MATH 2 part 2

Try this outA. Solve each equation and check your answers.1. _1_ _x_ 6. _2 _11_ 42 z 52. _2 _5_ 7. _6 _4_ 5 b 12 e e3. _9_ _3_ 8. _x _x_ 6 k4 244. _15_ _30_ 9. _3 + _2_ 5 g8 yy5. _7_ _8_ 10. _4 + _2_ 1 h3 c3B. Solve each equation and check your answers.1. _9 5 _1_ a a2. _3m_ + 2 _1_ 5 43. _2d_ 5 d 74. _1_ + _2_ 1 2q5. _b + 1_ _b + 2_ 2 36. _n – 4 _ n+2 37. _3p_ + p 5 2 7

8. _ 9 _ 3 x+19. _ 9 _ 3 x–210. _2a + 3_ _3_ a 2C. Solve each equation and check your answers.1. _2p + 8 _ _10p + 4_ 9 272. _5y – 3 _ _15y – 2 7 283. _2b – 1 + 2 _1_ b24. _d _ _17 + d 2d 255. _n – 2 + _n + 1 _10_ 4 336. _r + 5 _r – 1 _7_ 3 4 47. _4 _ _2__ 3 q q+18. _6 + _ _1__ 3 s s–29. __2_ + _1__ _5_ c–1 c+1 410. __5_ + _10_ 7 z–2 z+2 8

Lesson 2 Solve Problems Involving Rational Equations Rational Equations are equations containing rational algebraic expression. Wecan apply this knowledge in solving problems in real life like problems in work, uniformmotion and fractions.Work Problem:Example 1. The Mathematics Club of Matimyas High School is holding their annual car wash to raise funds for club projects. Eva can wash and wax one car in 3 hours. Ronnie can wash and wax one car in 4 hours. If Eva and Ronnie work together, how long will it take them to wash and wax one car?EXPLORE In one hour Eva can complete 1 of the job. 3In two hours, she can complete 1 (2) or 2 of the job. 33In t hours, she can complete 1 ( t ) or t of the job. 33 Use the following formula to solve the problem: rate of work (r) x time (t) = work done or rt = wPLANIn T hours, Eva can do t of the job. r tw 3 Eva 1 t tIn T hours, Ronnie can do t of the job. 33 4 Ronnie 1 t tTogether they wash and wax one car 44t + t =134 9

SOLVE12  t + t  = 12(1) Multiply each equation by the LCD, 12. 3 4 4t + 3t = 12 7t = 12 t = 12 7 Eva and Ronnie can wash and wax one car in 12 hours or about 1 hour and 43 7minutes.EXAMINE: Eva does  1  12  = 4 hours 3  7  7 Ronnie does  1  12  = 3 hours 4 7  7 Working together: 4 + 3 = 1 77Example 2. One pipe can fill a fire truck water tank in 10 minutes and another can fill it in 20 minutes. If the first pipe is open for 5 minutes before the second is opened, how long will it take them to finish filling the water tank?PLANLet x = time to finish filling the tankPipe 1 R t wPipe 2 1 x+5 1 (x + 5) 10 10 x 1 1 (x) 20 20SOLVE x+5+ x =1 10 202 ( x + 5) + x = 20 3x +10 = 20 10

3x = 10 x = 3 1 minutes or 3 minutes and 20 seconds 3Motion Problems: An object is said to be in uniform motion when it moves without changing itsspeed, or rate. distance = rate x time d = rtExample 3. Jack and his brother Jherz rented a boat to fish in Calawagan Island. The maximum speed of the boat in still water was 3 kilometers per hour. At this rate, a 9 km trip downstream with the current took the same amount of time as a 3 km trip upstream against the current. What was the rate of the current?EXPLORELet c = the rate of the current The rate of the boat when traveling downstream, or with the current, 3 km/hr plusthe rate of the current.3 + c = rate of the boat when traveling downstream The rate of the boat when traveling upstream, or against the current, 3 km/hrminus the rate of the current.3 – c = rate of the boat when traveling upstreamPLANTo represent the time, t, solve d = r (t) for tThus, t = d d r t t 9 3+c __9__ 3+c Downstream 3 3–c __3__ Upstream 3–c 11

SOLVE 9=9 where c ≠ 3 and -33+c 3−c3(3 – c) = 9(3 + c)9 – 3c = 27 + 9c- 12c = -18 c = 3 km/hr is the rate of the current 2 The three examples which follow illustrate three types of problems involvinguniform motion. Each is solved using a chart, a sketch, and the formula:Motion in opposite directionsExample 4. Boy and Jay leave the same point driving in opposite directions. Traffic conditions enable Boy to average 10 km/hr more than Jay. After two hours they are 308 km apart. Find the rate of each. 308 kilometers 2x 2(x + 10) Jay Boy d 2xPLAN 2(x + 10)The problem asks for Boy’s rate and Jay’s rateTo represent the rate r, solve d = r for r. tThus, r = d rt t Jay x 2 Boy x + 10 2SOLVEThe sketch will help you write the equationLet x = Jay’s rate 12

2x + 2(x + 10) = 308 Jay’s rate 2x + 2x + 20 = 308 Boy’s rate 4x = 288 x = 72 km/hr x + 10 = 82 km/hrMotion in the same directionExample 5. Joseph begins bicycling north at 20 km/hr at noon. Luis leaves from the ame place fifteen minutes later to catch him. If Luis bicycles north at 24 km/hr, when will he catch Joseph? Make a sketch of the given facts. Since the rates are given in terms of kilometersper hour, time must be expressed in hours, Fifteen minutes is 1 of an hour. 4Joseph 20xLuis 24(x – ¼)PLANThe problem asks for the time at which Luis will catch Joseph. rt d Joseph 20 x 20x Luis 24 x – 1 24(x – 1 ) 44SOLVEWhen Luis catches Joseph, the distances both have traveled will be equal.Let x = Joseph’s time20x = 24(x – 1 ) the equation to solve 420x = 24x – 6-4x = -6x = 1 1 Luis will catch Joseph 1 1 hours after noon, or at 1:30 P.M. 22 13

