1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number line? 4 How many even integers are between 7.5 and 3.5 on the number line? 5 How many odd integers are between 6.5 and 2.5 on the number line? 6 Write the inequality x > 9 using interval notation. 7 Write the inequality 6 x < 6 using interval notation. 8 Write the inequality as the union of two intervals. x 9 or x > 10 9 Write the expression without using absolute value symbols. 0 10 Write the expression without using absolute value symbols. 6 11 Find the distance between the following two points on the number line. 6 and 11 12 Write the number 12 2 without using exponents. 13 Simplify the expression x 2 x 3 PAGE 1
14 Simplify the expression (x 8 ) 4 15 Simplify the expression (x 5 ) 3 (x 2 ) 4 16 Simplify the expression x 2 y 7 8 Write the answer without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined. 17 Simplify the expression 1 x 6 Write the answer without using negative exponents. Assume that the variable is restricted to those numbers for which the expression is defined 18 Simplify the expression x 3 x 4 x 3 x Write the answer without using negative exponents. Assume that the variable is restricted to those numbers for which the expression is defined. PAGE 2
19 Simplify the expression x 10 x 3 ( x 4 ) 4 Write the answer without using negative exponents. Assume that the variable is restricted to those numbers for which the expression is defined 20 Simplify the expression a 5 b 4 3 Write the answer without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined. 21 Simplify the expression r 3 r 1 r 6 r 6 5 Write the answer without using negative exponents. Assume that the variable is restricted to those numbers for which the expression is defined. 22 Simplify the expression ( 8 2 z 5 y ) 1 ( 5y 5 z 2 ) 4 ( 5y z 2 ) 1 Write the answer without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined. 23 Simplify the expression 4 [ 2 2 + ( 22 6 ) ] 2 ( 4 5 ) PAGE 3
24 Let x = 3, y = 0, z = 1 and evaluate the expression. ( x 2 z 3 ) z 2 y 2 25 8.61 10 6 Express the number in standard notation. 26 Calculate the volume of a box that has dimensions of 4000 by 9300 by 4400 millimeters. Write the answer in scientific notation. 27 Simplify the expression. 9 16 1/2 28 Simplify the expression. ( 27 ) 1/3 29 Simplify the expression. Use absolute value symbols if necessary. ( 16y 4 ) 1/4 30 Simplify the expression. 27x 6 8y 3 1/3 31 Simplify the expression. 4 3/2 PAGE 4
32 Simplify the expression. 125 27 1/3 Write your answer without using negative exponents. 33 Simplify the expression ( 121s 4 ) 1/2 Assume that the variable represents a positive number. 34 Simplify the expression. a 3/7 a 2/7 a 4/7 Write the answer without using negative exponents. Assume that all variables represent positive numbers. 35 Simplify the expression 50 3 2 Your answer should be a radical expression. 36 Simplify the expression. Assume that all variables represent positive numbers, so that no absolute value symbols are needed. 9y 2 36y 3 8 100y 7 Your answer should be a radical expression. 37 Simplify the expression. Assume that all variables represent positive numbers, so that no absolute value symbols are needed. 4 162xy 5 + y 4 1250xy 4 512xy 5 Your answer should be a radical expression. PAGE 5
