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1. 1. Numerical Roots and Radicals Chapter Questions1. What are the properties of a square?2. What does taking the square root have to do with the area of a square?3. Why is it helpful to memorize perfect squares?4. What can be helpful when finding the square roots of numbers greater than 400?5. Why is it helpful to memorize perfect squares?6. Explain how to take the square root of a fraction or a decimal.7. Explain how to approximate a square root.8. What is the difference between an irrational and rational number?9. Why would we simplify a non-perfect square root instead of just estimating it?10. How do you solve an equation with perfect square and cube roots? Page 1 of 27 www.njctl.org
2. 2. Numerical Roots and Radicals Chapter ProblemsSquares, Square Roots & Perfect SquaresClasswork 21. A square has an area of 9 units . a. What is the side length of a square of this area? 2 b. Draw a square with an area of 9 units . c. What is the square root of 9? d. Explain why your answers in parts (a) and (c) are the same.2. Fill in the following table: Side Length Area of of a square the square 2 (units) (units ) 1 2 3 4 5 6 7 8 9 10 11 12 13 Page 2 of 27 www.njctl.org
3. 3. 3. Explain how the table above helps you find the square root of 121? a. √254. Simplify each square root. b. √64 c. √81 d. √49 e. √16Homework 25. A square has an area of 36 units . a. What is the side length of a square of this area? 2 b. Draw a square with an area of 36 units . c. What is the square root of 36? d. Explain why your answers in parts (a) and (c) are the same.6. Fill in the following table: Side Length Area of the of a square square 2 (units) (units ) 14 15 16 17 18 19 20 Page 3 of 27 www.njctl.org
4. 4. a. √2897. Simplify each square root. b. √400 c. √196 d. √361 e. √144Squares of Numbers Greater Than 20Classwork8. Fill in the following table: Side Length Area of the of a square square 2 (units) (units ) 10 20 30 40 50 60 70 80 90 1009. If you compare that to the table of side lengths from 1-10, what pattern do you notice? Page 4 of 27 www.njctl.org
5. 5. a. √280910. Simplify each square root. b. √7921 c. √484 d. √6400 e. √2025 f. √225 g. √841 h. √9409 i. √961 j. √4356Homework a. √504111. Simplify each square root. b. √1296 c. √8464 d. √3025 e. √3721 f. √6889 g. √576 h. √2401 i. √2500 j. √289 Page 5 of 27 www.njctl.org
6. 6. Simplifying Perfect Square Radical ExpressionsClasswork a. √2512. Simplify each square root. b. √64 c. −√81 d. √−81 e. √49 f. g. h. i. − √. 64 j. √. 0081 k. m. −√. 25 l. n. √. 0016 o. √−.04Homework a. √28913. Simplify each square root. c. √64 b. -√400 d. √361 e. √−10000 f. g. h. i. - √−.09 j. −√. 0196 k. √. 49 l. √. 0361 m. √. 25 n. o. Page 6 of 27 www.njctl.org
7. 7. Approximating Square RootsClasswork14. What two integers do the following square roots fall between? a. b. c. d. e.15. Draw and label a number line from 0 to 10. Place the following square roots on the number line. a. b. c. d. e.16. Estimate the following square roots. a. b. c. d. e. f. g. h. i. j.17. Approximate the square root to the nearest integer a. b. c. d. e. Page 7 of 27 www.njctl.org
8. 8. Homework18. What two integers do the following square roots fall between? a. b. c. d. e.19. Draw and label a number line from 0 to 10. Place the following square roots on the number line. a. b. c. d. e.20. Estimate the following square roots. a. b. c. d. e. f. g. h. i. j.21. Approximate the square root to the nearest integer a. b. c. d. e. Page 8 of 27 www.njctl.org
