5.1, 5.2, 5.3 Properites of Exponents last revised 12/4/2010

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1 48 5.1, 5.2, 5.3 Properites of Exponents last revised 12/4/2010 Properites of Exponents 1. *Simplify each of the following: a. b. 2. c. d. 3. e. 4. f. g. 5. h. i. j. Negative exponents are NOT considered to be simplified. Do NOT leave them in final answers! 48

2 49 k. s. l. m. 6. ALEKS PROPLEM TYPE: Simplify: n. o. p. q. r. Multiple choice: a. b. c. d. 49

3 Scientific Notation Scientific notationis a shorthand notation for writing extremely small or large numbers. Notation: *Multiply. Write your answers in scientific notation: 6. *Write each using scientific notation: ,374, trillion *Divide. Write your answers in scientific notation: 8. *Write each in standard form: **Suggested Assignment: pg. 389 #

4 Adding and Subtracting Polynomials monomial binomial *Add: trinomial polynomial *Find the perimeter: Vocabulary: 8x+2 6x - 4 5x+3 8x+2 6x - 4 *Given: the following:, find a. leading coefficient b. constant term c. degree of the second term *Subtract: d. degree of the polynomial If a term has more than one variable, its degree is the of its exponents. *What is the degree of the expression? *Subtract: *Add: 51

5 Multiplying Polynomials 8. *Multiply each of the following:

6 53 Mini-Review of Sections Multiply: 1. Simplify: 2. Multiply: 7. Multiply: 3. Divide: 8. Calculate the area of the larger rectangle in two ways. x 5 x 4. Multiply: 3 5. Multiply: 53

7 Dividing Polynomials A. Dividing by a monomial 3. Create separate fractions and then simplify each separately B. Dividing by a non-monomial Use long division. Recall... 54

8 In Math 90, you will learn another method for dividing called Synthetic Division. This process will only work when dividing by a linear factor (those without exponents). Problems like #7 above cannot be done using synthetic division since there is an exponent in the divisor. **Suggested Assignment: pg. 389 #

9 Introduction to Factoring Factoring is. Factoring is a process. B. Factoring by Grouping A. Factoring Out a Greatest Common Factor 8. *Factor each of the following completely

10 Simplify: 12. Review 5. Multiply: 1. Multiply: 6. Multiply: 7. Multiply: 2. Simplify: 3. Simplify: 57

11 58 8. Divide: 11. Factor: 12. Factor: 9. Divide: 13. Factor: 10. Divide: 14. Factor: 15. Factor: 58

12 Factoring Trinomials, part 1 * Factor each of the following:

13 Factoring Trinomials, part 2 5. pattern: 6. *Factor each of the following completely

14 Factoring Trinomials, part 3 The a-c Method

15 The Difference of Two Squares and Perfect Square Trinomials The Difference of Two Squares 9. *Factor each completely:

16 The Sum & Difference of Two Cubes Memorize: "SOAP" means... *Factor each completely: What if you forget these formulas? 63

17 64 A General Stategy for Factoring # Can I factor out a? 2. How many terms are there? a. if four, try. #30. b. If three, try. c. If two, try or try. # Can I factor further? The following exercises are from pages of your textbook, which are printed on the next page for your use. #45. #5. #15. #55. #25. #65. 64

18 65 65

19 Solving Equations Using the Zero Product Rule Quadratic equations are of the form 5. Zero Product Rule: If, Then 6. Solve each of the following equations

20 # # # # # # # # # #46 67

21 Quadratic Word Problems A. Number Problems 6.8 #16 The sum of the squares of two consecutive integers is 9 less than 10 times their sum. Find all such integers. 6.8 #14 If a number is added to two times its square, the result is 36. Find all such numbers. B. Consecutive Integer Problems Review... Consecutive Integers Consecutive Even Integers Consecutive Odd Integers The product of two consecutive odd integers is 63. Find all such integers. 6.8 #14 The product of two even consecutive integers is 48. Find all such numbers. 68

22 69 The perimeter of a rectangle is 22 cm and its area is 24cm 2. Find the length and width of this rectangle. The length of a rectangle is three times its width. Find the dimensions if the area is 48 cm 2. D. Height of a Projectile C. Area Problems 6.8 #24 A stone is dropped off a 256-ft. cliff. The height of the stone is given by, where t is the time (in seconds). When will it hit the ground? 6.8 #18 The width of a rectangular picture frame is 2 inches less than its length. The area is 120 in 2. Find its dimensions. 69

23 70 E. The Pythagorean Theorem Review of Chapters 5 & 6 Find x: x Simplify: 8 2. Simplify: 3. Simplify: 4. Simplify: 5. Simlify: 6.8 #40 The longer leg of a right triangle is 1 cm less than twice the shorter leg. The hypotenuse is 1 cm more than twice the shorter leg. Find the length of the shorter leg. 6. Simplify: 70