Round TripExample 6. Gil walks from his home to the mall at 8 km/hr and gets a ride back home at 40 km/hr. If his total traveling time is half an hour, how long did it take Gil to walk to the mall and how far was it? 8x 40( 0.5 – x )PLANThe problem asks for the time and distance of Gil’s walk To mall rt d 8x 8x Return home 40 0.5 – x 40(0.5 – x)SOLVELet x = time of Gil’s walk in hours.In all round trip problems, the distance going equals the distance returning. Thus, youhave this equation to solve:8x = 40(0.5 – x) It took Gil 25 minutes to walk8x = 20 – 40x48x = 20 x = 5 or 25 minutes 128  5  = 3 1 km about 3.3 km to the mall 12  3 14

Fraction ProblemExample 7. Twice the numerator of a fraction exceeds the denominator by 2. If 1 is added to both the numerator and the denominator, the resulting fraction is equal to 2 . Find the original fraction. 3PLANThe problem asks for the original fraction.SOLVELet n = numerator of the original fraction d = denominator of the original fractionThen n is the original fraction dTwice the numerator of a fraction exceeds the denominator by 2. 2n = d + 2If 1 is added to both the numerator and the denominator, the resulting fraction is equalto 2 . 3 n+1 = 2 d +1 3Simplify the equation: 2n = d + 2 2n – d = 2 Multiply by -2 -4n + 2d = -43(n + 1) = 2(d + 1) 3n – 2d = -1 3n – 2d = -1 - n = -5If n = 5, find d: n =52(5) = d + 2 d=8The original fraction is 5 8 15

Try this outA. Work ProblemsThe formula: rate x time = work done or rt = w is used to solve work problemsWork rates are often expressed in terms of the job to be done. 1. It takes you 2 days to paint your garage and it takes your friend 3 days. How long will it take you both work together? 2. It would take Mr. Go three hours to chop down the tree in front of his house. His son would need 6 hours to do the same job. How much time would it take them if they worked together? 3. Working alone, Betong can cultivate a field in 6 hours. If his brother Jelo helps him, it will only take 3 1 hours to cultivate the field. How long will it take Jelo to 2 do the job alone? 4. One pipe can fill a gasoline storage in 15 hours, while the second pipe can fill the same tank in 10 hours. How long will it take both pipes together to fill the tank? 5. Working alone, Joe can mow a large lawn in 3 hours, and Kevin can mow it in 4 1 hours. Suppose that they work together for one hour and then Kevin leaves. 2 How long will it take Joe to finish the job? 6. Valve A can drain a swimming pool in 3 hours and Valve B can drain it in 4 hours. Find x , the amount of time it will take to drain the pool using both valves. 7. Working alone, Valerie can paint the house in 3 days and Anne can paint it 4 days. Suppose that Valerie works alone for two days and then is joined by Anne. Find the time it will take the two of them to finish painting the house. 8. It takes Chuckie 5 hours to split a cord of wood, and it takes his sister 3 hours. How long will it take them working together? 9. Paula estimates it will take her 4 hours to type a report but only 3 hours for her brother to do it. If they both type at the same time, how long will it take? 10. Benny thinks it will take him 6 hours to replant the window boxes for his apartment, and Becky thinks it will take her 4 hours. How long will it take them working together? 16

B. Motion Problems 1. Jessica and Rosefer leave the same point traveling in opposite directions. Jessica is walking, but Rosefer is biking and averaging 8 km/h more than Jessica. After 2.5 h they are 40 km apart. Find the rate of each. 2. On a cross-country ski trail, exchange students from the Philippines Jaycel and Sophie average 5 km/hr skiing back to the end of the trail, and 3 km/hr skiing back to the beginning. If the total time spent skiing was 4 hours, how much time was spent on the return trip? What was the length of the round trip? 3. On a shooting the rapids trip, Jackie and Joy paddled 10 km downstream on the Pagsanjan Falls in two hours. However, It took them five hours to return upstream to their starting point paddling against the current. Find the paddling rate in still water and the rate of the current. 4. A commuter usually travels to work on the express train which averages 72 km/hr. She discovers that if she takes the local train, which averages 48 km/hr, it will take her fifteen minutes longer to get work. How far does she travel to work? How long does the trip take on the express train? 5. Tom and Lorena went to DOST – SEI, leave a toll booth at Bicutan Exit at the same time. Lorena drives east at 90 km/hr and Tom goes west at 70 km/hr. After how many hours are they 480 km apart? 6. A freight train leaves a railroad station at 7:00 A.M. traveling at 80 km/hr. One hour later an express train leaves the same station on a parallel track, traveling in the same direction at 120 km/hr. When will the express catch the freight? 7. Jose bicycles to school at 20 km/hr. After school finds that his bike has a flat tire and he must walk home. If he walks at 10 km/hr and his total traveling time for the round trip is one hour, how long did it take him to walk home? 8. Two planes leave Centennial Terminal at NAIA at the same time. Plane A is traveling north at 850 km/hr and the other traveling south at 750 km/hr. In how many hours will they be 4,800 apart? 9. Harry and Jun live 36 km apart. At noon each leaves his house and bicycles toward the other. Harry bicycles at 11 km/hr and Jun at 13 km/hr. At what time they will meet? 10. Using the butterfly stroke to swim the length of the pool, Carol averaged 1.5 m/s. Swimming back, she used the breast stroke and averaged 1.2 m/s. The entire swim took a minute and 15 seconds. How long was the pool? 17

