Roots and Radicals Chapter Questions

Transcription

1 Roots and Radicals Chapter Questions 1. 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. Explain how to take the square root of a fraction or a decimal. 6. Explain how to approximate a square root. 7. What is the difference between an irrational and rational number? 8. Explain how to convert between the different forms of a rational number. Roots and Radicals Chapter Problems Squares, Square Roots & Perfect Squares Classwork 1. A square has an area of 9 units 2. What is the side length of a square of this area? Draw a square with an area of 9 units 2. What is the square root of 9? Explain why your answers in parts (a) and (c) are the sam NJ Center for Teaching and Learning ~ 1 ~

2 2. Fill in the following table: Side Length of a square (units) Area of the square (units 2 ) Explain how the table above helps you find the square root of 121? 4. Simplify each square root. Homework 5. A square has an area of 36 units 2. What is the side length of a square of this area? NJ Center for Teaching and Learning ~ 2 ~

3 Draw a square with an area of 36 units 2. What is the square root of 36? Explain why your answers in parts (a) and (c) are the sam 6. Fill in the following table: Side Length of a square (units) Area of the square (units 2 ) Simplify each square root. NJ Center for Teaching and Learning ~ 3 ~

4 Squares of Numbers Greater Than 20 Classwork 8. Fill in the following table: Side Length of a square (units) Area of the square (units 2 ) If you compare that to the table of side lengths from 1-10, what pattern do you notice? 10. Simplify each square root. g. h. i. j. NJ Center for Teaching and Learning ~ 4 ~

5 Homework 11. Simplify each square root. g. h. i. j. Simplifying Perfect Square Radical Expressions Classwork 12. Simplify each square root. g. h. i. j. k. l. m. n. o. Homework 13. Simplify each square root. - NJ Center for Teaching and Learning ~ 5 ~

6 g. h. i. - j. k. l. m. n. o. Approximating Square Roots Classwork 14. What two integers do the following square roots fall between? 15. Draw and label a number line from 0 to 10. Place the following square roots on the number lin 16. Estimate the following square roots. g. h. NJ Center for Teaching and Learning ~ 6 ~

7 i. j. 17. Approximate the square root to the nearest integer 18. For what integer x is closest to 7.42? * 19. For what integer x is closest to 5.1? * 20. For what integer x is closest to 3.9? * Homework 21. What two integers do the following square roots fall between? 22. Draw and label a number line from 0 to 10. Place the following square roots on the number lin 23. Estimate the following square roots. g. * From Engage NY NJ Center for Teaching and Learning ~ 7 ~

8 h. i. j. 24. Approximate the square root to the nearest integer 25. For what integer x is closest to 5.2? * 26. For what integer x is closest to 6.1? * 27. For what integer x is closest to 6.9? * Rational & Irrational Numbers Classwork. 28. Circle the numbers below that are rational 3.5 π g. h i. j Given the statement If x is a rational number then is irrational. Which values of x make the statement false? * A) 15 B) 24 C) 4 D) Given the statement If x is a rational number then is irrational. Which values of x make the statement false? * A) 25 B) 49 C) 5 D) 59 * From Engage NY NJ Center for Teaching and Learning ~ 8 ~

9 Homework 31. Circle the numbers below that are irrational g. h. π i. j Given the statement If x is a rational number then is irrational. Which values of x make the statement false? * A) 36 B) 8 C) 16 D) Given the statement If x is a rational number then is irrational. Which values of x make the statement false? * A) 144 B) 12 C) 48 D)30 Converting Repeating Decimals to Fractions and Decimal Expansion Classwork 34. Write each repeating decimal as a fraction in simplest form * 35. Find the decimal expansion of the following * : * From Engage NY NJ Center for Teaching and Learning ~ 9 ~

10 Homework 36. Write each repeating decimal as a fraction in simplest form * 37. Find the decimal expansion of the following * : Properties of Exponents Classwork 38. Complete each equation for the missing value: (5 2 )(5 5 ) = 5? (12 7 )(12 3 ) = 12? (3-2 )(3 5 ) = 3? (4 9 )(4-3 ) = 4? (5 4 )(5? ) = 5 12 (10 7 )(10? )(10-6 ) = 10 3 g = 3? h = 5? i. 8 = 9? 9 j = 12? k ? = 10 3 * From Engage NY NJ Center for Teaching and Learning ~ 10 ~