24 71 7. Write in standard form. 12. Divide: 8. Write 365,000,000 in scientific notation. 9. Multiply: 13. Simplify: 10. Divide: 14. Add: 15. Subtract from. 11. Divide: 16. Multiply: 17. Multiply: 71

25 The square of a number is subtracted from 60, resulting in Find all such numbers. 20. The length of a rectangle is 1 ft. longer than twice its width. If the area is 78 ft 2, find the rectangle's deimensions. 19. The product of consecutive integers is 44 more than 14 times their sum. Find all such integers. 21. A right triangle has one leg that is 2 ft. longer than the other leg. The hypotenuse is 2 ft. less than twice the shorter leg. Find the lengths of all three sides of hte triangle. 72

26 Factor: 28. Factor: 23. Factor: ALEKS TOPIC - Domain of a rational function 24. Factor: The function h is defined below. Find all values of x that are NOT in the domain of h. If there is more than one value, separate them with commas. 25. Factor: 26. Factor: 27. Factor: same directions... g(x)= 73

27 74 Additional Review (Chapters 5&6) If Time f. 1. Write 463,000,000 in scientific notation. 2. Multiply: g. h. 3. Divide: 4. Simplify each: a. b. c. 5. Given, state find each: a. Leading coefficient: b. degree of the polynomial: d. 6. Find the degree of the polynomial: e. 74

28 75 7. Add: e. 8. Combine: 10. Divide: 9. Multiply each: a. 11. Divide: b. c. 12. Factor: d. 13. Factor: 75

29 Factor: 19. Solve: 15. Solve: 20. The length of a rectangle is 2 cm more than three times its width. Find the dimensions if the area is 56 cm Factor: 17. Factor: 18. Factor: 76

30 Introduction to Rational Expressions The Technique: Reduce: Simplify: When simplifying rational expressions, you are actually dividing both the numerator and the denominaror by common factors. Simplify: Simplify: Simplify: key phrase: Simplify: Simplify: Simplify: Simplify: Simplify: 77

31 Multiplying and Dividing Fractions *Multiply or divide as indicated. Simplify & 7.4 Adding and Subtracting Fractions Recall... when adding and subtracting fractions, you need to have a. Add: 2. *Find the least common multiple between each. Also state how many times each expression goes into the least common multiple. 1. and and and 78

32 79 4. and and 11. *Add or subtract. Simplify if possible

33

34 Complex Fractions

35 82 Review of Sections Simplify: 6. Add: 2. Multiply: 3. Add: 7. Simplify: 4. Subtract: 5. Add: 82

36 Rational Equations Main Idea: Clear out the fractions by multiplying through on each side of the equation by the least common denominator. 4. Be careful to check that your solution(s) work. Make sure that when you substitute your solution into the problem, you don't get a zero in the denominator. *Solve each of the following equations **Suggested Assignment: pg. 547 #

37 Proportions and Word Problems A. Geometrical Figures Similar triangles: 7.7 #26 Tim takes his pulse for 10 seconds and counts 12 beats. How many beats per minute is this? 7.7 #31 A map has a scale of 75 mi/in. If two cities measure 3.5 inches apart, how many miles does this represent? At the same time that a yardstick casts a 6 ft shadow, a tree casts a 30 ft shadow. How tall is the tree? C. Distance Problems B. Proportional Word Problems Key phrases: or. Key idea: Line up the units (either vertically or horizontally). 84

38 #41 A boat travels 54 miles upstream in the same time it takes to travel 66 miles downstream. If the current is 2 mph, what is the boat's speed in still water? D. Work Problems Fred can paint his house in 30 hours. Barney can paint the same house in 50 hours. How long would it take if they worked together? 7.7 #46 Devon can cross-country ski 5 km/hr faster than his sister Shanelle. Devon skis 45 km in the same time Shanelle skis 30 km. Find their speeds. Machine A can copy a report in 6 minutes. Machine B can copy the same report in 8 minutes. How long would it take if they worked together? 85

39 Variation Direct Variation 3. y varies jointly with x and w and inversely with z. y=4 when x=3, z=8 and w=12. Find y when x=6, z=10 and w=8. Inverse Variation "k" is called the Process for Solving: 1. Identify the type of variation. Write the "base" equation. 2. Substitute in the given values to solve for k. 3. Write the general variation equation for the problem using the value of k you just found. 4. Use the clues to find the requested value. 4. I varies inversely with the square of d, and I=30 when d=4. Find I when d=6. 1. y varies directly with x, and y=10 when x=2. Find the general variation equation and use it to find y when x= y varies inversely with x, and y=6 when x=10. Find the general variation equation and use it to find y when x= #42 The number of turkeys needed for a Thanksgiving dinner is directly proportional to the number of guests. If a cook knows that 3 turkeys will feed 42 guests, how many turkeys should he cook for 70 guests? 86