Fraction Problems: 1. The numerator of a fraction is three less than the denominator. If 4 is added to the numerator and to the denominator, the resulting fraction is equivalent to 3 . 4 Find the original fraction. 2. The numerator of a fraction is four less than the denominator. If 1 is subtracted from the numerator and from the denominator, the resulting fraction is equivalent to 2 . Find the original fraction. 3 3. The denominator of a fraction is 7 more than the numerator. If 5 is added to the numerator and to the denominator, the resulting fraction is equivalent to 1 . Find 2 the original fraction. 4. The denominator of a fraction is 1 less than twice the numerator. If 1 is subtracted from the denominator, the resulting fraction is 2 . Find the original 3 fraction. 5. If 1 is subtracted from the numerator of a fraction, the resulting fraction is 1 . If 5 2 is added to the numerator of the original fraction, the resulting fraction is 1 . Find 2 the numerator and the denominator of the original fraction.Lesson 3Solve Formulas for a Specified Variables Rational expressions and rational equations often contain more than onevariable. Sometimes it is useful to solve for one of the variables. Then you can find thevalues of any of the variables.Example 1. Solve for h in A = 1 h(a + b) This is the formula for the area of a 2 trapezoid The LCD is 2. A = 1 h (a + b) 2 Multiply each side by 2. 2A = 2 [ 1 h (a + b) ] 2 18

2A = h (a + b)2A = h(a + b) Divide each side by (a + b)a+b a+b 2A = ha+bThe formula below applies to camera and lens systems.Object lens image f ab 1 =1+1 f abIn the formula, f is the focal length of the lens, a is the distance from the object to thelens, and b is the distance from the image to the lens.Example 2. Solve for the formula above for f. 1 =1+1 f ab(abf ) 1  =  1 + 1 (abf ) Multiply each side by the LCD, abf. f a b Factor bf + af ab = bf + af ab = f (b + a) ab = f b+a 19

Try this outSolve each formula for the variable indicated.A. Science1. a = v , for t 4. v = r + at, for a t2. s = vt + 1 at2 , for v 5. f = W ⋅ V 2 , for R 2 gR3. F = G  Mn  , for M  d2 B. Business1. A = p + prt , for p 4. I = 365d _365d_ , for d 360 − dr2. I = 365 (100 − P) , for P PR 5. c = P −100 , for P P3. a = r – 0.25 , for y 2yC. Mathematics 4. S = n (A + t) , for n 1. y = mx + b , for m 2 2. m = y1 − y2 , for y2 x1 − x2 5. A = πr2 , for π 3. P = Q + R , for P DD 20

Lesson 4 Use Formulas that Involve Rational Expressions Electricity can be described as the flow of electrons through a conductor, such ascopper wire. Electricity flows more freely through some conductors than others. Theforce opposing the flow is called resistance. The unit of resistance commonly used is the ohm. You will discuss the conceptcompletely in your science and T.L.E. classes. conductor direction of flow resistance Resistance can occur one after another, that is, in series. Resistances can alsooccur in branches of the conductor going in the same direction, or in parallel.Formulas for the total resistance RT , are given at the diagram below.Series R1 R2 flowParallel RT = R1 + R2 R1 flow R2 1= 1 +1 RT R1 R2 21

Example 1. Assume that R1 = 5 and R2 = 6 ohms. Compute the total resistance of the conductor when the resistance are in parallel.1= 1 +1RT R1 R2 1 =1+1RT 5 61 = 6 + 5 = 11RT 30 30RT = 30 11 The total resistance is 30 or 2.727 ohms 11 A circuit, or path for the flow of electrons, often has some resistances connectedin series and others in parallel.Example 2. A parallel circuit has one branch in series as shown below. R2 R3 flow R1Given that the total resistance is 2.25 ohms, R1 = 3 ohms, and R2 = 4 ohms, find R3 .1= 1 +1RT R1 R2 + R3 1 =1+ 1 1= 12.25 3 4 + R3 9 4 + R3 9 = 4 + R3 1 −1= 12.25 3 4 + R3 R3 = 5 ohms 22

Try this outSolve the problems: 1. Resistance of 3 ohms, 6 ohms, and 9 ohms are connected in a series. What is the total resistance? 2. Eight lights on a Christmas tree are connected in series. Each has a resistance of 12 ohms. What is the total resistance? 3. Three coils with resistance of 3 ohms, 4 ohms, and 6 ohms are connected in parallel. What is the total resistance? 4. Three appliances are connected in parallel: a lamp with a resistance of 60 ohms, an iron with a resistance of 20 ohms, and a heating coil with a resistance of 80 ohms. Find the total resistance. 5. The formula in finding the Celsius temperature is C = 5 (F – 32). (a) Find C if F 9 is 59o. (b) Express the formula in Fahrenheit temperature. 6. Let’s Summarize Rational expression is the quotient of two polynomials with denominator notequal to zero. Rational equations are equations containing rational expressions. An object is said to be in uniform motion when it moves without changing itsspeed, or rate. distance = rate x time d = rt Work rates are often expressed in terms of the job to be done. The formula to solve work problem is rate of work ( r ) x time ( t ) = work done rt = w 23

What have you learnedFind or solve what is asked in the following:1. Solve _3_ + __1__ _1_ . The LCD is ______. x x–5 2x2. The value of x is _____.3. What is the distance traveled in 7 hours at 50 km/hr?a. 200 km b. 250 km c. 300 km d. 350 km4. How long does it take to travel 330 km at an average rate of 66 km/hr?Use this problem to solve # 5, 6 and 7. A fraction is represented by n. Express in terms of n and d the new fractionsobtained by following the given directions.5. Increase both numerator and denominator by 3.6. Decrease both numerator and denominator by 7.7. Interchange the numerator and denominator and then add 5 to both.Use this problem to solve # 8, 9, and 10. A jet can travel 3,900 km from Baguio City to Tarlac in 3 hours with the wind. Thereturn trip of jet against the same wind takes 4 hours and 20 minutes.Let j = rate of the jet in km/hr w = rate of the wind in km/hr8. Write the rate of the jet with the wind9. Write the first equation, the trip from Baguio City to Tarlac.10. Write the second equation, the return trip of the jet. Express 4 hours and 20 minutes in fraction form.11. The rate of the still air is _____.12. State the work rate: Lionel can vacuum his room in 8 minutes 24

