Algebra 2 Notes Powers, Roots, and Radicals Unit 07. a. Exponential equations can be solved by taking the nth

Algebra Notes Powers, Roots, and Radicals Unit 07 Exponents, Radicals, and Rational Number Exponents n th Big Idea: If b a, then b is the n root of a. This is written n a b. n is called the index, a is called the radicand. The square root has an implied index of. Rational exponents are another wa of expressing radicals. The radical n root of both sides. a m is the same as m n a. Exponential equations can be solved b taking the nth Objectives: N.RN.A. 1 Extend the properties of exponents to rational exponents. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/ to be the cube root of 5 because we want (5 1/ ) = 5 (1/) to hold, so (5 1/ ) must equal 5. N.RN.A. Extend the properties of exponents to rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents. A.REI.A. Reasoning with Equations and Inequalities. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions ma arise. Skills: The students will use properties of rational exponents to simplif and evaluate expressions. The student will solve equations containing radicals or rational exponents. Ex 1: 9, because 9 Ex : 8, because 8 Multiple Roots Ex : Find the real nth root(s) of a if a = 16 and n =. 16, because 16 and Roots as Rational Exponents: The 16. th n root, n a, can be written as an exponent 1 n a. a m n n a m Ex : Evaluate 1 81. 1 81, because 81. Note: This could also be written as 81. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 1 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Ex 5: Evaluate 15., because 15 5 Ex 6: Evaluate 16. 16 16 6 Ex 7: Evaluate 5. 5 15. 5 1 1 1 1 5 5 Finding Roots on the Calculator Ex 8: Use the graphing calculator to approximate 1. We will rewrite 1 with a rational exponent: 1. Tpe this into our calculator using the carrot ke for the exponent. Be sure to put the exponent in parentheses. Question: What expression is our calculator evaluating if ou do not use parentheses? Solving Equations with nth roots: To solve an equation in the form sides. 1 n 1 1 n n n n n n x a x a x a m m n n both sides to the reciprocal power: x a m x a Ex 9: Solve the equation n x 5. n x a, take the nth root of both. Note: To solve an equation in the form n m m n x a, raise Step One: Rewrite in the form n x a. x 5 x 7 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Step Two: Take the nth ( rd ) root of both sides. x 7 x Or Step Two: Take both sides to the 1 power. x x 7 1 1 x 7 Ex 10: Solve the equation x 7 81. x 7 81 Step One: Take the nth ( th ) root of both sides. (Or take both sides to the 1 power) x 7 Caution: 81 has real th roots! Step Two: Solve for x. Ex 11: Solve the equation Step One: Rewrite in the form n x 15 5x 5. a. x 7 x 10 15 5x 5 5 5 x 5 5 x 7 x Step Two: Raise both sides to the reciprocal power. 5 5 5 5 x 5 5 x Because there is no real square root of 5, this equation has no real solution. You Tr: Solve the equation x 1. Verif our answer on the graphing calculator. QOD: Can ou evaluate a radical if the radicand is negative and the index is odd? Explain. Can ou evaluate a radical if the radicand is negative and the index is even? Explain. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Sample CCSD Common Exam Practice Question(s): Write the radical expression 7 in exponential notation. A. B. C. 1 7 1 7 7 D. 7 Sample SAT Question(s): Taken from College Board online practice problems. 1. If x and are real numbers and the square of is equal to the square root of x, which of the following must be true? I. x II. x 0 III. 0 (A) I onl (B) I and II onl (C) I and III onl (D) II and III onl (E) I, II, and III x. If x 1 and x 7 (A) (B) 5 (C) (D) (E) m x, what is the value of m? Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Sample Problems: 1. Write the expression b using rational exponents. a. c. b. d. ANS: A DOK 1. Write the expression in radical form, and simplif. Round to the nearest whole number if necessar. a. ; c. ; b. d. ; ; 6 ANS: C DOK. Simplif. a. 5 c. 5 b. 156.5 d. 50 ANS: C DOK 1. Write as a radical expression. a. c. b. d. ANS: B DOK 1 5. Write two algebraic expressions for the square root of x. a. c. b. d. ANS: B DOK 1 6. Simplif. a. c. 0 b. x d. 1 ANS: B DOK 1 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 5 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 7. Write as a radical expression. a. c. b. d. ANS: A DOK 8. What is where is a nonzero integer? a. c. b. d. ANS: A DOK 9. Simplif. a. 10 c. b. 16 d. 6 ANS: C DOK 1 10. Simplif. a. 9 b. 18 ANS: A DOK 1 11. Simplif. a. 8 c. 1 b. 16 d. 56 ANS: D DOK 1 1. Simplif. a. c. 6 b. 16 d. 18 ANS: C DOK 1 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 6 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 1. What value of x makes a true statement? ANS: 5 DOK 1 1. Solve the equation, if possible. ANS: 9 DOK 1 15. Solve the equation, if possible. ANS: No solution DOK 1 16. Solve the equation, if possible. ANS: No solution DOK 1 17. Explain wh, when, is not a real number. ANS: DOK real number. in radical form. If x > 0, then. The square root of a negative number is not a 18. Rebecca tried to solve for the value of an integer with a fractional exponent. What mistake did Rebecca make? Identif her error and show the correct work. ANS: DOK Rebecca rewrote the expression using incorrect exponents. The value of should be. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 7 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 19. Use to show that. ANS: DOK 0. For all values of x such that,, and, compare when and when. Explain our reasoning. ANS: DOK If, then, so will be greater than x. If, then and will be less than x. 1. What tpe of number is if n is an even number and b < 0? ANS: DOK. Use the rules for exponents and roots to prove that = ANS: DOK Both expressions are equal to 6.. Use the rules for exponents and roots to prove =. ANS: DOK Both expressions are equal to 5. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 8 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 ESSAY 1. Are there an positive, whole number values of m and n that make the expression a real number when b < 0? Explain our reasoning and give examples. ANS: DOK Yes;, so with b < 0, the expression is a real number if the radicand is positive, or if the root is odd. So, m must be even or n must be odd. For an example of even m, number because it is a square root of a positive integer, is a real. For an example of n is odd,. Use the Power of a Power Propert to show that taking the square root of a number is the same as raising a number to the power. (Hint: Start with the fact that the square of the square root of a number is equal to that number, the two expressions equal to each other.). Then write x in a second wa using the Power of a Power Propert and set ANS: DOK The square of a square root of a number is that number itself:. Using the Power of a Power Propert: So, substituting the expression for x from the first equation into the second, we get: If ou raise both sides of the equation to the power of, ou get: Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 9 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Properties of Exponents and Rational Number Exponents Big Idea: The properties of exponents appl to real numbers with rational exponents and are used to simplif expressions that contain radicals or rational exponents. Objectives: N.RN.A. 1 Extend the properties of exponents to rational exponents. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/ to be the cube root of 5 because we want (5 1/ ) = 5 (1/) to hold, so (5 1/ ) must equal 5. N.RN.A. Extend the properties of exponents to rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Skills: The student will simplif radical expressions b appling properties of radicals. The student will use properties of rational exponents to simplif and evaluate expressions. Review: Properties of Exponents (Allow students to come up with these on their own.) We will now extend these properties for use with rational exponents. Let a and b be real numbers, and let m and n be integers. Product of Powers Propert Quotient of Powers Propert m n m n a a a m m a mn a 1 a or, a 0 n n nm a a a m Power of a Power Propert n a a mn ab a b Power of a Product Propert m m m Power of a Quotient Propert m a a, b 0 m b b m Negative Exponent Propert Zero Exponent Propert n m 1 a b a m or, a 0 a b a 0 a 1, a 0 n Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 10 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Note: The product and quotient properties for exponents can be extended to radicals, as we now know that a radical is simpl a rational exponent. Product Propert: a b a b n n n Quotient Propert: n a b n n a b 5. Ex 1: Use the properties of exponents to simplif the expression 1 1 1 Power of a Product: 5 5 Power of a Power: 1 1 1 1 5 5 10 10 Ex 1: Use the properties of exponents to simplif the expression 1. 1 1 Power of a Quotient: Rewrite with Rational Exponents: 1 Power of a Power: 1 Simplest Form of a Radical: all perfect nth powers are removed and all denominators are rationalized Ex 1: Simplif the expression 50. Step One: Factor out the perfect cube ( rd root). Step Two: Rewrite using the product propert. Step Three: Simplif b taking the cube root of the perfect cube. 15 15 5 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 11 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Ex 15: Simplif the expression 8. Step One: Multipl the numerator and denominator b a root that will make the denominator s radicand a perfect th root. (We can multipl b 7 for a product of 81, which is a perfect th root. Step Two: Simplif b taking the th root of the denominator. 7 7 5 7 7 81 5 Like Radicals: radical expressions that have the same index and same radicand Note: Onl like radicals can be added or subtracted. Ex 16: Simplif the following expression b adding/subtracting like radicals. 16 5 7 Step One: Simplif each radical b extracting an perfect cube roots. Step Two: Add/subtract the like radicals (cube roots of ). 8 5 7 5 7 7 5 5 5 Simplifing Variable Expressions Note: For the following exercises, we must assume all variables are positive. Ex 17: Simplif the expression 5 8 x. Step One: Rewrite the radicand extracting perfect th roots. x x x x x. Note: Powers of are perfect th roots 1 Step Two: Take the th root of an perfect roots. x x Step Three: Simplif. x x Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 1 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Ex 18: Simplif the expression x 18x 10 x. Step One: Multiplication (numerator). Step Two: Rewrite the radicands extracting perfect square roots. Step Three: Take the square root of an perfect roots. Step Four: Multipl and simplif. x 18x 10 x x 9 x x x x x xx xxxxx x 5 x x x Step Five: Rationalize the denominator. x x x Students should discover a shorter wa to extract perfect nth roots? Have students share their ideas. (Ex: The can divide the variable powers in the radicand b the index. The remainder is the power of the variable left in the radicand. Note This can be explained b illustrating using rational exponents.) You Tr: Simplif the expression. Assume the variable is positive. 