38 Rationalize the denominator and simplify. 3 5 3 Your answer should be a radical expression. 39 Rationalize each denominator and simplify. 3 t 54 3 t 250 + 3 t 686 Your answer should be a radical expression. 40 Simplify the radical expression. 8 49 Your answer should be a radical expression. 41 We can often multiply and divide radicals with different indexes. For example: 3 3 5 = 6 27 6 25 = 6 (27)(25) = 6 675 Use this idea to write the following expression as a single radical. 5 3 5 42 We can often multiply and divide radicals with different indexes. For example: 3 3 5 = 6 27 6 25 = 6 (27)(25) = 6 675 Use this idea to write the following expression as a single radical 6 2 6 43 Give the degree of polynomial 520x 5 + 7x 632 + 4 44 Perform the operations and simplify (8x 3 5x 2 ) + (6x 3 10x) PAGE 6
45 Perform the operations and simplify 6a 2 (a + 1) + 10a(a 2 4) a 2 (a + 9) 46 Perform the operation and simplify (a 5) 2 47 Perform the operation and simplify. (6r 9s) (5r 2 6rs 4s 2 ) 48 Perform the operations and simplify (a 15/2 + b 5/2 )(a 15/2 b 5/2 ) 49 Perform the operations and simplify (x 3/2 + y 5/2 ) 2 50 Rationalize the denominator 7 7 2 51 Rationalize the denominator. 11 5 5 7 PAGE 7
52 Rationalize the numerator. 13 2 7 53 Rationalize the numerator. x 11 6 54 Perform the division and write the answer without using negative exponents 14a 2 b 3 2ab 6 55 Perform the division and write the answer without using negative exponents 20r 2 s 5 t 3 24r 6 s 2 t 8 56 Perform the division and write the answer without using negative exponents 40x 6 y 4 z 9 8x 9 y 6 z 0 PAGE 8
57 Perform the division and write the answer without using negative exponents 225x 5 y 7 135x 2 y 5 + 45xy 9x 5 y 4 58 Perform the division. x + 3 3x 2 20x 87 59 Perform the division. x 2 + x 1 19x 3 8x 2 46x + 27 60 Perform the division. x 2 3 19x 3 8x 2 57x + 24 61 Perform the division. x 5 32 x 2 62 Complete the factoring formula. cx + nx = 63 Complete the factoring formula. x 2 2xy + y 2 64 Factor the expression completely. 8x 16 65 Factor the expression completely. 6x 2 + 3x 3 PAGE 9
66 Factor the expression completely. 5x 3 y 3 z 3 + 25x 2 y 2 z 2 125xyz 67 Factor the expression completely. 4x 3 y 3 z 3 + 16x 2 y 2 z 2 64xyz 68 Factor the expression completely. 4x 3 + 4x 2 7x 7 69 Factor the expression completely. 4ax + 8ay 9bx 18by 70 Factor the expression completely. 4z 2 25 71 Factor the expression completely. 64x 2 + 1 72 Factor the expression completely. z 4 1 73 Factor the expression completely. 36z 2 + 84z + 49 PAGE 10
74 Factor the expression completely. x 2 + 13x + 36 75 Factor the expression completely. 20x 2 7xy 40y 2 76 Factor the expression completely. x 4 + 2x 2 8 77 Factor the expression completely. 144y 2n 25z 8n 78 Factor the expression completely. (x + y) 3 125 79 Factor the expression completely. z 2 + 12z + 36 225y 2 80 Factor the expression completely. (a + b) 2 8(a + b) 9 81 Factor the expression completely. 36x 4 + 25 + 59x 2 PAGE 11
82 Simplify the fraction 9x 81 x 2 81 Assume that denominator is not 0. 83 Simplify the fraction xy + 5x + 2y + 10 x 3 + 8 Assume that denominator is not 0. 84 Perform the operations and simplify x 2 16 x x 2 x 2 + 8x + 16 85 Perform the operations and simplify x 2 + 7x x 7 x 2 49 x + 2 86 Perform the operations and simplify x 2 + 2x 15 x 2 10x + 25 x 2 9 x 2 25 Assume that denominators are not 0. PAGE 12
