following rdicl expressions. 7 x x 6 x x 6 x x x 0 16 x 0 x 16 4 x x Mrtin-Gy, Developmentl Mthemtics 17

18 Quotient Rule for Rdicls If nd re rel numbers, n n b n b n n b n b n n b n if b 0 Mrtin-Gy, Developmentl Mthemtics 18

19 Simplifying Rdicls Exmple Simplify the following rdicl expressions Mrtin-Gy, Developmentl Mthemtics 19

20 15. Adding nd Subtrcting Rdicls

21 Sums nd Differences Rules in the previous section llowed us to split rdicls tht hd rdicnd which ws product or quotient. We cn NOT split sums or differences. b b b b Mrtin-Gy, Developmentl Mthemtics 1

22 Like Rdicls In previous chpters, we ve discussed the concept of like terms. These re terms with the sme vribles rised to the sme powers. They cn be combined through ddition nd subtrction. Similrly, we cn work with the concept of like rdicls to combine rdicls with the sme rdicnd. Like rdicls re rdicls with the sme index nd the sme rdicnd. Like rdicls cn lso be combined with ddition or subtrction by using the distributive property. Mrtin-Gy, Developmentl Mthemtics

23 Adding nd Subtrcting Rdicl Expressions Exmple Cn not simplify 5 Cn not simplify Mrtin-Gy, Developmentl Mthemtics

24 Adding nd Subtrcting Rdicl Expressions Exmple Simplify the following rdicl expression Mrtin-Gy, Developmentl Mthemtics 4

25 Adding nd Subtrcting Rdicl Expressions Exmple Simplify the following rdicl expression Mrtin-Gy, Developmentl Mthemtics 5

26 Adding nd Subtrcting Rdicl Expressions Exmple Simplify the following rdicl expression. Assume tht vribles represent positive rel numbers. 45x x 5x 9x 5x x 5x 9x 5x x 5x x 5x x 5x 9x 5x x 5x x x 5x 9 10x 5x Mrtin-Gy, Developmentl Mthemtics 6

27 15.4 Multiplying nd Dividing Rdicls

28 Multiplying nd Dividing Rdicl Expressions If nd re rel numbers, n n b n n b n b n n b n b if b 0 Mrtin-Gy, Developmentl Mthemtics 8

29 Multiplying nd Dividing Rdicl Expressions Exmple Simplify the following rdicl expressions. y 5x 15xy 7 b b 6 7 b b 6 4 b 4 b Mrtin-Gy, Developmentl Mthemtics 9

30 Rtionlizing the Denomintor Mny times it is helpful to rewrite rdicl quotient with the rdicl confined to ONLY the numertor. If we rewrite the expression so tht there is no rdicl in the denomintor, it is clled rtionlizing the denomintor. This process involves multiplying the quotient by form of 1 tht will eliminte the rdicl in the denomintor. Mrtin-Gy, Developmentl Mthemtics 0

31 Rtionlizing the Denomintor Exmple Rtionlize the denomintor Mrtin-Gy, Developmentl Mthemtics 1

32 Conjugtes Mny rtionl quotients hve sum or difference of terms in denomintor, rther thn single rdicl. In tht cse, we need to multiply by the conjugte of the numertor or denomintor (which ever one we re rtionlizing). The conjugte uses the sme terms, but the opposite opertion (+ or ). Mrtin-Gy, Developmentl Mthemtics

33 Mrtin-Gy, Developmentl Mthemtics Rtionlize the denomintor Rtionlizing the Denomintor Exmple

34 15.5 Solving Equtions Contining Rdicls

35 Extrneous Solutions Power Rule (text only tlks bout squring, but pplies to other powers, s well). If both sides of n eqution re rised to the sme power, solutions of the new eqution contin ll the solutions of the originl eqution, but might lso contin dditionl solutions. A proposed solution of the new eqution tht is NOT solution of the originl eqution is n extrneous solution. Mrtin-Gy, Developmentl Mthemtics 5