40 87 Some Chapter 7 Review Problems 1. What values of x are NOT in the domain of? 5. Solve: 2. Add: 3. Simplify: 6. Find x and y. 4. Solve: 87

41 88 7. Hose A can fill a pool in 10 hours. Hose B can fill it in 12 hours. How long would it take if both hoses worked together? 10. Fred and Barney can complete a job in 12 hours if working together. If Barney works twice as fast as Fred, how long would each take, working alone? 8. The current in Lazy River is 2 mph. If one motors upstream 12 miles in the time it takes to motor 20 miles downstream, what is the speed of the boat in still water? 11. Bud goes 10 mph faster on his Harley than Ed goes on his Honda motorcycle. If Bud travels 105 miles in the same time that Ed travels 90 miles, what are the rates of the two bikers? 9. If T varies inversely as x and when x=40, T=200, find T when x=50. 88

42 Simplify: 16. Simplify: 13. Add: 17. Solve: 14. Subtract: 18. There are two machines that produce aluminum cans. The newer machine can produce 4200 cans in 210 minutes. It takes the older machine 280 minutes to produce that many cans. If the two machines work together, how long will it take them to produce 4200 cans? 15. Solve for h. 89

43 Introduction to Radicals Question: = since. 8.2 Simplifying Radicals *Evaluate or Simplify each: 1. Answer: Evaluate: Evaluate: 2. Evaluate: Evaluate: Property: 3. *Simplify each of the following: When, exactly, do you use absolute values? 90

44 91 5. Review. Simplify each Find x: 3 x 8 91

45 Adding and Subtracting Radicals 6. Property: *Combine. Simplify if possible x 7 45x 5. 92

46 Multiplying Radicals 7. Property: *Multiply. Simplify, if possible

47 Dividing Radicals Main idea: An expression is not considered to be simplified if it contains a radical in its denominator. The process of removing the radical from the denominator is called. *Simplify each Simplify: **Suggested Assignment: pg. 623 #

48 Solving Radical Equations 5. Steps: 1. Isolate the radical. 2. Square both sides. 3. Check your solutions! *Solve each:

49 The square root of twice a number is 4 less than the number. Find the number

50 97 The following problem from Aleks is a bit more challenging: 11. Solve: 97

51 98 Chapter 8 Review 6. Divide: 1. Simplify: 7. Divide: 2. Combine: 8. Divide: 3. Multiply: 4. Multiply: 9. Solve: 10. Solve: 5. Simplify: 98

52 Solving Quadratic Equations A. Factorable Quadratic Equations B. Taking the Square Root of each Side 4. These problems are just like those at the end of chapter 6. *Solve each equation

53 Completing the Square Process: 1. Write your equation in the form: 2. If there's a leading coefficient, divide both sides by it. 3. Cut the number in front of x in half; write this new value on the line below. 4. Square this new value and write the product in the blank on the line above; add this product to the right side of the equation also. 5. Insert an x, parentheses and exponent on the left to complete the square. 6. Add together the values on the right side. 7. Square root each side of the equation. Don't forget the plus-or-minus sign! 8. Solve for x. *Solve each by completing the square:

54

55 The Quadratic Formula *In each equation, identify the values of a, b and c: *Solve each equation using the quadratic formula. 3. *Find the value of k needed in order to complete the square:

56 Summary: a. If, use factoring. b. If, square root each side. 6. c. If, it's not quadratic! Solve in two ways. **Suggested Assignment: pg. 669 #

57 104 Chapter 9 Review: *Solve each of the following equations:

58

59 Solve: Final Exam Review 1. A 5% acid mixture is to be combined with a 40% acid solution. How much of each should be used to make 20 liters of a 25% acid solution? 13. Solve: 2. Find three consecutive odd integers who sum is

60 The product of consecutive integers is 72. Find all such numbers. 6. Half a number is 18 less than twice the number. Find the number. 7. Find the x- and y-intercepts of. 4. The length of a rectangle is 3 more than twice its width. Find the dimensions if the area is 65 square units. 8. Find the slope of 5. The length of a rectrangle is 1 cm less than twice its width. Find the dimensions if the perimeter is 22 cm. 9. Find the slope between and. 107

61 Find the equation of the line which passes through and is perpendicular to the line. 13. Evaluate: 14. Simplify: 11. Graph: y 15. Use the commutative property to finish this statement: 16. Use the associative property to finish this statement: x 17. Which property is illustrated? 18. Simplify: 12. Solve: 19. Add: 108

62 Simplify: 24. Bob bought two computers, one desktop and one laptop. Before finance charges, the laptop cost $500 less than the desktop. Bob paid for the computers using two different financing plans. For the desktop the interest rate was 3% per year, and for the laptop it was 5% per year. The total finance charges for one year were $335. How much did the desktop computer cost before finance charges? 21. Find the slope of the line perpendicular to 22. Combine: 23. Solve: 109

5.1, 5.2, 5.3 Properites of Exponents last revised 12/28/2010

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