Answer key 8. 5 + 5 = 1 x8How much do you know x = 40 which is b 1. k = 12 3 2. 16 9. P + W 3. a = 8 10. P - W 4. b 11. P – (W + 15) or P – W - 15 5. 316 2 km or 316.67 km 12. P + (W - 20) or P + W – 20 13. Q = P − R 3 6. 1 D x 6. z = 10 7. 1 11 8 7. e = 2Try this out 5Lesson 1A. 8. x = 241. x = 1 9. y = 1 2 52. b = 24 10. c = 12 5 6. n = –53. k =12 7. p = 24. g = 4 255. h = 8B.1. a = 22. m = − 25 12

3. d = –7 8. x = 24. q = 4 9. x = 55. b = 1 10. a = –6C. 6. r = –21. p = 5 7. q = − 4 , q = 12. y = 23. b = 2 3 8. s = 4 , s = 3 74. d = 34 3 9. c = − 3 , c = 35. h = 6 5 10. z = − 6 , z = 3 7Lesson 2A. 1. Let x = number of days required to do the job together You r tW Your friend 1 x 1x 2 2 1 x 1x 3 3Part of the job + Part of the job = whole jobYou do friend does1x + 1x =1236  x + x  = 6 (1) 2 3 3x + 2x = 6 5x = 6 x = 6 or 1.2 days working together 5 26

2. Let x = number of hours it took them to do the hob togetherMr. Go rt WSon 1x 1x 3 3 1x 1x 6 6x+ x = 1366  x + x  = 6(1) 3 62x + x = 63x = 6x = 2 hours working together3. r = number for Jelo r tw 1Betong 3.5 1 (3.5) Jelo 6 6 1 3.5 1 (3.5) r r 1  7  + 1  7  = 1 62 r27r + 42 = 12r 42 = 5r r = 8 2 hours or 8 hours and 24 minutes 5Jelo worked for 8 hours and 24 minutes4. Let x = time to finish filling the tankPipe 1 r t wPipe 2 1 x 1x 15 15 x 1x 1 10 10 27

x + x =115 102x + 3x = 130 302x + 3x = 30 5x = 30 x = 6 hours5. Let x = the number of hours it will take Joe to finish mowing the lawnKevin’s work rate : rt w Joe 1 x+1 1 (x + 1) Kevin 1 = 1 = 2 3341 9 9 21 222 9 91 (x + 1) + 2 = 1393( x + 1) + 2 = 9 3x + 5 = 9 3x = 4 x = 1 1 hours or 1 hour and 20 minutes 3It will take Joe 1 hour and 20 minutes to finish the job6. 1 1 or 1.71 hours 77. 4 days 78. 1 7 or 1.875 hours 89. 1 5 or 1.71 hours 710. 2.4 hours or 2 hours and 24 minutes 28

B. Motion Problems1. J.essica”s distance + Rosefer’s distance = Total distance Let x = Jessica’s rate.2.5x + 2.5(x+8) = 402.5x + 2.5x + 20 = 40 5x = 20 x=4Jessica’s rate is 4km/hrRosefer’s rate is 12 km/hr2. distance out = distance back r1 • t1 = r2 • t2 5(4 – t2) = 3 t2 20 – 5t2 = 3 t2 8t2 = 20 t2 = 2.52.5 hr for the return tripround trip equals15 km3. Let r = the paddling rate in still water c = the rate of the current2(r + c) =105(r – c) = 10 r+c=5 or 2r + 2c = 10 10r + 10c = 50 r–c=2 5r – 5c = 10 10r – 10c = 202r = 7 20r = 70 r = 3.5 r = 3.53.5 + c = 5 c = 1.5The rate of paddling in still water is 3.5 km/h.The rate of the current is 1.5 km/h4. Let t = time on the express train d = distance to work 72 t = d 48( t + 1 ) = d 4 48t + 12 = d 29

48t + 12 = 72t 12 = 24t t= 1 2 d = 72  1  = 36 2 She travels 36 km to work The trip takes 1 hour on express train 25. 90t + 70t = 480 160t = 480 t = 3 hours6. 8ot = 120 (t – 1) 80t = 120t – 120 t = 3 hours or 10 A.M.7. 20(1 – t) = 10t 20 – 20t = 10t t = 2 hour or 40 minutes 38. 850t + 750t = 4,800 1,600t = 4,800 t = 3 hours9. 11t + 13t = 36 24t = 36 t = 3 0r 1.5 hours 2 12:00 + 1.5 hours = 1:30 P.M.10. 15t = 1.2(75 sec – t) 15t = 90 – 1.2t 2.7t = 90 t = 33.33 hours or 33 hours and 20 minutes 1.5(33.33) = 50 30