16 8 11 QOD: When is n n x x, and when is n n x x? Sample CCSD Common Exam Practice Question(s): What is the value of the expression 1 1 1 6 6? A. B. C. 8 D. 16 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 1 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Sample Questions: 1. Simplif the expression. Assume that all variables are positive. a. c. b. d. ANS: B DOK 1. Simplif the expression. a. 79 c. 9 b. d. 7 ANS: D DOK 1. Simplif. a. c. b. d. 8 ANS: C DOK 1. Simplif. a. c. b. d. ANS: D DOK 1 5. Simplif. a. c. b. d. 81 ANS: D DOK 1 6. Simplif. a. c. 6 b. d. 8 ANS: D DOK 1 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 1 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 7. Simplif. All variables represent nonnegative numbers. a. c. b. d. ANS: B DOK 1 8. Simplif. All variables represent nonnegative numbers. a. c. b. d. ANS: A DOK 1 9. Simplif. All variables represent nonnegative numbers. a. b. c. d. ANS: B DOK 10. Simplif the expression. All variables represent nonnegative numbers. a. c. b. d. ANS: D DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 15 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 11. Simplif. All variables represent nonnegative numbers. a. c. b. d. ANS: B DOK 1. Frets are small metal bars positioned across the neck of a guitar so that the guitar can produce notes of a specific scale. To find the distance a fret should be placed from the bridge, multipl the where n is the number of notes higher than the string s root note. Determine where to place a fret to produce an A note on a C string (5 notes higher) that is 70 cm long. Round our answer to the nearest hundredth. a. 5. cm c. 58. cm b. 9. cm d. 7.9 cm ANS: A DOK 1. Simplif. All variables represent nonnegative numbers. a. x c. x (x) b. x d. x 6 ANS: A DOK 1. The surface area S of a cube with volume V is. What effect does increasing the volume of a cube b a factor of 7 have on the on the surface area? a. The surface area increases b a factor of 7. b. The surface area increases b a factor of. c. The surface area increases b a factor of. d. The surface area increases b a factor of. ANS: D DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 16 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 15. Which of the following is equivalent to for all values of x and? a. b. c. d. ANS: D DOK 16. Which expression is equivalent to for x > 0? a. b. c. d. ANS: A DOK 17. In the rectangle below, units and units. w l Part A: Find the area of the rectangle in square units. Part B: Find the ratio of the width to the length. Can ou be certain that this ratio is alwas greater than 1? Explain. a. Part A: Part B: b. Part A: ; This ratio must be greater than 1 because lengths are alwas positive. Part B: ; This ratio is less than 1 if x < 1. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 17 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 c. Part A: Part B: width. d. Part A:. This ratio must be greater than 1 because the length is longer than the Part B: ; This ratio is less than 1 if x < 1. ANS: B DOK 18. Use the power of a power propert to show that. ANS: DOK 19. When ou raise the nth root of a number k to the nth power, the result is k. Therefore,. Using the Power of a Power Propert, explain wh. ANS: DOK We know that. However, the Power of a Power Propert also tells us that. Therefore, for the Power of a Power Propert to hold,. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 18 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Big Idea: An two functions, Functions involving Powers, Roots and Radicals f x and g x, can be added, subtracted, multiplied, and divided. Functions can also be combined using composition of functions. In a composition function, the results of one function are used to evaluate a second function. The composition of f and g is f g. The Composition of two functions ma not exist. Given two functions f and g, f g( x ) is defined onl if the range of g x is a subset of the domain of f x. Objectives: F.BF.A.1b Build a function that models a relationship between two quantities. Write a function that describes a relationship between two quantities. Combine standard function tpes using arithmetic operations. For example, build a function that models the temperature of a cooling bod b adding a constant function to a decaing exponential, and relate these functions to the model. F.BF.A.1c Build a function that models a relationship between two quantities. Write a function that describes a relationship between two quantities. Compose functions. For example, if T() is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. Skills: The student will perform arithmetic operations of functions. The student will find a composition of functions. Function Operations: Let f and g be two functions, and h be the resulting function after performing the following operations. The domain of h will the intersection of the domains of f and g (all x-values that are in the domain of BOTH f and g). Addition: hx f x g x Ex 19: Let f x x and g x x 5. Find hx f x g x 5 h x x x h x x 5 and state its domain. The domain of f and g is All Real Numbers, so the domain of h is All Real Numbers. Subtraction: hx f x g x Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 19 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Ex 0: Let f x x and g x x 1. Find hx f x g x 1 h x x x h x x x 1 h x x 1 The domain of f and g are x 0, so the domain of h is x 0. Multiplication: hx f x g x Ex 1: Let f x x and h x x x 1 5 1 h x x x 1 g x x. Find hx f x g x and state its domain. and state its domain. The domain of f is All Real Numbers, and the domain of g x 0, so the domain of h is x 0. Division: f x Note: In the domain of gx h x h x, g x 0. Ex : Let f x x and g x x 5. Find hx x f and state its domain. g x h x x x 5 The domain of f is x, and the domain of g is All Real Numbers. Because gx is in the denominator, the domain of h must be restricted so that gx 0. So the domain of h is x, x 5. Composition of Two Functions: the composition h of the function f with the function g is. The domain of h is the set of all x-values such that x is in the domain of g and gx h x f g x is in the domain of f. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 0 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 1 x Ex : Let f x and g x x 1. Find f g x and state the domain. Step One: Substitute gx in for x in the function f x. f g x 1 x 1 Step Two: Determine the domain. The domain of gx is All Real Numbers. The domain of f g x is x, x x 1. Ex : Using the functions above, find g f x and state its domain. Step One: Substitute f x in for x in the function gx. 1 1 x x g f Step Two: Determine the domain. The domain of f x is x, x x 0. The domain of g f x is x, x x 0. Note: f g x g f x 1 x Ex 5: Let f x. Find f f x and state the domain. Step One: Substitute f x in for x in the function f x. f f x 1 1 x f f x x Step Two: Determine the domain. The domain of f x is x, x x 0. The domain of f f x is x, x x 0. Note: The function is restricted because the domain of f x has a domain of All Real Numbers. However, the domain of x was restricted. f f x x Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 1 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 You Tr: Let f x x and g x x. Find the following functions and state their domains. 