Rational Exponents and Radical Functions

Transcription

1 Rational Eponents and Radical Functions.1 nth Roots and Rational Eponents. Properties of Rational Eponents and Radicals. Graphing Radical Functions. Solving Radical Equations and Inequalities. Performing Function Operations. Inverse of a Function SEE the Big Idea Hull Speed (p. ) White Rhino (p. ) Concert (p. 0) Mars Rover (p. 0) Constellations (p. 0) Mathematical Thinking: Mathematicall proficient students can appl the mathematics the know to solve problems arising in everda life, societ, and the workplace.

2 Maintaining Mathematical Proficienc Properties of Integer Eponents (A.11.B) Eample 1 Simplif the epression. = + = 7 Eample Simplif the epression ( s t ). ( s t ) Product of Powers Propert Add eponents. = 7 Quotient of Powers Propert = Subtract eponents. = (s ) t = (s ) t = s t Simplif the epression. Power of a Quotient Propert Power of a Product Propert Power of a Power Propert 1.. n n... ( w z ). ( m7 m z m ) Rewriting Literal Equations (A.1.E) Eample Solve the literal equation = 10 for. = 10 + = 10 + = 10 + = 10 + Solve the literal equation for. Write the equation. Add to each side. Simplif. = Simplif. Divide each side b = 8. 1 = = = = = 1 1. ABSTRACT REASONING Is the order in which ou appl properties of eponents important? Eplain our reasoning. 87

3 Mathematical Thinking Using Technolog to Evaluate Roots Core Concept Evaluating Roots with a Calculator Eample Square root: = 8 Cube root: = Fourth root: = Fifth root: = Mathematicall profi cient students select tools, including real objects, manipulatives, paper and pencil, and technolog as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. (A.1.C) square root cube root fourth root fifth root () () () () 8 Approimating Roots Evaluate each root using a calculator. Round our answer to two decimal places. a. 0 b. 0 c. 0 d. 0 a Round down. b. c. d Round down. Round up. Round up. (0) (0) (0) (0) Monitoring Progress 1. Use the Pthagorean Theorem to find the eact lengths of a, b, c, and d in the figure.. Use a calculator to approimate each length to the nearest tenth of an inch. 1 in. 1 in. 1 in.. Use a ruler to check the reasonableness of our answers. 1 in. a b c d 1 in. 88 Chapter Rational Eponents and Radical Functions

4 .1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS Preparing for A.7.G nth Roots and Rational Eponents Essential Question How can ou use a rational eponent to represent a power involving a radical? Previousl, ou learned that the nth root of a can be represented as n a = a 1/n Definition of rational eponent for an real number a and integer n greater than 1. MAKING MATHEMATICAL ARGUMENTS To be proficient in math, ou need to understand and use stated definitions and previousl established results. Eploring the Definition of a Rational Eponent Work with a partner. Use a calculator to show that each statement is true. a. 9 = 9 1/ b. = 1/ c. 8 = 8 1/ d. = 1/ e. 1 = 1 1/ f. Writing Epressions in Rational Eponent Form 1 = 1 1/ Work with a partner. Use the definition of a rational eponent and the properties of eponents to write each epression as a base with a single rational eponent. Then use a calculator to evaluate each epression. Round our answer to two decimal places. Sample ( ) = ( 1/ ) = /. ^(/).1981 a. ( ) b. ( ) c. ( 9 ) d. ( 10 ) e. ( 1 ) f. ( 7 ) Writing Epressions in Radical Form Work with a partner. Use the properties of eponents and the definition of a rational eponent to write each epression as a radical raised to an eponent. Then use a calculator to evaluate each epression. Round our answer to two decimal places. Sample / = ( 1/ ) = ( ).9 a. 8 / b. / c. 1 / d. 10 / e. 1 / f. 0 / Communicate Your Answer. How can ou use a rational eponent to represent a power involving a radical?. Evaluate each epression without using a calculator. Eplain our reasoning. a. / b. / c. / d. 9 / e. 1 / f. 100 / Section.1 nth Roots and Rational Eponents 89