9. 9. Rational & Irrational NumbersClasswork22. Circle the numbers below that are rational b. √6 a. 3.5 c. π e. √10 d. f. −√49 g. √108 h. 0.25 i. j. 0.4Homework23. Circle the numbers below that are irrational. b. √7 a. c. √81 d. 6.75 √121 e. √61 f. g. √225 h. π i. j. 0.18 Page 9 of 27 www.njctl.org
10. 10. Radical Expressions Containing VariablesClasswork24. Simplify a. b. c. d. e. f. g. h.Homework25. Simplify a. b. c. d. e. f. g. h. Page 10 of 27 www.njctl.org
11. 11. Simplifying Non-Perfect Squre RadicandsClasswork26. Simplify a. b. c. d. e. f. g. h. i. j. k. l. m.Homework27. Simplify a. b. c. d. e. f. g. h. i. j. k. l. m. Page 11 of 27 www.njctl.org
12. 12. Simplifying Roots of VariablesClasswork28. Simplify a. b. c. d. e. f. g. h. i. j. k.Homework29. Simplify a. b. c. d. e. f. g. h. i. Page 12 of 27 www.njctl.org
13. 13. Properties of ExponentsClasswork30. Complete each equation for the missing value: 2 5 ? a. (5 )(5 ) = 5 7 3 ? b. (12 )(12 ) = 12 -2 5 ? c. (3 )(3 ) = 3 9 -3 ? d. (4 )(4 ) = 4 4 ? 12 e. (5 )(5 ) = 5 7 ? -6 3 f. (10 )(10 )(10 ) = 10 4 2 ? g. 3 ÷ 3 = 3 59 ? h. =5 56 95 ? i. =9 98 4 6 ? j. 12 ÷ 12 = 12 8 ? 3 k. 10 ÷ 10 = 10 2? 4 l. =2 23Homework31. Complete each equation for the missing value: 2 7 ? a. (12 )(12 ) = 12 5 2 ? b. (2 )(2 ) = 2 -3 5 ? c. (5 )(5 ) = 5 8 -5 ? d. (15 )(15 ) = 15 7 ? 15 e. (6 )(6 ) = 6 -6 ? 8 5 f. (11 )(11 )(11 ) = 11 7 3 ? g. 7 ÷ 7 = 7 1110 ? h. 6 = 11 11 7 9 ? i. 3 ÷3 =3 Page 13 of 27 www.njctl.org
14. 14. 26 ? j. =2 210 13 6 2 k. ? = 13 13 ? 6 3 l. 5 ÷5 =5Solving Equations with Perfect Square and Cube Roots a. 4 = 32Classwork32. Solve. = 28 = −25 b. d. 6 = 864 c. e. 3 = 147 a. 21 = −21Homework33. Solve. = 125 c. 7 = 252 b. d. −6 = 162 e. =4 Page 14 of 27 www.njctl.org
15. 15. Numerical roots & Radicals Multiple choice QuestionsDetermine whether the given numbers are perfect squares. Circle your answer.1) 1 Yes No2) 8 Yes No3) 16 Yes No4) 25 Yes No5) 82 Yes NoCircle the simplified version of each square root:6) √144 a. 14 b. 12 c. 72 d. 217) a. 10 b. 6 c. 0.6 d. 188) −√. 0049 a. -7 b. 0.7 c. 0.07 d. -0.07 Circle whether the given number is rational or irrational9) π rational irrational10) 0.875 rational irrational11))√39 rational irrational Page 15 of 27 www.njctl.org
16. 16. Between what two integers do the following square roots fall?12) √45 a. 4&5 b. 6&7 c. 7&8 d. 5&613)√125 a. 11 & 12 b. 12 & 13 c. 13 & 14 d. 14 & 1514)Simplify: √45 a. 40√5 b. 2√5 c. 3√5 d. 9√515)(47)(43) = 4? a. 10 b. 24 c. 4 d. 5Short Constructed Response – Write the correct answer for each question. No partial credit will begiven.16) Approximate √47 ≈ ________17) Approximate √230 ≈ ________18) Solve: 5 2 = 180 Page 16 of 27 www.njctl.org
17. 17. x3  10819) Solve: 220) 75 ___________________21) ∶ 200 _____________________22) Find the missing value 114 ÷ 116 = 11? ____________________23) = ? ________________24) (67)(6-2) = 6?25) Simplify √ ____________ Page 17 of 27 www.njctl.org
18. 18. Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given.26) Write two exponential expressions with like bases. Leave all answers in simplified exponential form. a. Expression 1: Expression 2: b. Multiply your expressions. c. Divide your expressions. th d. Raise your first expression to the 5 power. Page 18 of 27 www.njctl.org
