Numbers, Operations, and Expressions. 1) Determine the classification(s) for each number below. List all that apply. 3

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1 Numbers, Operations, and Expressions Review of Natural Numbers, Whole Numbers, Integers, and Rational Numbers 1) Determine the classification(s) for each number below. List all that apply. a) 11 b) 9.8 c) 21 0 d) Review of Natural Numbers, Whole Numbers, Integers, and Rational Numbers 2) Determine the classification(s) for each number below. List all that apply a) 2 b) 7 c) 0 d) 72 6 e) 45. Review of Exponents, Squares and Square Roots ) Simplify a) 121 b) 1 2 c) 225 d) ) Determine the square root of each number. If the square root does not exist, write no real solution a) 49 h) 121 b) c) 289 d) 64 e) 15 2 f) ( 6) 2 g) ) Estimate each square root to the nearest integer a) 9 b) 24 c) 226 d) 10 e) 10 f) 292 i) j) 0.64 k) l) m).24 NJ Center for Teaching and Learning ~ 1 ~

2 Review of Exponents, Squares and Square Roots 6) Simplify a) 169 b) 19 2 c) 625 d) ) Determine the square root of each number. If the square root does not exist, write no real solution a) 100 h) 144 b) c) 24 d) 6 e) 8 2 f) ( 9) 2 g) ) Estimate each square root to the nearest integer a) 96 b) 7 c) 578 d) 116 e) 200 f) 411 i) j) 0.25 k) l) m) 2.25 Review of Irrational Numbers & Real Numbers 9) Determine the classification(s) for each real number below. List all that apply. a. 100 b. 15 c. 4 9 d. 0 e f. 11 g. π h ) Determine whether each statement is true or false. Justify your answer. a) The sum of two rational numbers is rational b) The sum of a rational number and irrational number is rational. c) The product of a nonzero rational number and an irrational number is irrational. NJ Center for Teaching and Learning ~ 2 ~

3 Review of Irrational Numbers & Real Numbers 11) Determine the classification(s) for each real number below. List all that apply. a) 65 b) 25 c) 12 2 d) 5 e) 0 f) π 6 g) 12,85.9 h) ) Determine whether each statement is true or false. Justify your answer. a) The sum of two rational numbers is irrational b) The sum of a rational number and irrational number is irrational. c) The product of a nonzero rational number and an irrational number is rational. Properties of Exponents 1) Simplify each expression using the properties of exponents. a) g 7 g 6 l) w 7 u 9 b) h 8 h m) x 4 y 7 z c) j 2 j 4 8a n) 4 b 5 c 6 k 7 d) k 2 e) x 5 x 11 f) y 8 y 10 g) 9 0 h) 7(2 0 ) i) j) 11 (6 0 ) k) x 5 y 8 Properties of Exponents 2a b 2 c 10d 2 e 5 f 7 o) 25d e 1 f 2 p) (a 4 ) 5 q) (d 7 ) 4 r) (bc ) 2 s) (2e 2 f g 5 ) 4 t) ( 8h4 j 5 k h 2 j k ) 2 14) Simplify each expression using the properties of exponents. a) p 4 p l) c 8 d 10 b) q 7 q 4 m) e 5 f 7 g 4 r 9 c) r d) t t 4 e) u 5 u 11 f) v 8 v 10 g) 5 0 h) i) 8(4 0 ) j) (7 0 ) k) a 5 b 7 NJ Center for Teaching and Learning ~ ~ n) 9h 4 j 5 k 6 27h j 2 k 18x 1 y 5 z 7 o) 42x 4 y 1 z 2 p) (u ) 9 q) (v 5 ) 6 r) (a 4 b) s) (r 2 s 4 t 2 ) t) ( 5c d 5 e 7c 4 d 2 e ) 2

4 Like Terms Create a like term for the given term. 15) 4x 16) 1y 17) 15x 2 18) 16xy 19) x Simplify the expression if possible. 20) 7x + 8x 21) 6x + 8y + 2x 22) 15x 2 + 5x 2 2) 5x +2(x + 8) 24) -10y + 4y 25) 9(x + 5) + 7(x ) 26) 8 + (x 4)2 27) 7y + 8x + y + 2x 28) x + 2x 29) x 2 + 5x 2 0) 2x + 4x + 1) 6y y 2) 9y + 4y 2y + y ) x + 5x + x ) 8x x + 2x + 15 Like Terms Create a like term for the given term. 5) 6x 6) Y 7) 10x 2 8) 14xy 9) -5x Simplify the expression if possible. 40) 17x + 18x + 41) 6x + 8y - 2x y 42) 15x 2 + 5x 2 + 2x 4) 5x +2(x + 8) + 44) -10y + 4y 5 45) 9(x - 5) + 7(x + ) 46) 18 + (x 4)2 4 47) 7y + 8x + y + 2x ) x + 2x + x + 5x 49) 6x 2 + 5x 2 50) 12x + 14x + y 51) 6y y + 6xy + 4xy 52) 9y + 4y 2y + y + y 2 5) x + 5x + x x 54) 8x x + 2x y Evaluating Expressions Evaluate the expression for the given value 55) (2n + 1) 2 for n = 56) 2(n + 1) 2 for n = 4 57) 2n for n = 58) 4x + x for x = 5 59) (x ) for x = 7 60) 8(x + 5)(x 2) for x = 4 61) x 2 for x = 2 NJ Center for Teaching and Learning ~ 4 ~