5 .1 Lesson What You Will Learn Core Vocabular nth root of a, p. 90 inde of a radical, p. 90 Previous square root cube root eponent Find nth roots of numbers. Evaluate epressions with rational eponents. Solve equations using nth roots. nth Roots You can etend the concept of a square root to other tpes of roots. For eample, is a cube root of 8 because = 8. In general, for an integer n greater than 1, if b n = a, then b is an nth root of a. An nth root of a is written as n a, where n is the inde of the radical. You can also write an nth root of a as a power of a. If ou assume the Power of a Power Propert applies to rational eponents, then the following is true. (a 1/ ) = a (1/) = a 1 = a (a 1/ ) = a (1/) = a 1 = a (a 1/ ) = a (1/) = a 1 = a Because a 1/ is a number whose square is a, ou can write a = a 1/. Similarl, a = a 1/ and a = a 1/. In general, n a = a 1/n for an integer n greater than 1. UNDERSTANDING MATHEMATICAL TERMS When n is even and a > 0, there are two real roots. The positive root is called the principal root. Core Concept Real nth Roots of a Let n be an integer (n > 1) and let a be a real number. n is an even integer. a < 0 No real nth roots a = 0 One real nth root: n 0 = 0 a > 0 Two real nth roots: ± n a = ±a 1/n n is an odd integer. a < 0 One real nth root: n a = a 1/n a = 0 One real nth root: n 0 = 0 a > 0 One real nth root: n a = a 1/n Finding nth Roots Find the indicated real nth root(s) of a. a. n =, a = 1 b. n =, a = 81 a. Because n = is odd and a = 1 < 0, 1 has one real cube root. Because ( ) = 1, ou can write 1 = or ( 1) 1/ =. b. Because n = is even and a = 81 > 0, 81 has two real fourth roots. Because = 81 and ( ) = 81, ou can write ± 81 = ± or ±81 1/ = ±. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the indicated real nth root(s) of a. 1. n =, a = 1. n =, a = 9. n =, a = 1. n =, a = 90 Chapter Rational Eponents and Radical Functions

6 Rational Eponents A rational eponent does not have to be of the form 1. Other rational numbers, such n as and 1, can also be used as eponents. Two properties of rational eponents are shown below. Core Concept Rational Eponents Let a 1/n be an nth root of a, and let m be a positive integer. a m/n = (a 1/n ) m = ( n a ) m a m/n 1 = a m/n = 1 (a 1/n ) m = 1 ( n a ) m, a 0 Evaluating Epressions with Rational Eponents Evaluate each epression. a. 1 / b. / COMMON ERROR Be sure to use parentheses to enclose a rational eponent: 9^(1/) 1.. Without them, the calculator evaluates a power and then divides: 9^1/ = 1.8. Rational Eponent Form a. 1 / = (1 1/ ) = = b. / 1 = / = 1 ( 1/ ) = 1 = 1 8 Radical Form 1 / = ( 1 ) = = / 1 = / = 1 ( ) = 1 = 1 8 When using a calculator to approimate an nth root, ou ma want to rewrite the nth root in rational eponent form. Approimating Epressions with Rational Eponents Evaluate each epression using a calculator. Round our answer to two decimal places. a. 9 1/ b. 1 /8 c. ( 7 ) a. 9 1/ 1. b. 1 /8. c. Before evaluating ( 7 ), rewrite the epression in rational eponent form. ( 7 ) = 7 /.0 9^(1/) ^(/8) ^(/) Monitoring Progress Evaluate the epression without using a calculator. Help in English and Spanish at BigIdeasMath.com. /. 9 1/ / /8 Evaluate the epression using a calculator. Round our answer to two decimal places when appropriate. 9. / 10. / 11. ( 1 ) 1. ( 0 ) Section.1 nth Roots and Rational Eponents 91

7 Solving Equations Using nth Roots To solve an equation of the form u n = d, where u is an algebraic epression, take the nth root of each side. Solving Equations Using nth Roots Find the real solution(s) of (a) = 18 and (b) ( ) = 1. COMMON ERROR When n is even and a > 0, be sure to consider both the positive and negative nth roots of a. a. = 18 Write original equation. = Divide each side b. = Take fifth root of each side. = Simplif. The solution is =. b. ( ) = 1 Write original equation. = ± 1 Take fourth root of each side. = ± 1 Add to each side. = + 1 or = 1 Write solutions separatel..1 or 0.8 Use a calculator. The solutions are.1 and 0.8. Real-Life Application A hospital purchases an ultrasound machine for $0,000. The hospital epects the useful life of the machine to be 10 ears, at which time its value will have depreciated to $8000. The hospital uses the declining balances method for depreciation, so the annual depreciation rate r (in decimal form) is given b the formula r = 1 ( S C) 1/n. In the formula, n is the useful life of the item (in ears), S is the salvage value (in dollars), and C is the original cost (in dollars). What annual depreciation rate did the hospital use? The useful life is 10 ears, so n = 10. The machine depreciates to $8000, so S = The original cost is $0,000, so C = 0,000. So, the annual depreciation rate is r = 1 ( S C) 1/n = 1 ( 0,000) /10 = 1 ( ) 1/ The annual depreciation rate is about 0.17, or 1.7%. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the real solution(s) of the equation. Round our answer to two decimal places when appropriate = 1. 1 = 1 1. ( + ) = 1 1. ( ) = WHAT IF? In Eample, what is the annual depreciation rate when the salvage value is $000? 9 Chapter Rational Eponents and Radical Functions