1. f x g x. f x g x. f g x. g f x QOD: Give an example of two distinct functions f x and gx such that f g x g f x. Sample CCSD Common Exam Practice Question(s): 1. Let f x x and f x g x? A. B. C. D. x x x x x x x x g x x x. What is the difference of the two functions,. If f x x 1 and g x x, what is the product of f and 1 A. 55 B. 1 C. 10 D. 8 g?. Let f x x and g x x x. What expression is equal to A. B. C. D. x x x x x x 6x x x f g x? Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Sample SAT Question(s): Taken from College Board online practice problems. 1. The graphs of the functions f and g in the interval from x and x are shown above. Which of the following could express g in terms of f? (A) g x f x 1 (B) g x f x 1 (C) g x f x (D) g x f x 1 1 1 (E) g x f x 1. Let the function f be defined b f x 5x a, where a is a constant. If f f what is the value of a? (A) 5 (B) 0 (C) 5 (D) 10 (E) 0. Let the function f be defined b f x x 7x 10 and value of t? Grid-In 10 5 55, f t1 0. What is one possible Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Sample Problems: 1. Perform the indicated operation. Let and. a. b. c. d. ANS: A DOK 1. Given and, find. a. = c. = b. = d. = ANS: A DOK. Given and, what are and? a. ; c. ; b. ; d. ; ANS: D DOK. What is if and? a. d. b. e. c. ANS: C DOK 5. Perform the indicated operation. Let and. a. b. c. d. ANS: D DOK 1 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 6. Given and, which is the rule for? a., b., c. d.,, ANS: D DOK 1 7. If and, what is? a. c. b. d. ANS: A DOK 8. Let and. Perform the indicated operation and state the domain. ANS: ; all real numbers DOK 9. Let and. Perform the indicated operation and state the domain. ANS: ; all real numbers DOK 10. Let, and. Perform the indicated operation and state the domain. ANS: ; all real numbers DOK 11. Let and. Perform the indicated operation and state the domain. ANS: ; all real numbers except DOK 1. Let and. Perform the indicated operation and state the domain. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 5 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 ANS: ; all real numbers except 0 DOK 1. Let and. Perform the indicated operation and state the domain. ANS: ; all real numbers except 0 DOK 1. Given and, find. a. = c. = b. = d. = ANS: B DOK 15. Given and, write the composite function and state its domain. a. b. ANS: C DOK, 16. Let. Find. a. 5 c. 65 b. 15 d. 15 ANS: D DOK 1 17. Let and. Find. a. c. b. d. ANS: A DOK 18. Let and. Find. a. c. b. d. ANS: C DOK, c. d.,, Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 6 of 6 01/16/01,

Algebra Notes Powers, Roots, and Radicals Unit 07 19. Given and, what are and? a. ; c. ; b. ; d. ; ANS: C DOK 0. Which expression is equal to if and? a. d. b. e. c. ANS: D DOK 1. What is if and? a. c. b. d. ANS: B DOK. Given and what is the domain of? a. All real numbers c. b. d. ANS: D DOK. Given and, which is? a. c. b. d. ANS: B DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 7 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07. The general manager of a professional baseball team uses the function, where w is the number of wins the team had the previous season, to predict the number of season tickets the team will sell this season. The team president uses the function, where t is the number of season tickets sold, to determine the amount of mone available to acquire new plaers. Find, a function that gives the amount of mone available to acquire new plaers in terms of the number of wins in the previous season. a. b. c. d. ANS: A DOK 5. The area of a square is given b the function where x is the length of a side. When p centimeters of wire are bent to create a square, the function can be used to find the length of each side of that square. Which function gives the area of a square created b bending p centimeters of wire in terms of p? a. b. c. d. ANS: C DOK 6. A rental car costs $5 for a 1-da rental, plus $.05 per mile. The total cost (in dollars) is given b the equation where m represents the mileage. The car averages miles per gallon, and so the total mileage is given b the equation. Find, and find the total cost after using 8 gallons of gas. ANS: DOK ; $7.80 7. The function converts degrees Fahrenheit F to degrees Celsius C. The function converts degrees Celsius C to degrees Kelvin K. Find the composition of the function K with the function C and explain what it represents. ANS: DOK Kelvin K. ; The composition is a function that converts degrees Fahrenheit F to degrees Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 8 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Inverse Relations/Functions involving Powers, Roots and Radicals Big Idea: The inverse relation is the set of ordered pairs obtained b exchanging the coordinates of each ordered pair. The domain of a relation becomes the range of its inverse, and the range of the relation becomes the domain of its inverse. To determine if the original function has an inverse function it must pass the vertical and horizontal line tests showing it is both a function and one-to-one. To verif functions are inverses show that [ f g]( x) [ g f ]( x) x. The graph of a function and its inverse are smmetrical to the graph x. When a function does not have an inverse, restrict its domain in order to find an inverse function. Objectives: F.BF.B.a Build new functions from existing functions. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = x or f(x) = (x+1)/(x 1) for x 1. F.BF.B.b Build new functions from existing functions. Verif b composition that one function is the inverse of another. F.BF.B.c Build new functions from existing functions. Read values of an inverse function from a graph or a table, given that the function has an inverse. F.BF.B.d Build new functions from existing functions. Produce an invertible function from a noninvertible function b restricting the domain. Skill: The student will derive and verif inverses of functions. Inverse Relation: a mapping of the output values of a relation to its input values Ex 6: Find the inverse relation of the relation. x 1 0 1 7 1 5 11 To find the inverse relation, we will switch the input (x) values and the output () values. x 7 1 5 11 1 0 1 Note: Looking at the graph of the relation (solid points) and its inverse relation (open points), we can see that the inverse relation includes all of the points reflected over the line x. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 9 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Finding the Equation of an Inverse Relation: recall that the inverse of a relation is its reflection over the line x. Therefore, to find the equation of an inverse relation, we will reverse the x and variables and solve for. Note: If both the relation and its inverse relation are functions, then the two relations are called inverse functions. Notation for Inverse Functions: The inverse of a function f is denoted 1 f. Caution: This is not to be confused with the exponent 1!! Ex 7: Find the equation of the inverse function of f x x. Step One: Switch the x and variables. Note: Step Two: Solve for. Step Three: Write in inverse notation. f x. x x 1 x 1 1 f x x We can verif our answer using the graph of the ordered pairs, as in our first example. However, a function and its inverse have another special relationship. Verifing Inverse Functions: To verif that two functions are inverses, we must show that f f x f f x x. The function x is the identit function, so the composition of a 1 1 function and its inverse is the identit. Ex 8: Show algebraicall and graphicall that the functions f x x and 1 1 f x x are inverses. Step One: Show that 1 f f x x. 1 f f x x 1 f f x x 1 f f x x 1 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 0 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Step Two: Show that 1 f f x x. 1 f f x x 1 f f x x 1 f f x x 1 Step Three: Graph the functions and show that the are a reflection over the line In this graph, the line x is bold. We have shown that these functions are inverses. x. Ex 9: Find the inverse of the function x. x for 0 Step One: Switch the x and. x Step Two: Solve for. x Because x 0, we onl need the positive square root. x Step Three: Rewrite in inverse notation. 1 x Let s take a look at the graphs of these two functions. The are reflections of each other over the line x. Calculator Note: To graph x on its restricted domain, use parentheses after the function. What if the domain of x was not restricted? Let s take a look at the graphs. You can see that the inverse of x is not a function. B looking at a function s graph, we can see if it has an inverse function using the Horizontal Line Test. Horizontal Line Test: If a horizontal line intersects the graph of a function f not more than once, then the inverse of f is a function. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 1 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Ex 0: Determine whether x 1 is a function. If it is, determine if it has an inverse function. If it does, find the inverse function and graph to verif. Step One: Look at the graph of to verif it is a function. x 1 and use the vertical line test It passes the vertical line test, so it is a function. Step Two: Look at the graph of x 1 and use the horizontal line test to verif it has an inverse function. It passes the horizontal line test, so it has an inverse function. Step Three: Find the inverse function. 1 x 1 x 1 x1 x1 Step Four: Graph the two functions. The inverse is a function, and it is the reflection of the original function over the line x. Power Function: a function of the form b ax, where a is a real number and b is a rational number Note: As shown in the examples above, the inverse of a power function is a radical function, which will be discussed in the next section. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Graphing Inverses on the Graphing Calculator Your calculator cannot find the inverse of a function, but it can draw the inverse. Ex 1: Use a graphing calculator to graph the inverse of x. Step One: Enter the function into Y1. (For absolute value, use abs in the MATH menu.) Step Two: On the home screen choose 8:DrawInv from the Draw menu, then choose Y1 from the VARS menu. Press Enter, and the calculator will automaticall draw the inverse. Is the inverse a function? Did ou know before the calculator graphed it? You Tr: Find the inverse of the function algebraicall and graphicall. Is the inverse a function? f x x, x 0. Verif that these are inverses both QOD: What is the difference between the vertical and horizontal line tests? Explain wh each test is used in each case. Sample CCSD Common Exam Practice Question(s): Which is the inverse of the function x? A. B. C. D. x, where x 0 x, where x 0 9 x, where x 0 x, where x 0 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Sample Questions: 1. Which is the inverse of? a. c. b. d. ANS: D DOK 1. Which is the inverse of? a. c. b. d. ANS: C DOK 1. Use inverse operations to write the inverse of 5. a. 5x c. + 5 b. x + 5 d. (x + 5) ANS: D DOK. What is the inverse of the function? a. c. b. d. ANS: C DOK 5. Which is the inverse of? a. c. b. d. ANS: C DOK 6. Which is the inverse of? a. c. b. d. ANS: D DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 7. Solve to find the inverse of. State an domain restrictions. a. c. ; ; b. ANS: C DOK 1 ; 8. Find the inverse of and state its domain. a. ; c. ; b. ; d. ; ANS: A DOK 1 d. 9. Find the inverse for and the domain restrictions necessar for the inverse g to be a function. a. ; c. ; ; b. ; d. ANS: A DOK 1 ; 10. What function and domain restriction has an inverse function of? a. ; c. ; b. ANS: B DOK ; d. ; 11. Which function, or, must have a restricted domain in order for its inverse to be a function? Find the domain restriction and its inverse. a. c. ; ; b. d. ; ; ANS: A DOK 1. Which statement must be true if f and g are inverses of one another? a. b. c. d. ANS: C DOK 1 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 5 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 1 1. Determine b composition whether and are inverses. 