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1 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 Write the expression without using absolute value symbols Find the distance between the following two points on the number line. 6 and Write the number 12 2 without using exponents. 13 Simplify the expression x 2 x 3 PAGE 1

2 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

3 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 [ ( 22 6 ) ] 2 ( 4 5 ) PAGE 3

4 24 Let x = 3, y = 0, z = 1 and evaluate the expression. ( x 2 z 3 ) z 2 y 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 /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

5 32 Simplify the expression /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 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 y 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 xy 5 + y xy 4 512xy 5 Your answer should be a radical expression. PAGE 5

6 38 Rationalize the denominator and simplify Your answer should be a radical expression. 39 Rationalize each denominator and simplify. 3 t 54 3 t t 686 Your answer should be a radical expression. 40 Simplify the radical expression Your answer should be a radical expression. 41 We can often multiply and divide radicals with different indexes. For example: = = 6 (27)(25) = Use this idea to write the following expression as a single radical We can often multiply and divide radicals with different indexes. For example: = = 6 (27)(25) = Use this idea to write the following expression as a single radical Give the degree of polynomial 520x 5 + 7x Perform the operations and simplify (8x 3 5x 2 ) + (6x 3 10x) PAGE 6

7 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 Rationalize the denominator PAGE 7

8 52 Rationalize the numerator Rationalize the numerator. x 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

9 57 Perform the division and write the answer without using negative exponents 225x 5 y 7 135x 2 y xy 9x 5 y 4 58 Perform the division. x + 3 3x 2 20x Perform the division. x 2 + x 1 19x 3 8x 2 46x Perform the division. x x 3 8x 2 57x 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 Factor the expression completely. 6x 2 + 3x 3 PAGE 9

10 66 Factor the expression completely. 5x 3 y 3 z x 2 y 2 z 2 125xyz 67 Factor the expression completely. 4x 3 y 3 z x 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 Factor the expression completely. 64x Factor the expression completely. z Factor the expression completely. 36z z + 49 PAGE 10

11 74 Factor the expression completely. x x Factor the expression completely. 20x 2 7xy 40y 2 76 Factor the expression completely. x 4 + 2x Factor the expression completely. 144y 2n 25z 8n 78 Factor the expression completely. (x + y) Factor the expression completely. z z y 2 80 Factor the expression completely. (a + b) 2 8(a + b) 9 81 Factor the expression completely. 36x x 2 PAGE 11

12 82 Simplify the fraction 9x 81 x 2 81 Assume that denominator is not Simplify the fraction xy + 5x + 2y + 10 x Assume that denominator is not Perform the operations and simplify x 2 16 x x 2 x 2 + 8x Perform the operations and simplify x 2 + 7x x 7 x 2 49 x 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

13 87 Perform the operations and simplify ax + 6bx + 8a + 48b a ab + 36b 2 x 2 64 x 2 16x Perform the operations and simplify x x 2 4 x 2 + 9x + 14 x 2 + 2x x 2 + 5x 14 x 2 7x Simplify the expression 2 x x + 1 x + 4 Assume that denominator is not Perform the operations and simplify 8x x 3 24 x 3 Assume that denominator is not Perform the operations and simplify x + 2 x 2 + 3x x x 2 1 PAGE 13

14 92 Perform the operations and simplify 9x x x Perform the operations and simplify 1 x x + 8 3x 8 x Perform the operations and simplify 16 2x 3y x 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

15 96 Simplify the complex fraction. x 2 13x x 2 y x x 2 y 97 Simplify the complex fraction x x x x 98 Write the expression 2y 1 8x y 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

16 ANSWER KEY 1 ( ) x y z x 2 y 2 z 2 +4x y z 16 a ( x+1) ( 4x 2 7) y 3 y 69. ( x+2y) ( 4a 9b) 5 4 ( 2z+5) ( 2z 5) y 2x y 70. ( 2z 5) ( 2z+5) prime ( z 2 +1) ( z 1) ( z+1) ( z 2 +1) ( z+1) ( z 1) 29 ( 9, ) t ( z 1) ( z 2 +1) ( z+1) t ( z+1) ( z 2 +1) ( z 1) ( z 1) ( z+1) z 2 +1 ( ) ( ) ( z+1) ( z 1) z 2 +1 ( ) ,6) 40. 6z ( 6z+7) ( 6z+7) (, 9 ( 10, 6 x+4 ) ( ) ( x+9) ( x+9) ( x+4) ( 5x 8y) ( 4x+5y) ( 4x+5y) ( 5x 8y) ( x 2 +4) ( x 2 2) ( x 2 2) ( x 2 +4) ( 17 14x 3 5x 2 12y n 5z 4n ) 12y n +5z 4n x 77. ( 12y n +5z 4n ) 12y n 5z 4n a 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) a ( z+6 15y) ( z+6+15y) x 32 30r 3 81r 2 s+30r s 2 +36s 3 ( a+b 9) ( a+b+1) a+b+1) ( a+b 9) 15. x a 15 b ( 6x 2 +x+5) 6x 2 x+5 x y +y y 56 x 3 +2x ( ) 3 7( 7+2) x x+9 ( ) ( y+5) ( x 2 2x+4) ( ) PAGE 1

17 ANSWER KEY ( x ( x 4) ) ( x ) ( x+4) ( x 2 4x) ( x+4) 19. x x ( x+7) ( 7( 13+2) ) ( x+2) ( x 3 +14x 2 +x 49) ( ) ( x+2) ( x+5) a 15 ( x 121) ( x+3) ( x 5) ) 20. b ( 6 x+66) ( x 2 +10x+25) ( x 2 2x 15) 21. r 10 7a ( x 8) 54. b ( a+6b) z 11 ( ( 5 3 y 20 ) 5s 3 x 2 +7x) ( x+2) 64z ( 6r 4 t ) x ( x+7) 125y 20 ( x+2) ( ) z 9 ( 4x+3) ( x 3 y ) ( x+4) ( 9) 25y 3 15y x 3 ( x 4 y ) ( 2x x ( x 2 1) ( x 3) ( x+3) x ( x 2 9) 3 19x ( x+8) x 4 +2x 3 +4x 2 +8x x ( c+n) y x ( n+c) ( c+n) x 2 x3 y 2 z 4 ( n+c) x 3x 2 ( x y) ( x 7) ( 2y) ( x+6) 8( x 2) ( x 3) ( x 2) 8 ( x+3) PAGE 2

18 ANSWER KEY 3x 2 ( 2+x) 3 3x 2 ( x+2) ( 2+x) 3x ( x+2) 3x 2 ( ) s x y z x 2 y 2 z 2 +5x y z x 8y+11x ( ) ( 2x+5) ( x 1) PAGE 3

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