36 Solving Rdicl Equtions Exmple Solve the following rdicl eqution. x 1 5 x 1 5 Substitute into the originl eqution. x 1 5 x 4 So the solution is x = true Mrtin-Gy, Developmentl Mthemtics 6

37 Solving Rdicl Equtions Exmple Solve the following rdicl eqution. 5x 5 5x 5 5x 5 x 5 So the solution is. Substitute into the originl eqution Does NOT check, since the left side of the eqution is sking for the principl squre root. Mrtin-Gy, Developmentl Mthemtics 7

38 Solving Rdicl Equtions Steps for Solving Rdicl Equtions 1) Isolte one rdicl on one side of equl sign. ) Rise ech side of the eqution to power equl to the index of the isolted rdicl, nd simplify. (With squre roots, the index is, so squre both sides.) ) If eqution still contins rdicl, repet steps 1 nd. If not, solve eqution. 4) Check proposed solutions in the originl eqution. Mrtin-Gy, Developmentl Mthemtics 8

39 Solving Rdicl Equtions Exmple Solve the following rdicl eqution. x x 1 1 x 1 1 x 11 x Substitute into the originl eqution So the solution is x =. Mrtin-Gy, Developmentl Mthemtics true

40 Solving Rdicl Equtions Exmple Solve the following rdicl eqution. x x 1 8 x 1 8 x x 1 8 x x 1 64 x 4x 0 6x 4x 0 ( x)(1 4x) 1 x or 4 Mrtin-Gy, Developmentl Mthemtics 40

41 Solving Rdicl Equtions Exmple continued Substitute the vlue for x into the originl eqution, to check the solution. ( ) true So the solution is x = Mrtin-Gy, Developmentl Mthemtics flse 8

42 Solving Rdicl Equtions Exmple Solve the following rdicl eqution. y 5 y 4 y 5 y 4 y y 4 y y y 4 y 4 y y Mrtin-Gy, Developmentl Mthemtics 4

43 Solving Rdicl Equtions Exmple continued Substitute the vlue for x into the originl eqution, to check the solution flse So the solution is. 4 4 Mrtin-Gy, Developmentl Mthemtics 4

44 Solving Rdicl Equtions Exmple Solve the following rdicl eqution. x 4 x 4 x 4 x 4 x 4 x 4 x x 4 x 4 x 4 8x 4 x 4 x x 1 4 x 4 x 1 4 x 4 4x (x 4) 48x 64 x 4x 80 0 x 0x 4 0 x 4 or 0 Mrtin-Gy, Developmentl Mthemtics 44

45 Solving Rdicl Equtions Exmple continued Substitute the vlue for x into the originl eqution, to check the solution. ( 4) 4 ( 4) true 4 ( 0) 4 ( 0) true 68 So the solution is x = 4 or 0. Mrtin-Gy, Developmentl Mthemtics 45

46 15.6 Rdicl Equtions nd Problem Solving

47 The Pythgoren Theorem Pythgoren Theorem In right tringle, the sum of the squres of the lengths of the two legs is equl to the squre of the length of the hypotenuse. (leg ) + (leg b) = (hypotenuse) leg hypotenuse leg b Mrtin-Gy, Developmentl Mthemtics 47

48 Using the Pythgoren Theorem Exmple Find the length of the hypotenuse of right tringle when the length of the two legs re inches nd 7 inches. c = + 7 = = 5 c = 5 inches Mrtin-Gy, Developmentl Mthemtics 48

49 The Distnce Formul By using the Pythgoren Theorem, we cn derive formul for finding the distnce between two points with coordintes (x 1,y 1 ) nd (x,y ). d x x y 1 y1 Mrtin-Gy, Developmentl Mthemtics 49

50 The Distnce Formul Exmple Find the distnce between (5, 8) nd (, ). d x x y 1 y1 d d 5 ( ) 8 6 d Mrtin-Gy, Developmentl Mthemtics 50