C. Fraction Problems 1. n = numerator d = denominator d – 3 = numerator of original fractionn+4 = 3d+4 4d −3+4 = 3 d+4 44(d + 1) = 3(d + 4)4d + 4 = 3d + 12d=8 denominator8–3=5 numerator5 is the original fraction82. n = numerator d = denominator d – 4 = numerator of original fractionn −1 = 2d −1d − 4−1 = 2 d −1 32(d – 1) = 3(d – 5)2d – 2 = 3d – 15d = 13 denominator13 – 4 = 9 numerator9 is the original fraction133. n = numerator d = denominator 7 + n = denominator of original fractionn+5 = 1d +5 2 n+5 =17+n+5 2 31

n+5 = 1n + 12 22(n + 5) = (n + 12)2n + 10 = n + 12 n=2 numerator7+2=9 denominator2 is the original fraction94. n = numerator d = denominator 2n – 1 = denominator of original fraction n =22n −1−1 3 n =2 2n − 2 3 3n = 2(2n – 2) 3n = 4n – 4 n=4 numerator2(4) – 1 = 7 denominator4 is the original fraction75. n = numerator d = denominatorn −1 = 1 or 2(n – 1) = d d2n+5 = 1 or 3(n + 5) = d d32n – 2 = 3n + 15 n = -17 numerator2(-17) - 2 = -36 denominator 32

Lesson 3 4. v = r + at v−r =aA. Science t 1. t = v a 5. f = WV 2 gR 2. 2s = 2vt + at2 R = WV 2 2s – at2 = 2vt fg 2s − at2 = v 3. 2ya = r – 0.25(2y) 2t 2ya = r – 0.5y 3. Fd2 = G(Mm) 2ya + 0.5y = r Fd 2 = M y(2a + 0.5) = r Gm y= rB. Business 2a + .0.5 1. A = p + prt A = p(1 + rt) p = 1 + rt A 2. I = 365(100 − P) PR PRI = 36,500 – 365P P = 36000 IR + 365 4. I(360 – dr) = 365d 360I – drI = 365d 360I = 365d – drI 360I = d(365 – rI) 3601 = d 365 − rl 5. cP = P – 100 100 = P – cP 100 = P (1 – c) 100 = P 1− c 33

C. Mathematics1. y = mx + b 4. 2S = n(A + t) m = y−b n = 2s x A+t2. m (x1 – x2) = y1 – y2 5. A = πr2 y2 = mx1 – mx2 – y1 π= A r23. P = QD + RLesson 41. RT = 3 + 6 + 9 = 18 ohms 92. 8(12) = 96 ohms 123. 1 = 1 + 1 + 1 = RT 3 4 6 RT = 12 94. 1 = 1 + 1 + 1 = 19 RT 60 20 80 240 RT = 240 or 12.632 ohms 195. a. 15o b. F = 9 C + 32 5What have you learned1. 2x(x - 5) 7. d + 5 n+52. x = 25 7 8. j + w3. d 9. 3(j + w) = 3,9004. 5 hours 10. 4 1 (j – w) = 3,900 or 13( j − w) = 3,9005. n + 3 33 d +3 11. 1,100 km/hr6. n − 7 12. 1 job/minute d −7 8 34

Module 5 Radical ExpressionsWhat this module is about Just as you did in the case of adding and subtracting radical expressions,this module will allow you to multiply and divide them by applying the same basicprocedures in dealing with algebraic expressions. You will constantly be usingproperties of radicals which is in the box for easy reference.Property 1: ab = a · bProperty 2: a= a bbWhat you are expected to learn 1. Recognize basic radical notation 2. apply the basic properties of radicals to obtain an expression in simplest radical form. 3. multiply and divide radical expressions.How much do you know A. Multiply the following expressions.. 1. 4 3 · 3 3 2. 5 7 · 2 7 3. 2 5 · 7 4. 5 2 · 5 1

5. (2 x2b ) (5 b ) B. Divide the following expressions. 1. 2 ÷ 3 2. 3 4 ÷ 3 6 3. 2 ÷ 3 2 4. 2 ÷ (2 + 3 5. xy ÷ ( x - y )What will you do Lesson 1 Multiplication of Radical Expression In multiplying radical, there are three cases to be considered. These are: a. Indices are the same. When multiplying radicals having the same index, apply n x ⋅ n y = n xy and then if necessary, simplify the resulting radicand.b. Indices are different but radicands are the same. To find the product of radicals with different indices, but the same radicand, apply the following steps: 1. transform the radical to fractional exponents. 2. multiply the powers by applying: xm · xn = xm+n (law of exponent) 3. rewrite the product as a single radical. 4. simplify the resulting radicand if necessary.c. Indices and radicands are different. To find the product with different indices and radicands, follow the following steps: 1. transform the radicals to powers with fractional exponents. 2

2. change the fractional exponents into similar fractions.3. rewrite the product as a single radical4. Simplify the resulting radicand if necessary.Multiplying monomial radicalsRules to follow:Rule 1. If radicals to be multiplied have the same indices, follow the stepsin the examples. ___Example 1. Multiply: √2 · √3 · √5Solution: Write the product of two or more radicals as a single expression. _ _ _ ____ √2 · √3 · √5 = √ 2 · 3 · 5 __ = √30 __ __Example 2. Find the product: √12 · √18Solution: There are two approaches to solve. __ __ _____ √12 · √18 = √12·18 by property 1 ___ Look for the largest perfect square = √216 factor of 216, which is 36. __ _ = √36 · √6 _ = 6√6Second approach: First put each radical into simplest form. __ __ _ _ _ _ √12 · √18 = √4 · √3 · √9 · √2 __ = 2√3 · 3√2 Rearrange the factors. __ = 2 · 3√3 √2 _ = 6 √6Note that the second approach used kept numbers much smaller. Thearithmetic was easier when the radical is simplified first. _ __Example 3. Find the product: √7 · √ 14Solution: _ __ ____ √7 · √14 = √ 7·14 3