8 .1 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY Rewrite the epression a s/t in radical form. Then state the inde of the radical.. COMPLETE THE SENTENCE For an integer n greater than 1, if b n = a, then b is a(n) of a.. WRITING Eplain how to use the sign of a to determine the number of real fourth roots of a and the number of real fifth roots of a.. WHICH ONE DOESN T BELONG? Which epression does not belong with the other three? Eplain our reasoning. (a 1/n ) m ( n a ) m ( m a ) n a m/n Monitoring Progress and Modeling with Mathematics In Eercises 10, find the indicated real nth root(s) of a. (See Eample 1.). n =, a = 8. n =, a = 1 7. n =, a = 0 8. n =, a = 9. n =, a = 10. n =, a = 79 In Eercises 11 18, evaluate the epression without using a calculator. (See Eample.) 11. 1/ / 1. / / 1. ( ) 1/ 1. ( ) / / / ERROR ANALYSIS In Eercises 19 and 0, describe and correct the error in evaluating the epression / = (7 1/ ) = 9 = 81 USING STRUCTURE In Eercises 1, match the equivalent epressions. Eplain our reasoning. 1. ( ) A. 1/. ( ) B. /. 1 C. 1/. D. / In Eercises, evaluate the epression using a calculator. Round our answer to two decimal places when appropriate. (See Eample.)., / / 9. 0,7 / 0. 8 / 1. ( 187 ). ( 8 ) 8 MATHEMATICAL CONNECTIONS In Eercises and, find the radius of the figure with the given volume.. V = 1 ft. V = 1 cm 0. / = ( ) = r r 9 cm = Section.1 nth Roots and Rational Eponents 9

9 In Eercises, find the real solution(s) of the equation. Round our answer to two decimal places when appropriate. (See Eample.). = 1. = ( + 10) = ( ) = 9. = = 1. + = =. 1 = 7. 1 =. MODELING WITH MATHEMATICS When the average price of an item increases from p 1 to p over a period of n ears, the annual rate of inflation r (in decimal form) is given b r = ( p 1/n p 1 ) 1. Find the rate of inflation for each item in the table. (See Eample.) Item Price in 191 Price in 01 Potatoes (lb) $0.01 $0.7 Ham (lb) $0.1 $.9 Eggs (dozen) $0.7 $1.9. HOW DO YOU SEE IT? The graph of = n is shown in red. What can ou conclude about the value of n? Determine the number of real nth roots of a. Eplain our reasoning. = a 7. NUMBER SENSE Between which two consecutive integers does 1 lie? Eplain our reasoning. 8. THOUGHT PROVOKING In 119, Johannes Kepler published his third law, which can be given b d = t, where d is the mean distance (in astronomical units) of a planet from the Sun and t is the time (in ears) it takes the planet to orbit the Sun. It takes Mars 1.88 ears to orbit the Sun. Graph a possible location of Mars. Justif our answer. (The diagram shows the Sun at the origin of the -plane and a possible location of Earth.) (1, 0) Not drawn to scale 9. PROBLEM SOLVING A weir is a dam that is built across a river to regulate the flow of water. The flow rate Q (in cubic feet per second) can be calculated using the formula Q =.7 h /, where is the length (in feet) of the bottom of the spillwa and h is the depth (in feet) of the water on the spillwa. Determine the flow rate of a weir with a spillwa that is 0 feet long and has a water depth of feet. spillwa h 0. REPEATED REASONING The mass of the particles that a river can transport is proportional to the sith power of the speed of the river. A certain river normall flows at a speed of 1 meter per second. What must its speed be in order to transport particles that are twice as massive as usual? 10 times as massive? 100 times as massive? Maintaining Mathematical Proficienc Simplif the epression. Write our answer using onl positive eponents. (Skills Review Handbook) (z ). ( ) Write the number in standard form. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons 9 Chapter Rational Eponents and Radical Functions

