Chapter 8 RADICAL EXPRESSIONS AND EQUATIONS

Name: Instructor: Date: Section: Chapter 8 RADICAL EXPRESSIONS AND EQUATIONS 8.1 Introduction to Radical Expressions Learning Objectives a Find the principal square roots and their opposites of the whole numbers from b Approximate square roots of real numbers using a calculator. c Solve applied problems involving square roots. d Identify radicands of radical expressions. e Determine whether a radical expression represents a real number. f Simplify a radical expression with a perfect-square radicand. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1 4. principal radical radicand square 1. The number c is a(n) root of a if c = a.. The square root is the positive square root. 3. The symbol is called a(n) symbol. 4. In the expression 17, 17 is called the. 0 to Objective a Find the principal square roots and their opposites of the whole numbers from 0 to 5. Find the square roots. 5. 49 5. 5. 6. 81 6. 7. 400 7. 8. 196 8. 67

Simplify. 9. 100 9. 10. 11 10. 11. 144 11. 1. 34 1. Objective b Approximate square roots of real numbers using a calculator. Use a calculator to approximate each square root. Round to three decimal places. 13. 10 13. 14. 43 14. 15. 14.65 15. 16. 1.654 16. 17. 13 0 17. 18. 345 8 18. 68

Name: Instructor: Date: Section: Objective c Solve applied problems involving square roots. 19. The attendants at a parking lot park cars in temporary spaces before the cars are taken to permanent parking stalls. The number N of such spaces needed is approximated by the formula N =.5 A, where A is the average number of arrivals during peak hours. Find the number of spaces needed when the average number of arrivals is (a) 49; (b) 60. 19. a) b) 0. The formula r = 5L can be used to approximate the speed r, in miles per hour, of a car that has left a skid mark of length L, in feet. What was the speed of a car that has left skid marks of length (a) 0 feet? (b) 100 feet? 0. a) b) Objective d Identify radicands of radical expressions. Identify the radicand. 1. 17w. 5t + 18 3. 6 xy 7 4. x x+ y 3 69

Objective e Identify whether a radical expression represents a real number. Determine whether each expression represents a real number. Write yes or no. 5. 100 5. 6. 100 6. 7. 49 7. 8. ( 64) 8. Objective f Simplify a radical expression with a perfect-square radicand. Simplify. Remember that we have assumed that radicands do not represent the square of a negative number. 9. w 9. 30. 36y 30. 31. ( ) 3p 31. 3. ( ) uv 3. 33. ( a 10) 33. 34. x + 1x+ 36 34. 70

Name: Instructor: Date: Section: Chapter 8 RADICAL EXPRESSIONS AND EQUATIONS 8. Multiplying and Simplifying with Radical Expressions Learning Objectives a Simplify radical expressions. b Simplify radical expressions where radicands are powers. c Multiply radical expressions and, if possible, simplify. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1 4. half nonnegative perfect squares radicands 1. For any radicands A and B, A B = A B.. The product of square roots is the square root of the product of the. 3. A square-root radical expression is simplified when its radicand has no factors that are. 4. To take the square root of an even power, take the exponent. Objective a Simplify radical expressions. Simplify by factoring. 5. 4 5. 6. 150 6. 7. 5t 7. 8. 17a 8. 9. 180 p 9. 71

10. 11. 5x m 10m+ 5 10. 11. 1. x x 90 + 60 + 10 1. Objective b Simplify radical expressions where radicands are powers. Simplify by factoring. 13. 14 x 13. 14. 13 t 14. 15. ( ) 6 5 y + 15. 16. 5( a + 9) 16 16. 17. 9 300b 17. 18. 10 5 30x y 18. 7

Name: Instructor: Date: Section: Objective c Multiply radical expressions and, if possible, simplify. Multiply and then, if possible, simplify by factoring. 19. 10 19. 0. 1 14 0. 1. 0y 6x 1.. 13c 6c. 3. 3t 3t 3. 4. 6xy 3x y 4. 73

5. 7 3x + 5. 6. x 7 x 7 6. 7. 4 0ab 1a b 7. 8. s t 10 1 54 6 st 8. 9. 98pq 14 pq 3 5 6 9. 30. 18xyz 150xyz 3 4 4 7 6 30. 74

Name: Instructor: Date: Section: Chapter 8 RADICAL EXPRESSIONS AND EQUATIONS 8.3 Quotients Involving Radical Expressions Learning Objectives a Divide radical expressions. b Simplify square roots of quotients. c Rationalize the denominator of a radical expression. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1 4. perfect square radicands rationalizing separately A A 1. The statement = states that the quotient of two square roots is the square root B B of the quotients of the. A A. The statement = states that we can take the square roots of the numerator and B B the denominator. 3. When we find an equivalent expression without a radical in the denominator, we are the denominator. 4. To rationalize a denominator, we can multiply by 1 under the radical to make the denominator of the radicand a(n). Objective a Divide radical expressions. Divide and simplify. 5. 80 5 5. 75

6. 5 0 6. 7. 7 11 7. 8. 3x 8x 8. 9. 99c 3 11c 9. Objective b Simplify square roots of quotients. Simplify. 10. 49 11 10. 11. 81 169 11. 76

Name: Instructor: Date: Section: 1. 50 640 1. 16 13. y 13. 14. 4x 5 14. 15. 7t t 11 5 15. Objective c Rationalize the denominator of a radical expression. Rationalize the denominator. 16. 3 5 16. 77

17. 5 8 17. 18. 9 0 18. 19. 7 19. 0. 5 y 0. 1. x 8 1.. 16 15. 78

Name: Instructor: Date: Section: 3. 5 5 3. 4. 3 4 4. 5. 11 t 5. 6. s 3 18 6. 7. 5x 3 75x 7. 79