__ = √98 express the radicand as product of the largest perfect square factor. __ _ = √ 49 · √2 _ = 7√2 __Example 4. Multiply: a√3 · b√6 Solution: __ ___ a√3 · b√6 = ab√3 · 6 simply multiply the radicand having the same index. __ = ab√18 express the radicand as product of the largest square factor __ = ab√9 · √2 _ = 3ab√2 ___ ____Example 5. Get the product: √2ab3 · √12ab Solution: ____ ____ ___________ √2ab3 · √12ab = √ (2ab3) · (12ab) applying the lawof exponent _____ = √24a2b4 expressing the radicand as the largest square factors _ __ _ = √4 · √6 √a2 √b4 _ = 2ab2 √6Rule 2. If the radicals have different indices but same radicands, transform the radicals to powers with fractional exponents, multiply the powers by applying the multiplication law in exponents and then rewrite the product as single radical. __Example 6. √5 · 4√ 5 __ Solution: √5 · 4√5 = 51/2 · 51/4 = 5½+¼ 4

= 53/4 __ ___ = 4√53 or 4√125 ____ ____Example 7. (4√2x – 1) ( 3√2x – 1 Solution: _____ _____ (4√2x – 1 ) ( 3√2x – 1) = (2x -1 )1/4 (2x – 1)1/3 = (2x – 1) ¼ + 1/3 = (2x – 1) 7/12 _______ = 12√(2x – 1)7Rule 3: If radicals have different indices and different radicands, convert the radicals into powers having similar fraction for exponents and rewrite the product as a single radical. Simplify the answer if possible. __Example 8. √2 3√3 Solution: _ _ √2 3√3 = 21/2 · 31/3 = 23/6 · 32/6 __ __ = 6√23 · 6√32 ____ = 6√ 8 · 9 __ = 6√72 __Example 9. 4√2 · 3√5 Solution: _ _ 4√2 · 3√5 = 21/4 · 51/3 = 23/12 · 54/12 __ __ = 12√23 · 12√54 _ ___ = 12√8 · 12√625 _____ = 12√ 5000 Multiplying a radical by a binomial In each of the following multiplication, you are to use the distributiveproperty to expand the binomial terms. 5

_ __Example 10. Multiply: √3 ( 2√3 + √5)Solution: Using the distributive law, then __ _ _ _ __ √3 (2√3 + √5) = √3 · 2√3 + √3 · √5 _ _ ____ = 2√3 · √3 + √ 3· 5 ___ = 2.3 + √15 __ = 6 + √15 __ _Example 11. Multiply and simplify: 2√x (√x - 3) – 4(3 - 5√x)Solution: Proceed as if there are no radicals- using the distributive law toremove the parentheses;__ _ __ _ _2√x (√x - 3) – 4(3 - 5√x) = 2√x √x - 6√x – 12 + 20√x __ = 2 x -6√x – 12 + 20√x __ = 2x - 6√x–12 + 20√x combine like terms _ = 2x + 14√x – 12 Binomial Multiplication. This method is very much similar to the FOIL method. The terms areexpanded by multiplying each term in the first binomial by each term in thesecond binomial. ___ _Example 12. (4√3 + √2) (√3 -5√2 ___ _ Solution: (4√3 + √2) (√3 -5√2) Use the FOIL method, that is multiplying the first terms, outer terms, inner terms and the last terms. _ _ _ _ __ _ _ = 4(√3)(√3) -4√3(5√2) + √2(√3) - √2(5√2) _ __ _ = 4(√3)2 - 20√6 + √6 -5(√2)2 __ = 4 · 3 - 20√6 + √6 – 5 · 2 6

__= 12 - 20√6 + √6 – 10 _= 2 - 19√6 _ __ _Example 13. (√a + √3) (√b + √3) _ __ _ Solution: (√a + √3) (√b + √3) FOIL these binomial then simplify. __ _ _ _ _ _ = √a√b + √3√a + √3√b + (√3)2 __ __ __ = √ab + √3a + √3b + 3 __Example 14. Multiply and simplify: (√7 - √3 )2Solution. Watch out! Avoid the temptation to square them separately.Remember: (a+b)2 ≠a2+ b2__ _ ___(√7 - √3)2 = (√7 - √3) (√7 - √3)__ _ _ _ _ __= √7 √7 - √7 √3 - √7 √3 + √3 √3__ __= 7 - √21 - √21 + 3 Combine like terms__= 10 - 2√21_ ____Example 15. (√a – 3)2 – (√a – 3 )2Solution: Note the difference between the two expressions being squared.The first is a binomial; the second is not. _ ___ _ _ ____ ___(√a – 3)2 – (√a – 3 )2 = (√a – 3)(√a – 3) - √a – 3 √a-3 __ _ _ = √a√a - 3√a -3√a + 9 – (a - 3)Note that the parentheses around a – 3 is essential. _ = a – 6 √a + 9 – a + 3 _ 7

= -6√a + 12 Multiplying Conjugate Binomials The product of conjugates are always rational numbers. The product of apair of conjugates is always a difference of two squares (a2 – b2), multiplication ofa radical expression by its conjugate results in an expression that is free ofradicals. __ __ Example 16. (√13 -3) (√13 + 3)Solution: Multiply out using FOIL.__ __ __ __ __ __(√13 -3) (√13 + 3) = √13 √13 + 3√13 - 3√13 – 9 The middle terms combine to 0. = 13 – 9 =4 This answer does not involve radical. _ _ __Example 17. (√5 + √7 ) (√5 - √7) A difference of squares A square of a root is the __ original integer = (√5)2 – (√7)2 = -2 Simplified _ __ _Example 18. (√7 + 2√3)(√7 - 2√3) __ = (√7 )2 – (2√3 )2= 7 – 12= -5 8