19. 19. Answer Key1. a. 3 units b. 3 units 3 units c. 3 2 2 d. Area of a Square = Side and 9 = 32. Side Length Area of the of a Square square 2 (units) (units ) 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 23. Since Area of a square = side , the square root of the area = side. So, √121 = 11. Page 19 of 27 www.njctl.org
20. 20. 4. a. 5 b. 8 c. 9 d. 7 e. 45. a. 6 units b. 6 units 6 units c. 6 2 2 d. Area = Side and 36 = 66. Side Length Area of the of a square square 2 (units) (units ) 14 196 15 225 16 256 17 289 18 324 19 361 20 4007. a. 17 b. 20 c. 14 d. 19 e. 12 Page 20 of 27 www.njctl.org
21. 21. 8. Side Length Area of the of a square square 2 (units) (units ) 10 100 20 400 30 900 40 1600 50 2500 60 3600 70 4900 80 6400 90 8100 100 10,0009. Each answer in this table is 100 times greater than the corresponding answer in the other table. (or 2 10 times greater).10. a. 53 b. 89 c. 22 d. 80 e. 45 f. 15 g. 29 h. 97 i. 31 j. 6611. a. 71 b. 36 c. 92 d. 55 e. 61 f. 83 g. 24 h. 49 i. 50 j. 17 Page 21 of 27 www.njctl.org
22. 22. 12. a. 5 b. 8 c. -9 d. No real solution e. 7 5 f. 7 g. No real solution h. ½ 5 i. 7 j. –½ k. 0.8 l. 0.09 m. -0.5 n. 0.04 o. No real solution13. a. 17 b. -20 c. 8 d. 19 e. No real solution f. 1/3 g. ½ h. No real solution i. -3/4 j. 1/10 k. No real solution l. -0.14 m. 0.7 n. 0.19 o. 0.514. a. 8 and 9 b. 12 and 13 c. 2 and 3 d. 7 and 8 e. 10 and 1115. Page 22 of 27 www.njctl.org
23. 23. 16. a. 2.45 b. 8.37 c. 7.42 d. 3.74 e. 10.29 f. 6.4 g. 8.94 h. 8.06 i. 2.83 j. 15.2617. a. 7 b. 6 c. 8 d. 3 e. 918. a. 12 and 13 b. 3 and 4 c. 9 and 10 d. 8 and 9 e. 13 and 1419.20. a. 8.83 b. 2.65 c. 7.94 d. 5.39 e. 6.48 f. 11.75 g. 17.32 h. 12.17 i. 4.58 j. 7.2121. a. 4 b. 6 c. 4 d. 6 e. 7 Page 23 of 27 www.njctl.org
24. 24. 22. a. Rational b. Irrational c. Irrational d. Rational e. Irrational f. Rational g. Irrational h. Rational i. Rational j. Rational23. a. Rational b. Irrational c. Rational d. Rational e. Rational f. Rational g. Irrational h. Irrational i. Rational j. Rational24. 3 a. b b 3 b. b c. b b 2 d. b b 2 e. b f. b g. b4 4 h. b b25. a. x3 x 2 b. x x c. x d. x2 e. x4 f. x3 g. x x 4 h. x x Page 24 of 27 www.njctl.org
25. 25. 26. a. 5 2 b. 4 3 c. 2 5 d. 2 2 e. 5 3 f. 10 3 g. 5 5 h. 2 7 i. 3 7 j. 6 2 k. 105 3 l. 33 2 m. 66 627. a. 4 6 b. 5 6 c. 4 5 d. 2 6 e. 10 5 f. 6 3 g. 2 30 h. 7 3 i. 2 21 j. 8 5 k. 56 5 l. 120 3 m. 112 228. a. 3 x 2 y 2 z 2x x y z 4 41 z b. x 2 y 3 z 30 y c. 7 x3 y 4 z 4 yz d. 3 x3 yz 3 yz e. Page 25 of 27 www.njctl.org
26. 26. f. 2x2 y 2 z 4 7 y 2 x 2 y z 4 14 x g. 5x y 3 z 3 x h. i. x 3 y z 3 31y x 2 y3 z 4 58xz j. x y 4 z 2 17 y k.29. 2 4 2 a. 2 x y z 10 xz x3 y3 z 3 yz b. 2 x2 y3 z 2 6 c. x3 y 2 z 3 65y d. 2x y z 14 x e. x 2 y 2 z 3 10 yz f. g. 2 x 2 yz 2 7 yz x y 4 z 22 z h. 3x4 y 3 z 3xy i.30. a. 7 b. 10 c. 3 d. 6 e. 8 f. 2 g. 2 h. 3 i. -3 j. -2 k. 5 l. 731. a. 9 b. 7 c. 2 d. 3 e. 8 f. 3 Page 26 of 27 www.njctl.org
27. 27. g. 4 h. 4 i. -2 j. -4 k. 4 l. 932. a. 2 b. ±14 c. -5 d. ±12 e. ±733. a. -1 b. ±25 c. ±6 d. -3 e. 4 20. 5x7y√3Review answers 1. Yes 10. rational 19. -6 21. 10xyz2√2 2. No 11. irrational 3. Yes 12. b 4. Yes 13. a 22. -2 5. No 14. c 23. 4 6. B 15. a 24. 5 7. C 16. 7 25. x5 8. D 17. 15 26. Expressions will vary 9. Irrational 18. 6 Page 27 of 27 www.njctl.org