10 . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.7.G Properties of Rational Eponents and Radicals Essential Question How can ou use properties of eponents to simplif products and quotients of radicals? Reviewing Properties of Eponents Work with a partner. Let a and b be real numbers. Use the properties of eponents to complete each statement. Then match each completed statement with the propert it illustrates. Statement Propert a. a =, a 0 A. Product of Powers b. (ab) = B. Power of a Power c. (a ) = C. Power of a Product d. a a = D. Negative Eponent e. ( a b ) =, b 0 E. Zero Eponent f. a =, a 0 F. Quotient of Powers a SELECTING TOOLS To be proficient in math, ou need to consider the tools available to help ou check our answers. For instance, the following calculator screen shows that and 8 are equivalent. ( ())( ()) (8) g. a 0 =, a 0 G. Power of a Quotient Simplifing Epressions with Rational Eponents Work with a partner. Show that ou can appl the properties of integer eponents to rational eponents b simplifing each epression. Use a calculator to check our answers. a. / / b. 1/ / c. ( / ) d. (10 1/ ) e. 8/ 8 1/ f. 7/ 7 / Simplifing Products and Quotients of Radicals Work with a partner. Use the properties of eponents to write each epression as a single radical. Then evaluate each epression. Use a calculator to check our answers. a. 1 b. c. 7 d. 98 e. 10 Communicate Your Answer f.. How can ou use properties of eponents to simplif products and quotients of radicals?. Simplif each epression. a. 7 b. 0 1 c. ( 1/ 11/ ) Section. Properties of Rational Eponents and Radicals 9

11 . Lesson What You Will Learn Core Vocabular simplest form of a radical, p. 97 conjugate, p. 98 like radicals, p. 98 Previous properties of integer eponents rationalizing the denominator absolute value COMMON ERROR When ou multipl powers, do not multipl the eponents. For eample, 10. Use properties of rational eponents to simplif epressions with rational eponents. Use properties of radicals to simplif and write radical epressions in simplest form. Properties of Rational Eponents The properties of integer eponents that ou have previousl learned can also be applied to rational eponents. Core Concept Properties of Rational Eponents Let a and b be real numbers and let m and n be rational numbers, such that the quantities in each propert are real numbers. Propert Name Definition Eample Product of Powers Power of a Power Power of a Product Negative Eponent Zero Eponent Quotient of Powers Power of a Quotient a m a n = a m + n (a m ) n = a mn (ab) m = a m b m a m = 1 a m, a 0 a 0 = 1, a 0 a m a n = am n, a 0 ( a b m, b 0 b) m = am 1/ / = (1/ + /) = = ( / ) = (/ ) = = (1 9) 1/ = 1 1/ 9 1/ = = 1 1/ 1 = 1/ = = 1 / 1/ = (/ 1/) = = 1 ( 7 1/ = ) 71/ 1/ = 9 Chapter Rational Eponents and Radical Functions Using Properties of Eponents Use the properties of rational eponents to simplif each epression. a. 7 1/ 7 1/ = 7 (1/ + 1/) = 7 / b. ( 1/ 1/ ) = ( 1/ ) ( 1/ ) = (1/ ) (1/ ) = 1 / = / c. ( ) 1/ = [( ) ] 1/ = (1 ) 1/ = 1 [ ( 1/)] = 1 1 = 1 1 d. 1 = 1/ ) 1/ e. ( 1/ 1/ = (1 1/) = / = [ ( 1/ ) Monitoring Progress Simplif the epression. ] = (7 1/ ) = 7 (1/ ) = 7 / 1. / 1/. Help in English and Spanish at BigIdeasMath.com 1/. ( 01/ 1/ ). ( 1/ 7 1/ )