Try this outPerform the indicated operations. Simplify all answers as completely as possible.A. _ __ _ __1. √3√11 6. √5 √45 _ _ __ _ _ __2. √3√5√13 7. √2√6√10 __ ___3. √6 √24 8. √3 √5 √6 __ __ __ __4. √18 √32 9. √24 √28 _ _5. (-4√2 )2 10. ( 3√5 )2B. _ _ __ __ _ 16. (2√3 - √7)(2√3 + √7)11. 2√5c . 5√5 __ _ ____12. 2√5 (5√3 + 3√5) 17. ( 1 + √x + 2 )2 __ ___13. (2√5 -4)( 2√5 + 4) ____ 18. √3 ( 2√3 - 3√2)14. (3√3 - √2) ( √2 + √3) __ __ __ 19. 3√2 (√2 – 4)+ √2 (5 - √2)15. (√3 + 2) (√3 -5) __ 20. (√x + √3 )2C. What’s Message? Do you feel down with people around you? Don’t feel low. Decode themessage by performing the following radical operations. Write the wordscorresponding to the obtained value in the box provided for. 9

10

are not __and irreplaceable √2 · 5√8consider yourselfDo not __Each one 3√7 · 4√7for peopleis unique __more or less 4√3 · 3√3nor even equalof identical quality __to others √9 · √4 ____ _____ 3√9xy2 · 3 3√3x4y6 (4√3a3)2 _ __ √3 · 3√18 __ _ √27 · √3 __ √a (√a3 – 7) __ 5√7 · 2√7 __ __ ___ (√5a)(√2a)(3√10a2)6 36 9 _ 30a2 a2-7√a48a3 20 70 20 ___ ___ 9xy2 3√x2y2 6√54 _ 12√7 11

Lesson 2 Division of radicalsDividing a radical by another radical, follows the rule similar tomultiplication. When a rational expression contains a radical in its denominator,you often want to find an equivalent expression that does not have a radical inthe denominator. This is rationalization. Study the following examples. __Example 1. Simplify: √72 √6Solution: You are given two solutions: __b. Simplify √72. b. Make one radical expression___ __ __ __√72 = √36 √2 √72 = 72√6 √6 √6 6 _ __ = 6√2 Rationalize = √12 √6 _ _ __ = √4 √3 = 6√2 . √6 _ √6 √6 = 2√3 __ = 6√12 6 __ = √12 __= √4 √3 _= 2√3Note: Clearly the second method is more efficient. If you have the quotient oftwo radical expressions and see that there are common factors which can bereduced, it is usually method 2 is a better strategy, first to make a single radicaland reduce the fraction within the radical sign. then proceed to simplify theremaining expression.___Example 2. √6b7_√30ab ___Solution: √6b7__ = 6b7 Reduce √30ab 30ab 12

b6 = 5a _ = √b6 √5a = b3 √5a __ = b3 . √5a √5a √5a __ = b3√5a 5a Rationalizing binomial denominators The principle used to remove such radicals is the familiar factoringequation. If a or b is square root, and the denominator is a + b, multiply thenumerator and the denominator by a – b and if a or b is a square root and thedenominator is a + b, multiply the numerator and the denominator by a – b. (a +b) (a – b) = a2- b2Example 3. ___2___√7 - √5 __ __Solution: the denominator is √7 - √5, is the difference, so multiply thenumerator and the numerator by the sum √7 + √5: __ _____2___ x √7 + √5 = 2(√7 + √5)√7 - √5 √7 + √5 (√7)2 – (√5)2 __ = 2(√7 + √5 )_ 7–5 __ = 2(√7 + √5) Simplify 2 __ = √7 + √5Example 4. ___20___ √10 + √6 __ _ Solution: ___20___ = ___20___ . √10 - √6 √10 + √6 √10 + √6 √10 - √6 13

__ _ = 20(√10 - √6) 10 – 6 __ _ = 20(√10 - √6 ) 4 __ __ __ = 5(√10 - √6) or 5√10 - 5√6Example 5. Simplify as completely as possible: ___8___ - 10 3 - √5 √5 Solution: Begin by rationalizing each denominator. Keep in mind that each fraction has sits own rationalizing factor. _____8___ - 10 = ___8___ . 3 + √5 - 10 · √53 - √5 √5 3 - √5 3 + √5 √5 · √5 __ = 8(3 + √5) - 10√5 Reduce each fraction 9–5 5 __ = 8(3 + √5) - 10√5 Simplify the numerator 45 and denominator which __ are not radicand. = 2(3 + √5) - 2√5 Combine similar radicands. __ = 6 + 2√5 - 2√5 =6 __Example 6. Simplify: 12 + √186Solution: Begin by simplifying the radical.__ _ _12 + √18 = 12 +√9 √266 _= 12 + 3√2 Factor out the common factor 6 of 3 in the numerator. _= 3(4 + √2) simplify 6 __= 4 + √2 or 2 + √2 22 14

__Example 7. √2 ÷ 3√2__ _Solution: √2 ÷ 3√2 = __√2__ 3√2 = 21/2 Change the radicals to fractional exponent. 21/3 = 23/6 Change the fractional exponents to similar 22/6 fractions = 6 23 Transform the expression as a single radical. 22 and simplify. _ = 6√2 _____ _______Example 8. Express as a single radical: √4xy2z2 ÷ 6√16xy2z4 _____ _______ Solution: √4xy2z2 ÷ 6√16xy2z4 Transform to fraction _____ = __√4xy2z2__ 6√16xy2z4 = (4xy2z3)1/2 Change to fractional exponent (16xy2z4)1/6 = (4xy2z3)3/6 Change the fractional (16xy2z4)1/6 exponent to similar fractions. _______ Rewrite as radical expressions = 6√(4xy2z2)3 the radicand to powers. 6√16xy2z4 = 6 64x3y6z6 Simplify. 16xy2z4 = 6 4x2y4z2 __Example 9. Perform: √2 ÷ (2 + √3) __ _Solution: √2 ÷ (2 + √3) = __√2__ rewrite the expression 2 + √3 15