5 62) 5x + 45 for x = 6 6) 4x for x = ) 4y + x for x = 2 and y = 65) x + 17 for x = 12 and y = ½ y 66) 6x + 8y for x = 9 and y = ¼ 67) x + (2x 8) for x = 10 68) 5(x) + 8y for x = 2 and y = 10 Evaluating Expressions Evaluate the expression for the given value 69) (2n + 1) 2 for n = 1 70) 2(n + 1) 2 for n = 71) 2n for n = 5 72) 4x + x for x = 6 7) (x ) for x = 74) 8(x + 5)(x 2) for x = 6 75) x 2 for x = 8 76) 5x + 45 for x = 77) 4x for x = ) 4y + x for x = 12 and y = 1 79) x + 17 for x = 2 and y = ½ y 80) 6x + 8y for x = 8 and y = ¾ 81) x + (2x 8) for x = 11 82) 5(x) + 8y for x = 12 and y = 5 Ordering Expressions Order the terms by the degree of the variable in each expression. 8) x x 5x 2 84) w w 8w + 5w 4 85) 60 12xy + 2x 2 7y 2 86) 4uv 8u 6 v u 2 v u 4 v 4 87) 18xy 2 x x 2 y + 8y Ordering Expressions Order the terms by the degree of the variable in each expression. 88) 11x 2 4x + 17 x 89) 2w 20w + 8w w 2 90) 1pq 19 + p 2 8q 2 91) 6 14u 4 v + 2u 5 v 4 54uv 5u v 2 92) 9 + 5y 2x + xy 2 20x 2 y NJ Center for Teaching and Learning ~ 5 ~

6 a. Rational b. Rational c. Rational, Integer d. Rational, Integer, Whole, Natural a. Rational, Integer, Whole, Natural b. Rational c. Rational, Integer, Whole d. Rational, Integer e. Rational a. 11 b. 169 c. 15 d. 289 a. 7 b. no real solution c. -17 d. no real solution e. 15 f g. 12 h. no real solution i j. 0.8 k l. no real solution m. 1.8 a. 6 b. 5 c. -15 d. - e. 11 f. -17 a. 1 b. 61 c. 25 d. 144 a. no real solution b. 25 c. -18 d. no real solution e. 8 f. -9 g. 26 h i. no real solution Answer Key 8. j. 0.5 k. no real solution l m. 1.5 a. -10 b. 6 c. -24 d. -11 e. -14 f a. Rational, Integer, Whole, Natural b. Irrational c. Rational d. Rational, Integer, Whole e. Rational f. Rational, Integer g. Irrational h. Rational, Integer, Whole, Natural 10. a. True: If two rational numbers (or fractions) are added together, then the result has to be another rational number (or fraction). For example, + 5 = = 8 = , which is still a rational number. In general, if a b + c d = ad ad+bc bd bc = bd bd, where a, b, c and d are integers, b 0, d 0, then the sum is rational. b. False: Counterexample = 1 + π cannot 2 be simplified. If you perform the sum with the decimal equivalents, = c. True: If a rational number, not equal to 0, and an irrational number are multiplied together, the result has to be irrational. For example, 2 = 2( ) =.46410, which is still irrational 11. a. Irrational b. Rational, Integer c. Rational, Integer, Whole, Natural d. Rational e. Rational, Integer, Whole f. Irrational g. Rational h. Rational, Integer 12. NJ Center for Teaching and Learning ~ 6 ~

7 a. False: Counterexample: = = 8 24 = , which is a rational number, not an irrational number. b. True: If a rational number and irrational number are added together, the result is an irrational. For example, 1 + π cannot 2 be simplified. If you perform the sum with the decimal equivalents, =.64159, which is an irrational number. c. False: For example, 2 = 2( ) =.46410, which is an irrational number. 1. a. g 1 b. h 5 c. j 6 d. k 5 e. x f. y 2 g. 1 h. 7 i. 9 j. 8 x 5 k. y 8 l. u 9 w 7 m. n. y 7 x 4 z a 4 c 4b 7 2e 6 o. 5d 5 f 5 p. a 20 q. d 28 r. b 2 c 6 s. t. 16e 8 g 20 f 12 9k 8 64h 12 j 16 a. p b. q 11 c. r 6 d. t 7 1 e. u 6 f. v 18 g. 1 h. 14 i. 8 j. 19 b 7 k. a 5 l. c 8 d 10 m. e5 g 4 f 7 j 7 n. h 7 k 9 z 9 o. 7x 5 y 4 p. u 27 q. v 0 r. a 12 b s. 27s 12 t 6 r 6 49c 14 t. 25d 14 e Multiple Answers ex:2(2x) 16. Multiple Answers ex:26y/2 17. Multiple Answers ex:(x)(5x) 18. Multiple Answers (4x)(4y) 19. Multiple Answers ex:x 2 /x x 21. 8x+8y x x y x x y+10x 28. x 29. 6x x+ 1. y 2. 12y. 7x x Multiple Answers ex: (2x) 6. Multiple Answers ex. 5y 4y 7. Multiple Answers ex. 5x(2x) 8. Multiple Answers ex. 7x(2y) 9. Multiple Answers ex. 5x 10x 40. 5x x+7y x 2 +2x 4. 7x y x x y+10x x x x+y 51. y+10xy y+y x+15-7y NJ Center for Teaching and Learning ~ 7 ~

8 x 5x 2 + x w 4 8w + w w 86. 2x 2 12xy 7y or 7y 2 12xy + 2x u 6 v 5 15u 4 v u 2 v 2 + 4uv x 7x 2 y + 18xy 2 + 8y + 81 or 8y + 18xy 2 7x 2 y x x + 11x 2 4x w 4 + 2w + 9w 2 20w p 2 1pq 8q 2 19 or 8q 2 1pq + p u 5 v 4 14u 4 v 5u v 2 54uv x 20x 2 y + xy 2 + 5y + 9 or 5y + xy 2 20x 2 y 2x + 9 NJ Center for Teaching and Learning ~ 8 ~