8. 7 y xy 8. 9. 90ab 35 3 4 ab 9. 80

Name: Instructor: Date: Section: Chapter 8 RADICAL EXPRESSIONS AND EQUATIONS 8.4 Addition, Subtraction, and More Multiplication Learning Objectives a Add or subtract with radical notation, using the distributive law to simplify. b Multiply expressions involving radicals, where some of the expressions contain more than one term. c Rationalize denominators having two terms. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1. conjugates like radicals 1. have the same radicands.. Expressions such as 1 + and 1 are known as. Objective a Add or subtract with radical notation, using the distributive law to simplify. Add or subtract. Simplify by collecting like radical terms, if possible. 3. 3 5+ 8 5 3. 4. 1 7 10 7 4. 5. 3 n 15 n 5. 6. 45+ 6 5 6. 7. 80 45 7. 8. 54 + 4 8. 81

9. 3 8 18 + 00 9. 10. 3 0 4 15 + 45 10. 11. 4 54x 11. 1. 7x+ 45 + 3x+ 5 1. 13. 5n mn + n m n 6m m 3 4 5 13. 14. 1 + 5 14. Objective b Multiply expressions involving radicals, where some of the expressions contain more than one term. Multiply. 15. 6( 7 3 15. ) 8

Name: Instructor: Date: Section: 16. ( 3+ )( 5 5 16. ) 17. ( ) 6 10 + 17. 18. ( 13 + )( 13 18. ) 19. ( 3+ 17)( 3 17 19. ) w z 0. ( ) 0. 83

Objective c Rationalize denominators having two terms. Rationalize the denominator. 6 1. + 5 1.. 10 15 +. 3. 7 x 3. 4. 4+ 5 6 + y 4. 84

Name: Instructor: Date: Section: Chapter 8 RADICAL EXPRESSIONS AND EQUATIONS 8.5 Radical Equations Learning Objectives a Solve radical equations with one or two radical terms isolated, using the principle of squaring once. b Solve radical equations with two radical terms, using the principle of squaring twice. c Solve applied problems using radical equations. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1. principle of squaring radical equation 1. A(n) has variables in one or more radicands.. The states that if a = b is true, then a = b is true. Objective a Solve radical equations with one or two radical terms isolated, using the principle of squaring once. Solve. 3. x = 10 3. 4. x = 7.5 4. 5. t + = 14 5. 6. x + 3 = 1 6. 7. 7+ y 3 = 11 7. 85

8. 10 5n = 0 8. 9. 3x 5 = x 1 9. 10. y = 3 10. 11. 4t+ 7 = 4t 3 11. 1. 7+ x+ 1= x 1. 13. x 5= x+ 1 13. 14. x 4= x 4 14. 15. 3x+ 13 = x+ 1 15. 16. 5x 1+ 3= x 16. 17. x + 5 x+ = 0 17. 18. x 1 x= 5 18. 86

Name: Instructor: Date: Section: y+ y 5 3= y+ 7 19. 19. ( )( ) 0. 5x 11= 1 x 0. Objective b Solve radical equations with two radical terms, using the principle of squaring twice. Solve. Use the principle of squaring twice. 1. 5x+ 1= 1+ 4x 3 1.. 7+ 10 z = 8+ 1 z. 3. t 7 = 1+ t+ 3 3. 4. x 1= x 4 4. 87

Objective c Solve applied problems using radical equations. Solve. 5. How far to the horizon can you see through an airplane window at a height, or altitude, of 9,000 ft? Use the formula D= h. 5. 6. A person can see 50 mi to the horizon through an airplane window. How high above sea level is the airplane? Use the formula D= h. 6. 7. The formula r = 5L can be used to approximate the speed r, in miles per hour, of a car that has left a skid mark of length L, in feet. How far will a car skid at 60 mph? At 80 mph? 7. 88

Name: Instructor: Date: Section: Chapter 8 RADICAL EXPRESSIONS AND EQUATIONS 8.6 Applications with Right Triangles Learning Objectives a Given the lengths of any two sides of a right triangle, find the length of the third side. b Solve applied problems involving right triangles. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1 4. hypotenuse legs Pythagorean equation right triangle 1. A(n) is a triangle with a 90 angle.. In a right triangle, the longest side is called the. 3. In a right triangle, the sides forming the right angle are called the. 4. The equation a + b = c is called the. Objective a third side. Given the lengths of any two sides of a right triangle, find the length of the Find the length of the third side of each right triangle. Where appropriate, give both an exact answer and an approximation to three decimal places. 5. 5. 6. 6. 89

7. 7. 8. 8. In a right triangle, find the length of the side not given. Where appropriate, give both an exact answer and an approximation to three decimal places. Standard lettering has been used. 9. a = 16, b= 30 9. 10. a = 30, c= 50 10. 11. b= 3, c= 10 11. 1. a = 4, c= 1 1. 90

Name: Instructor: Date: Section: 13. c= 1, b= 11 13. 14. a = 11, b= 5 14. Objective b Solve applied problems involving right triangles. Solve. Don t forget to make a drawing. Give an exact answer and an approximation to three decimal places. 15. Students have worn a path diagonally across a large grassy rectangle between classroom buildings. The rectangular area is 00 ft long and 100 ft wide. How long is the diagonal path? 15. 16. An airplane is flying at an altitude of 3300 ft. The slanted distance to the airport is 14,600 ft. How far is the airplane horizontally from the airport? 16. 91

17. A 15-ft guy wire reaches from the top of a pole to a point on the ground 6 ft from the pole. How tall is the pole? 17. 18. Find the length of a diagonal of a square whose sides are 10 cm long. 18. 9