__ = __√2__ . 2 -√3 rationalize 2 + √3 2 - √3 __ = 2√2 - √6 simplify 4-3 __ = 2√2 - √6 __ _ _Example 10. Simplify: √xy ÷ (√x - √y) Solution: __ _ _ __ √xy ÷ (√x - √y) = __√xy__ rewrite the expression (√x - √y) __ _ _ = __√xy__ . √x + √y rationalize √x - √y √x + √y ___ ___ = √x2y + √xy2 x–y __ = x √y + y√x x–yTry this outA. Divide and simplify __ __ 1. 6√18 ÷ 12√40 __ __ 2. 8√19 ÷ 4√38 __ 3. 20√3 ÷ 5√3 __ 4. 42√6 ÷ 3√6 __ _ 5. -4√20 ÷ √2 __ 6. 10√18 ÷ 2√9 __ __ 7. 5√96 ÷ 2√24 __ __ 8. 3/7 √30 ÷ 1/3 √15 __ __ 9. 20√46 ÷ 5√23 16

_ __10. 6√3 ÷ √18 _ __11. 12√2 ÷ 2√27 _ __12. 12√6 ÷ ¼ √72 __ ___13. √50 ÷ √125 __ ___14. √45 ÷ √40015. 3 3x2b ÷ 4 25xy2B. Simplify __ 1. √10 ÷ 3 2 _ 2. 3 3 ÷ √3 3. 4 3 ÷ 3 3 4. 3 6 ÷ 4 6 5. 3 36 ÷ 4 6 __ 6. √9 ÷ √3 7. 4 27 ÷ 3 2 8. __1__ 2 + √5 9. __1__ 3 - √11 10. __3__ √3 – 1D. Why is tennis a noisy game?Solve the radicals by performing the indicated operation. Find the answer belowand exchange it for each radical letter. 17

__ __ __ _E √2 ÷ √3 P 3√8 ÷ 3√6 R 4√36 ÷ 4√6 __ ___7___ _ __I 3√4 ÷ 3√6 E √6 + √5 K 4√6 - 3√21 __ _ __ √3 ____ ____L √2 ÷ 3√2 C √5 ÷ √15 Y 3√ 3x2b ÷ 3√25xy2 __ V _1_ ___ _S 2√2 ÷(2+√3) √x E 3 √108 ÷ 3√2 __ __ A 400 R _1_A 5√63 ÷ 6√7 20 __ _ √5 __ __ R √80 ÷ √5E 6√28 A 20√46 __ 3√4 5√23 Y 10√18 ÷ 2√9 __ __R __1__ __ ___ T 5√96 ÷ 2√24 2+√5 E √25 ÷ √625 65A __3__ __ √3 - 1 S 3√3 ÷ 3√5√6 √x _ 3√36 _______ __32 7 4 5√2 3 6√2 5 3√15bxy √5 √5 2 5y 5 5_ 3√3+3 3√18 _ 3√75 _ _ _ √3 _ _ _ 254√ 6 2 3 2 2√7 5 4√2 -2+√5 2√5 3 4√7-3√7 3 3√2 6Let us summarize Definition: The pairs of expressions like x - √y and x + √y or √x - √y and √x + √y are called conjugates. The product of a pair of conjugates has no radicals in it. Hence, when we rationalize a denominator that has two terms where one or more of them involve a square-root radical, we multiply by an expression equal 1, that is, by using the conjugate of the denominator. 18

What have you learnedA. Fill in the blanks. 1. For a = b2, _______is the square root of ______. 2. When no index is indicated in a radical, then it is understood that the index is _____. 3. In radical form, 169 3/2 is written as ____ or ____ __ 4. In simplest form. √54 is ____ __ 5. In simplest form 3√16 is ____ __ 6. In simplest form 4√64 is ____ __ 7. in simplest form, 6√16 is ___ _____ 8. In simplest form √50x7y11 ____ ____ 9. The product of (3√ 2 + 4)(3√2 – 4) __ __ 10. The product √26 . 4√4 _ __ __ 11. The combined form 5√7 -2√28 - 3√48 is ___________. __ 12. In simplest form, the quotient √27 = _______ √48 ___ 13. In simplest , the quotient 3√135 = _____ 3√40 _ 14. In simplest form, the quotient __√7__ √3 - √2 ______ 15. In simplest form, the quotient 4√162x6y7 = ____ 432x8y 19

Answer KeyHow much do you know: A. 1. 3·4·3 = 36 2· 5·2·7 = 70 __ 3. 2√35 4.5√10 5. 10bx _ B. 1. √6/3 __ 2. 3√18/3 _ 3. 6√2 __ 4. 2√2 - √6 __ 5. x√y + y√x x-yTry this outLesson 1 __A. 1. √33 ___ 2. √199 3. 12 __ 4. 12√12 5. 32 6. 15 _ 7. 2√3 __ 8. 3√10 __ 9. 2√42 20

10. 45 _B. 11. 50√c __ 12. 10√15 + 30 13. 4 __ 14. 2√6 + 7 or 7 + 2√6 __ 15. -3√3 – 7 or -7 - 3√3 16. 5 ____ 17. 3 + x + 2 √x + 2 _ 18. 6 - 3√6 _ 19. 4 - 7√2 __ 20. 3 + x + 2√3xC.Do not consider more or less nor even to others yourself equalfor people are not of identical each one Is unique quality and are irreplaceable 21

Lesson 2 _ 11. 2√6Try this out. _ 3 __A. 1. 3√5 12. √12 20 4 __ __ 13. √10 2. 2√19 5 19 _ 3. 4 14. 3√5 4. 14 _____ 15. 4√75bxy2 __ 5. - 4√10 5y _ _ 6. 5 √2 8. -2 +√5 7. 10 __ _ 9. 3 + √11 8. 9√2 -2 2 _ _ 10. 1 + 3√3 9. 4√2 2 _ 2210. √6 ___B. 1. 6√250 ___ 2. 6√243 3 __ 3. 12√311 3 _ 4. 12√6 __ ____ 5. 12√65 or 12√7776 _ 6. √3 ___