12 Simplifing Radical Epressions The Power of a Product and Power of a Quotient properties can be epressed using radical notation when m = 1 for some integer n greater than 1. n Core Concept Properties of Radicals Let a and b be real numbers and let n be an integer greater than 1. Propert Name Definition Eample Product Propert Quotient Propert n a b = n a n b n a a n b =, b 0 n b = 8 = 1 = 1 = 81 = Using Properties of Radicals Use the properties of radicals to simplif each epression. a = 1 18 = 1 = Product Propert of Radicals b. 80 = 80 = 1 = Quotient Propert of Radicals A radical with inde n is in simplest form when these three conditions are met. No radicands have perfect nth powers as factors other than 1. No radicands contain fractions. No radicals appear in the denominator of a fraction. To meet the last two conditions, rationalize the denominator. This involves multipling both the numerator and denominator b an appropriate form of 1 that creates a perfect nth power in the denominator. Write each epression in simplest form. Writing Radicals in Simplest Form a. 1 b. 7 8 a. 1 = 7 Factor out perfect cube. = 7 Product Propert of Radicals = Simplif. 7 7 b. = = 8 = Make denominator a perfect fifth power. Product Propert of Radicals Simplif. Section. Properties of Rational Eponents and Radicals 97

13 For a denominator that is a sum or difference involving square roots, multipl both the numerator and denominator b the conjugate of the denominator. The epressions a b + c d and a b c d are conjugates of each other, where a, b, c, and d are rational numbers. 1 Write in simplest form Writing Radicals in Simplest Form 1 = + The conjugate of + is. = 1 ( ) ( ) Sum and Difference Pattern = Simplif. Radical epressions with the same inde and radicand are like radicals. To add or subtract like radicals, use the Distributive Propert. Adding and Subtracting Like Radicals and Roots Simplif each epression. a b. (8 1/ ) + 10(8 1/ ) c. a = (1 + 7) 10 = 8 10 b. (8 1/ ) + 10(8 1/ ) = ( + 10)(8 1/ ) = 1(8 1/ ) c. = 7 = = ( 1) = Monitoring Progress Simplif the epression Help in English and Spanish at BigIdeasMath.com (9 / ) + 8(9 / ) The properties of rational eponents and radicals can also be applied to epressions involving variables. Because a variable can be positive, negative, or zero, sometimes absolute value is needed when simplifing a variable epression. Rule Eample 98 Chapter Rational Eponents and Radical Functions When n is odd n n = 7 7 = and 7 ( ) 7 = When n is even n n = = and ( ) = Absolute value is not needed when all variables are assumed to be positive.

14 Simplifing Variable Epressions STUDY TIP You do not need to take the absolute value of because is being squared. Simplif each epression. a. b. 8 a. = ( ) = ( ) = b. 8 = 8 = ( ) = Writing Variable Epressions in Simplest Form COMMON ERROR You must multipl both the numerator and denominator of the fraction b so that the value of the fraction does not change. Write each epression in simplest form. Assume all variables are positive. a. a 8 b 1 c b. a. a 8 b 1 c = a a b 10 b c b. 8 8 c. 11/ / z Factor out perfect fifth powers. = a b 10 c a b Product Propert of Radicals = ab c a b Simplif. = 8 = 9 = c. 11/ / z = 7 (1 /) 1/ z ( ) = 7 1/ 1/ z Make denominator a perfect cube. Product Propert of Radicals Simplif. Adding and Subtracting Variable Epressions Perform each indicated operation. Assume all variables are positive. a. + b. 1 z z z a. + = ( + ) = 11 b. 1 z z z = 1z z z z = (1z z) z = 9z z Monitoring Progress Section. Properties of Rational Eponents and Radicals 99 Help in English and Spanish at BigIdeasMath.com Simplif the epression. Assume all variables are positive. 1. 7q / 1/ 1/ 1. 9w w w

15 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING How do ou know when a radical epression is in simplest form?. WHICH ONE DOESN T BELONG? Which radical does not belong with the other three? Eplain our reasoning Monitoring Progress and Modeling with Mathematics In Eercises 1, use the properties of rational eponents to simplif the epression. (See Eample 1.). (9 ) 1/. (1 ) 1/. 1/ / 7. ( 8 1/ 10 ) 8. ( 9 1/ ) 9. ( / 1/ ) ( 1/ / ) 1/ 11. / 1 / / 1. 9 /8 9 7/8 7 / In Eercises 1 0, use the properties of radicals to simplif the epression. (See Eample.) In Eercises 9, write the epression in simplest form. (See Eample.) In Eercises 7, simplif the epression. (See Eample.) (11 1/ ) + 9(11 1/ ) 0. 1(8 / ) (8 / ) In Eercises 1 8, write the epression in simplest form. (See Eample.) ( 1/ ) ( 1/ ). ( 1/ ) + (0 1/ ) 7. ERROR ANALYSIS Describe and correct the error in simplifing the epression = ( + ) = 8 = 8 8 = 8 = 1 00 Chapter Rational Eponents and Radical Functions