5 a. Yes,. b. No,. ANS: A DOK 1 1. Lisa is using composition to show that f(x) = 1 x + 17 is the inverse of. Her work is shown below. Where did Lisa make her first error? a. She made an error when distributing. b. She found f(g(x)) instead of g(f(x)). c. She incorrectl substituted g(x) into the equation for x. d. Lisa did not make an errors. ANS: A DOK 1 15. Pat is using composition to show that is the inverse of. His work is shown below. What should Pat write for Step of his work? a. b. c. d. ANS: A DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 6 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 16. Which of the following pairs of functions are NOT inverses of one another? a. ; c. ; b. d. ; ; ANS: C DOK 17. A student is going to show c(x) and d(x) are inverse functions b composition. Under what conditions is it sufficient to show that c(d(x)) = x and unnecessar to also show d(c(x)) = x? a. when one of the functions has onl prime coefficients and constant terms b. when one of the functions is a radical function c. when both c(x) and d(x) are one-to-one functions d. It is never sufficient to onl show c(d(x)) = x. ANS: C DOK 18. Find the inverse of f. x 0 1 6 f(x) 0 1 5 a. b. c. d. ANS: C DOK 1 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 7 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 19. Graph. Then, write and graph the inverse. a. c. 8 8 8 8 x 8 8 x 8 8 b. d. 8 8 8 8 x 8 8 x 8 8 ANS: D DOK 0. What is? x 0 1 10 f(x) 0 6 a. 0 c. b. 10 d. ANS: A DOK 1 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 8 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 1. Make a table with the two ordered pairs that correspond to the inverse of the graphed function f. 5 1 f 5 1 1 5 x 1 5 a. c. b. d. ANS: C DOK 1. Consider the graph of and its inverse. Notice that its inverse is not a function. 10 8 6 inverse 10 8 6 6 8 10 x 6 8 f 10 Describe two different domain restrictions for invertible, without changing its range. a. or c. or b. or d. or ANS: C DOK that would make it Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 9 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07. The graph of and its inverse are shown. Notice that its inverse is not a function. f 10 8 6 10 8 6 6 8 10 x 6 8 inverse 10. Describe two different domain restrictions for that would make it invertible, without changing its range. a. or c. or b. or d. or ANS: A DOK Essa: 1. Use composition to show that is the inverse of. Show our work. ANS: If f(x) is the inverse of g(x), then f(g(x)) = x. f(x) = DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 0 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07. A person s foot size appears to be a function of his or her height, as shown in the table. Part A: List the ordered pairs that represent the inverse of the function in the table. What does the inverse of this function describe? Part B: How might the information in the function and its inverse be useful? Explain. Part C: Describe the relationship between an individual s height and foot size? Would a graph be useful in communicating this relationship? ANS: Part A: Sample answer: (9, 60), (9.5, 6), (9.9, 66), (10.5, 69), (10.8, 7), (11.5, 75), (11.7, 78) This inverse function describes the person s height as it depends on his or her foot size. Part B: Sample answer: The height function as dependent on foot size could be used, for example, b law enforcement to estimate the height of a suspect from his or her footprint. The foot size function as dependent on height could be used, for example, for fitting a uniform for a new emploee if onl the height is known. Part C: Sample answer: There appears to be a linear relationship between foot length and height for people who are full grown. A graph would make the linear nature of this relationship more easil or quickl recognizable. DOK. Draw examples of various functions and their inverses. Use our drawings to answer the following questions. Part A: Is the inverse of a function alwas a function? Wh or wh not? Part B: Describe the cases in which the inverse of a function is also a function. ANS: Drawings will var. Part A: No, some functions do not have unique -values. Part B: The original function would have to pass a vertical line test and a horizontal line test in order for its inverse to be a function. DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 1 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Special Functions Square Root, Cube Root Big Idea: A function that contains a variable inside a square root smbol is called a square root function. The domain and range is limited to values for which the function is defined. The parent function is f ( x) x. The cube root function has a parent graph of square root and cube root functions can be transformed. f ( x) x. As with other functions, Objectives: F.IF.C.7b Analze functions using different representations. Graph functions expressed smbolicall and show ke features of the graph, b hand in simple cases and using technolog for more complicated cases. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. F.BF.B. Build new functions from existing functions. Identif the effect on the graph of replacing f(x) b f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technolog. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Skills: The student will graph square root and cube root equations. The student will find the domain and range of square root and cube root equations. Radical Function: a function of the form n a x h k In the previous section, we graphed square root and cube root functions. (These are two tpes of radical functions.) The square root function is the inverse of a quadratic function, and the cube root function is the inverse of a cubic function. (These are both power functions.) We will use the parent functions Graph of x : Domain: x 0 ; Range: 0 End Behavior: As x and x, f x Y Intercept (0,0) x. Graph of x Domain: all real numbers; Range: all real numbers End Behavior: Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 As x, f x As x, f x Y Intercept (0,0) Explore: Graph the following on the graphing calculator and make note of the changes to the appropriate parent function. x x x 1 x x x x 5 x Do the same activit with cube roots in place of the square roots. Ex : Using our findings from the exploration, predict how the graph of would compare to the graph of calculator to verif our conjecture. x. Use the graphing 1 5 x 8 Sample answer: It would be reflected over the x-axis, stretched verticall, and shifted to the left and up 8. Summar of the Transformations on Square Root and Cube Root Functions: a x h k a x h k a: If a 0, the graph is reflected over the x-axis. If a 1, the graph shrinks verticall. If a 1, the graph stretches verticall. h: The graph is shifted h units horizontall. k: The graph is shifted k units verticall. Ex : Describe how the graph of x1 compares to x. Then sketch the graph and state its domain and range. h 1 and k, so the graph will be shifted left 1 and up. a 1, so the graph will not be stretched. Note: It helps to sketch the graph of the parent function first. The graph of x1 is the bold graph. Domain: x 1; Range: (Note: These can be found b looking at the graph.) Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 You Tr: Sketch the graph the function 1 in the form a x h k first.) QOD: If f x? f x x, what would the graph of f x Sample CCSD Common Exam Practice Question(s): x and state its domain and range. (Hint: Write it look like? What is the domain and range of What is the graph of x? Alg II Unit 07 Notes Powers Roots & Radicalsrev Page of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Sample Problems: 1. Graph: f(x) = 1 a. 8 7 6 5 1 8 7 6 5 1 1 1 5 6 7 8 x 5 6 7 8 c. 8 7 6 5 1 8 7 6 5 1 1 1 5 6 7 8 x 5 6 7 8 b. 8 7 6 5 1 8 7 6 5 1 1 1 5 6 7 8 x 5 6 7 8 d. 8 7 6 5 1 8 7 6 5 1 1 1 5 6 7 8 x 5 6 7 8 ANS: A DOK 1. Graph the radical function and then find the domain and range. a. c. 10 10 10 10 x 10 10 x 10 10 b. 10 d. 10 10 10 x 10 10 x 10 10 ANS: A DOK 1 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 5 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07. Using the graph of as a guide, describe the transformation and graph. a. Compress f verticall b a factor of. c. Stretch f horizontall b a factor of. 10 10 8 8 6 6 10 8 6 6 8 10 x 6 8 10 b. Compress f horizontall b a factor of. 10 8 6 10 8 6 6 8 10 x 6 8 10 d. Stretch f verticall b a factor of. 10 8 6 10 8 6 6 8 10 x 6 8 10 10 8 6 6 8 10 x 6 8 10 ANS: B DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 6 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07. Using the graph of as a guide, describe the transformation and graph. a. Stretch f verticall b a factor of and c. Compress f horizontall b a factor of translate it right units. and translate it down units. 10 10 8 8 6 6 10 8 6 6 8 10 x 6 8 10 b. Stretch f verticall b a factor of and translate it left units. 10 8 6 10 8 6 6 8 10 x 6 8 10 10 8 6 6 8 10 x 6 8 10 d. Compress f horizontall b a factor of and translate it up units. 10 8 6 10 8 6 6 8 10 x 6 8 10 ANS: A DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 7 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 5. Which is the graph of? a. 10 c. 10 10 10 x 10 10 x 10 10 b. 10 d. 10 10 10 x 10 10 x 10 10 ANS: B DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 8 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 6. Graph: = 1 ANS: DOK 1 8 7 6 5 1 8 7 6 5 1 1 5 6 7 8 x 5 6 7 8 7. Graph: f(x) = ANS: Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 9 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 8. Sketch the graph of. ANS: 6 5 1 6 5 1 1 5 6 x 5 6 9. Sketch the graph of. ANS: 6 5 1 6 5 1 1 5 6 x 5 6 10. Sketch the graph of. ANS: 6 5 1 6 5 1 1 5 6 x 5 6 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 50 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 11. Graph and identif its domain and range. Compare the graph with the graph of. ANS: The domain is and the range is. The graph of is a vertical stretch (b a factor of 6) with a reflection in the x-axis of the graph of. DOK 1. Graph and identif its domain and range. Compare the graph with the graph of. ANS: The domain is and the range is. The graph of is a vertical stretch (b a factor of ) with a reflection in the x-axis of the graph of. DOK Image: 1. Graph the function. Then state the domain and range. ANS: domain: all real numbers. range: all real numbers. DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 51 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 1. Graph Then state the domain and range. ANS: domain: all real numbers. range: all real numbers. DOK 15. Graph. Then state the domain and range. ANS: domain: range: DOK 16. Graph. Then state the domain and range. ANS: domain: range: DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 5 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 17. The velocit (in meters per second) of a moving object is given b the function, where x is the object's kinetic energ (in joules). Part A: Use a graphing calculator to graph the function. Part B: Use the graph to find the approximate kinetic energ of the object when its velocit is meters per second. Explain our method. ANS: Part A: Part B: about 00 joules; Use the trace feature to find the value of x when is. DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 5 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Solving Radical Equations Big Idea: In solving radical equations to undo an nth root, raise each side to the nth power. When solving radical equations, the result ma be a number that does not satisf the original equation. Such a number is called an extraneous solution. Objectives: (Common Core) A.REI.A. Reasoning with Equations and Inequalities. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions ma arise. A.CED.A.1 Create equations that describe numbers or relationships. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Skill: The student will solve equations containing radicals or rational exponents. Solving a Radical Equation Step One: Isolate the radical. Step Two: Raise both sides to the nth power, where n is the index of the radical. Step Three: Isolate the variable. Step Four: Check our solution in the original equation. This is crucial, as ou ma obtain extraneous solutions. Ex : Solve the equation 5 x 8. Step One: x x Step Two: x x 7 Step Three: 7 x Step Four: 5 8 5 7 8 7 5 8 The solution works in the original equation, so 7 x Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 5 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Ex 5: Solve the equation 6x 5 0 6. Step One: 6x 5 1 6x 5 7 Step Two: 6x 5 7 6x 5 9 Step Three: 6x 5 x 9 Step Four: 6 9 5 0 6 9 0 6 1 0 6 The solution does not work in the original equation. Therefore, it is an extraneous solution, and this equation has NO SOLUTION. Question: Could ou have determined earlier in the process of solving that this equation had no solution? Explain. Ex 6: Solve the equation x x. Step One: Done (radical is isolated) Step Two: x x x x x Step Three: Because this is a quadratic equation, ou ma use one of the methods for solving quadratic equations (quadratic formula, factoring, or completing the square). 0 x 5x 6 x x 0 x, This can be factored, so we will solve using this method. Step Four: 0 0 1 1 Solution Set:, Check: x x x : x x x : Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 55 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Equations with Two Radicals: To solve, we will move the radicals to opposite sides, then raise both sides to the nth power, where n is the index of the radical. Ex 7: Solve the equation x10 x 0. Step One: Move one of the radicals to the other side. x10 x Step Two: Raise both sides to the th power. x10 x x10 16x Step Three: Solve for x. 10 1x 5 7 x Step Four: 5 5 10 0 7 7 80 5 0 7 7 These are not perfect th roots, so we will check on the calculator. The solution is 5 x. 7 Solving Equations with Rational Exponents Step One: Isolate the expression with the rational exponent. Step Two: Raise both sides to the reciprocal power. Step Three: Isolate the variable. Step Four: Check our solution in the original equation. This is crucial, as ou ma obtain extraneous solutions. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 56 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Ex 8: Solve the equation x 1 8. x 18 Step One: x 1 16 Step Two: x 1 16 Step Three: x 7 Step Four: 7 1 8 8 8 16 8 The solution is x 7. Solving a Radical Equation on the Graphing Calculator: We will solve equations b graphing. You ma either graph both sides of the equation as two functions and find the x-coordinate of the point of intersection, or set the equation equal to zero and find the x-intercept of the resulting function. Ex 9: Solve the equation x x b graphing. Method 1: Graph both sides of the equation. The solution is x 8. Question: What does the -value represent in the point of intersection? Method : Set the equation equal to zero. x x 0 The solution is x 8. You Tr: Solve the equation x 5 x 6. Verif our answer on the graphing calculator. QOD: Explain using rational exponents wh raising a radical to the nth power, where n is the index of the radical, will eliminate the radical. Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 57 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Sample CCSD Common Exam Practice Question(s): What is the value of x in the equation x 16? A. x = 0 B. x = 8 C. x = D. x = 80 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 58 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 Sample Questions: 1. Solve the equation. a. c. b. d. ANS: D DOK 1. Solve. a. x = 1 c. x = 9 b. x = 11 d. x = 18 ANS: C DOK 1. Solve: a. c. no real number solutions b., -9 d. -9 ANS: C DOK 1. Solve: a. 07 b. 1 6 c. 07, 09 d. 599 6 ANS: A DOK 1 5. Solve: a. 9 b. no solution c. 9, 8 d. 8 ANS: A DOK 1 6. Solve the equation. Is the solution rational or irrational? a. ; irrational c. ; irrational b. x = ; rational d. x = 8; rational ANS: B DOK 1 7. Solve the equation a. c = 00 c. c = b. c = 0 d. c = ANS: A DOK 8. Solve the equation a. z = 6 c. z = 676 b. z = d. z = 6 ANS: C DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 59 of 6 01/16/01

Algebra Notes Powers, Roots, and Radicals Unit 07 9. Solve the equation. a. 9 8 c. 9 b. 096 d. 81 ANS: D DOK 10. Solve the equation. a. z = c. z = 5 b. z = 1 d. No solution. ANS: A DOK 11. Solve a. z = 5 c. No solution. b. z = d. z = 1 ANS: A DOK 1. The area of a rectangle is 11. The length is 7, and the width is. What is the value of x? What is the width of the rectangle? a. The value of x is 16. The width is. b. The value of x is 16. The width is 56. c. The value of x is 56. The width is 16 or 16. d. The value of x is 56. The width is 16. ANS: D DOK 1. Solve. a. No solution. c. x = 5 is not defined for. b. or d. x = 6 ANS: C DOK 1. Solve. a. or c. x = b. or d. ANS: C DOK 15. The following equation describes the number of meters, x, which must be added to a string that measures 1 meters so that a pendulum will have a complete swing (back and forth) that lasts 8 seconds. How much longer should the string be so that the complete swing of the pendulum will be 8 seconds? a. 1.1 m b. 1 m c. 10.1 m d. 1.51 m ANS: A DOK Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 60 of 6 01/16/01