87 Perform the operations and simplify ax + 6bx + 8a + 48b a 2 + 12ab + 36b 2 x 2 64 x 2 16x + 64 88 Perform the operations and simplify x 3 + 343 x 2 4 x 2 + 9x + 14 x 2 + 2x x 2 + 5x 14 x 2 7x + 49 89 Simplify the expression 2 x + 4 + 4x + 1 x + 4 Assume that denominator is not 0. 90 Perform the operations and simplify 8x x 3 24 x 3 Assume that denominator is not 0. 91 Perform the operations and simplify x + 2 x 2 + 3x + 2 + x x 2 1 PAGE 13
92 Perform the operations and simplify 9x x 2 9 9 x + 3 93 Perform the operations and simplify 1 x 8 + 3 x + 8 3x 8 x 2 64 94 Perform the operations and simplify 16 2x 3y + 16 2x 5z 80z 48y ( 2x 3y )( 2x 5z ) 95 Simplify the complex fraction 2x 5 y 4 4x 2 z 4 y 2 Assume that the denominators are not 0. PAGE 14
96 Simplify the complex fraction. x 2 13x + 42 9x 2 y x 2 36 9x 2 y 97 Simplify the complex fraction x + 2 15 x x + 8 + 15 x 98 Write the expression 2y 1 8x 1 + 11y 1 without using negative exponents, and simplify the resulting complex fraction. 99 Write the expression ( x + 6 ) 1 + ( x 1 ) 1 ( x + 6 ) 1 without using negative exponents, and simplify the resulting complex fraction. PAGE 15
ANSWER KEY 1 ( ) 1. 4 34. 7 67. 4x y z x 2 y 2 z 2 +4x y z 16 a 2. 6 35. 2 2 68. ( x+1) ( 4x 2 7) 3. 8 36. 26y 3 y 69. ( x+2y) ( 4a 9b) 5 4 ( 2z+5) ( 2z 5) 4. 37. 4y 2x y 70. ( 2z 5) ( 2z+5) 5. 4 38. 5 81 71. prime ( z 2 +1) ( z 1) ( z+1) ( z 2 +1) ( z+1) ( z 1) 29 ( 9, ) 210 3 4t ( z 1) ( z 2 +1) ( z+1) 6. 39. 72. 0.138095 3 4t ( z+1) ( z 2 +1) ( z 1) ( z 1) ( z+1) z 2 +1 ( ) ( ) ( z+1) ( z 1) z 2 +1 ( ) 2 7. 4 6,6) 40. 6z+7 7 73. ( 6z+7) ( 6z+7) (, 9 ( 10, 6 x+4 ) ( ) ( x+9) 8. 41. 3125 74. ( x+9) ( x+4) 6 0 432 ( 5x 8y) ( 4x+5y) 9. 42. 75. 6 ( 4x+5y) ( 5x 8y) ( x 2 +4) ( x 2 2) 10. 6 43. 632 76. ( x 2 2) ( x 2 +4) ( 17 14x 3 5x 2 12y n 5z 4n ) 12y n +5z 4n 11. 44. 10x 77. ( 12y n +5z 4n ) 12y n 5z 4n 12. 144 45. 3a 3 15a 2 40a 78. ( ) ( ) ( ) ( x+y 5) x 2 +2x y+25+y 2 +5x+5y x 5 a 2 ( z+6+15y) ( z+6 15y) 13. 46. 10a+25 79. ( z+6 15y) ( z+6+15y) x 32 30r 3 81r 2 s+30r s 2 +36s 3 ( a+b 9) ( a+b+1) 14. 47. 80. a+b+1) ( a+b 9) 15. x 23 48. a 15 b 5 81. ( 6x 2 +x+5) 6x 2 x+5 x 16 3 5 16. 49. 2 2 5 82. y +y y 56 x 3 +2x ( 7 7+14) 3 7( 7+2) 3 17. x 6 50. 83. 9 x+9 ( ) ( y+5) ( x 2 2x+4) ( ) PAGE 1
ANSWER KEY ( x ( x 4) ) ( x 3 5 11 5 5+ 77 35 ) ( x+4) 18. 51. 84. 18 ( x 2 4x) ( x+4) 19. x 23 52. 9 x ( x+7) ( 7( 13+2) ) ( x+2) 85. 9 ( x 3 +14x 2 +x 49) ( 7 13 +14) ( x+2) ( x+5) a 15 ( x 121) ( x+3) ( x 5) ) 20. b 12 53. 86. ( 6 x+66) ( x 2 +10x+25) ( x 2 2x 15) 21. r 10 7a ( x 8) 54. b 3 87. ( a+6b) 22. 8 2 z 11 ( ( 5 3 y 20 ) 5s 3 x 2 +7x) ( x+2) 64z 11 55. ( 6r 4 t 5 88. ) x ( x+7) 125y 20 ( x+2) ( ) 23. 40 56. 5z 9 ( 4x+3) ( x 3 y 2 89. ) ( x+4) ( 9) 25y 3 15y + 5 24. 57. 9 x 3 ( x 4 y 3 90. 8 ) ( 2x 25. 8610000 58. 3x 29 91. ( x 2 1) 27 1.6368 10 11 ( x 3) ( x+3) 26. 59. 19x 27 92. 27 ( x 2 9) 3 19x 24 1 27. 60. 93. 4 3 ( x+8) 28. 3 61. x 4 +2x 3 +4x 2 +8x+16 94. 0 x ( c+n) 29. 2 y x ( n+c) 1 62. 95. ( c+n) x 2 x3 y 2 z 4 ( n+c) x 3x 2 ( x y) ( x 7) 30. 63. 96. ( 2y) ( x+6) 8( x 2) ( x 3) 31. 8 64. 97. ( x 2) 8 ( x+3) PAGE 2
ANSWER KEY 3x 2 ( 2+x) 3 3x 2 ( x+2) 32. 65. 5 ( 2+x) 3x 2 98. ( x+2) 3x 2 ( ) 33. 11s 2 66. 5x y z x 2 y 2 z 2 +5x y z 25 99. 2x 8y+11x ( ) ( 2x+5) ( x 1) PAGE 3