11 l.? 2 3 = A rectangle has a length of mm and a width of mm. Write an expression for the area of the rectangle as a power of 5. * 40. Express the volume of a cube with a side length of inches as a power of 7. * 41. a) Write an exponential expression for the area of a rectangle with a length of meters and a width of meters. b) Evaluate the expression to find the area of the rectangl * Homework 42. Complete each equation for the missing value: (12 2 )(12 7 ) = 12? (2 5 )(2 2 ) = 2? (5-3 )(5 5 ) = 5? (15 8 )(15-5 ) = 15? (6 7 )(6? ) = 6 15 (11-6 )(11? )(11 8 ) = 11 5 g = 7? h. 6 = 11? 11 i = 3? j. 10 = 2? 6 13 k.? = l. 5? 5 6 = A rectangle has a length of mm and a width of mm. Write an expression for the area of the rectangle as a power of 4. * 44. Express the volume of a cube with a side length of inches as a power of 2. * * From Engage NY * From Engage NY NJ Center for Teaching and Learning ~ 11 ~

12 45. a) Write an exponential expression for the area of a rectangle with a length of meters and a width of meters. b) Evaluate the expression to find the area of the rectangl * Roots & Radicals Review Determine whether the given numbers are perfect squares. Circle your answer. 1) 1 Yes No 2) 8 Yes No 3) 16 Yes No 4) 25 Yes No 5) 82 Yes No Circle the simplified version of each square root: 6) ) ) NJ Center for Teaching and Learning ~ 12 ~

13 Circle whether the given number is rational or irrational 9) π rational irrational 10) rational irrational 11) ) rational irrational 12) Between what two integers does the following square root fall? 4 & 5 6 & 7 7 & 8 5 & 6 13) (4 7 )(4 3 ) = 4? ) Approximate 15) Find the missing value = 11? 16) 17) (6 7 )(6-2 ) = 6? 18) A rectangle has a length of cm and a width of cm. What is the area of the rectangle written as a power of 4? 19) Draw and label a number line from Place the following numbers on the number line: *,, * From Engage NY NJ Center for Teaching and Learning ~ 13 ~

14 20) Write as a fraction in simplest form. * 21) Write as a fraction in simplest form. * 22) Write two exponential expressions with like bases. Leave all answers in simplified exponential form. Expression 1: Expression 2: Multiply your expressions. Divide your expressions. Raise your first expression to the 5 th power. NJ Center for Teaching and Learning ~ 14 ~

15 Answer Key 1. 3 units 3 units 3 units units 2. 3 Area of a Square = Side 2 and 9 = 3 2 Side Length of a Square Area of the square 6 units 6 units (units) (units 2 ) Area = Side 2 and 36 = 6 2 Side Length of a square Area of the square 4 16 (units) (units 2 ) Since Area of a square = side 2, the square root of the area = sid So, 121 = NJ Center for Teaching and Learning ~ 15 ~

16 8. Side Length of a square (units) Area of the square (units 2 ) , Each answer in this table is 100 times greater than the corresponding answer in the other tabl (or 10 2 times greater) g. 29 h. 97 i. 31 j g. 24 h. 49 i. 50 j No real solution g. No real solution h. ½ 5 i. 7 j. ½ k. 0.8 l m n o. No real solution No real solution 1/3 g. ½ h. No real solution i. -3/4 j. 1/10 k. No real solution l m. 0.7 n o NJ Center for Teaching and Learning ~ 16 ~

17 g h i j g h i j NJ Center for Teaching and Learning ~ 17 ~

18 Rational Irrational Irrational Rational Irrational Rational g. Irrational h. Rational i. Rational j. Rational 29. c 30. a,b 31. Rational Irrational Rational Rational Rational Rational g. Irrational h. Irrational i. Rational j. Rational 32. a,c,d 33. a x y z 10xz x y z yz x y z x y z 65y 2x y z 14x x y z yz 2 2 g. 2x yz 7 yz h. i. 4 x y z 22z 4 3 3x y z 3 xy g. 2 h. 3 i. -3 j. -2 k. 5 l mm in x m NJ Center for Teaching and Learning ~ 18 ~

19 g. 4 h. 4 i. -2 j. -4 k. 4 l mm in x in 2 Review answers 1. Yes 2. No 3. Yes 4. Yes 5. No 6. B 7. C 8. D 9. Irrational 10. rational 11. irrational 12. b 13. a 14. ~ cm = =5.48 = Expressions will vary NJ Center for Teaching and Learning ~ 19 ~