16 8. MULTIPLE REPRESENTATIONS Which radical epressions are like radicals? A ( /9 ) / B C E D ( ) F In Eercises 9, simplif the epression. (See Eample.) r t m k n. 1 1z g. h h 8 n p 7 n p 1. ERROR ANALYSIS Describe and correct the error in simplifing the epression. h 1 g h = 1 g = (h ) g = h g In Eercises 70, perform the indicated operation. Assume all variables are positive. (See Eample 8.) z z 7. 7/ 7/ 8. 7 m 7 + m 7/ 9. 1w 10 + w w 70. (p 1/ p 1/ ) 1p MATHEMATICAL CONNECTIONS In Eercises 71 and 7, find simplified epressions for the perimeter and area of the given figure. 71. / 7. 1/ 1/ 7. MODELING WITH MATHEMATICS The optimum diameter d (in millimeters) of the pinhole in a pinhole camera can be modeled b d = 1.9[(. 10 ) ] 1/, where is the length (in millimeters) of the camera bo. Find the optimum pinhole diameter for a camera bo with a length of 10 centimeters. pinhole film. OPEN-ENDED Write two variable epressions involving radicals, one that needs absolute value in simplifing and one that does not need absolute value. Justif our answers. In Eercises 7, write the epression in simplest form. Assume all variables are positive. (See Eample 7.) 7. 81a 7 b 1 c r s 9 t 7 10m 9. n w w w 1. v 7 v 18w 1/ v / 7w / v 1/. 7 / / z / 1/ 1/ tree 7. MODELING WITH MATHEMATICS The surface area S (in square centimeters) of a mammal can be modeled b S = km /, where m is the mass (in grams) of the mammal and k is a constant. The table shows the values of k for different mammals. Mammal Rabbit Human Bat Value of k a. Find the surface area of a bat whose mass is grams. b. Find the surface area of a rabbit whose mass is. kilograms (. 10 grams). c. Find the surface area of a human whose mass is 9 kilograms. Section. Properties of Rational Eponents and Radicals 01

17 7. MAKING AN ARGUMENT Your friend claims it is not possible to simplif the epression because it does not contain like radicals. Is our friend correct? Eplain our reasoning. 7. PROBLEM SOLVING The apparent magnitude of a star is a number that indicates how faint the star is in relation to other stars. The epression.1m1 tells m.1 how man times fainter a star with apparent magnitude m 1 is than a star with apparent magnitude m. Star Apparent magnitude Constellation Vega 0.0 Lra Altair 0.77 Aquila Deneb 1. Cgnus a. How man times fainter is Altair than Vega? b. How man times fainter is Deneb than Altair? c. How man times fainter is Deneb than Vega? Deneb Cgnus Altair Vega Lra Aquila 77. CRITICAL THINKING Find a radical epression for the perimeter of the triangle inscribed in the square shown. Simplif the epression. 78. HOW DO YOU SEE IT? Without finding points, match the functions f() = and g() = with their graphs. Eplain our reasoning. A B REWRITING A FORMULA You have filled two round balloons with water. One balloon contains twice as much water as the other balloon. a. Solve the formula for the volume of a sphere, V = πr, for r. b. Substitute the epression for r from part (a) into the formula for the surface area of a sphere, S = πr. Simplif to show that S = (π) 1/ (V) /. c. Compare the surface areas of the two water balloons using the formula in part (b). 80. THOUGHT PROVOKING Determine whether the epressions ( ) 1/ and ( 1/ ) are equivalent for all values of. 81. DRAWING CONCLUSIONS Substitute different combinations of odd and even positive integers for m and n in the epression n m. When ou cannot assume is positive, eplain when absolute value is needed in simplifing the epression. 8 Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Identif the focus, directri, and ais of smmetr of the parabola. Then graph the equation. (Section.) 8. = 8. = 8. = Write a rule for g. Describe the graph of g as a transformation of the graph of f. (Section.7) 8. f() =, g() = f() 8. f() =, g() = f() 87. f() =, g() = f( ) 88. f() = +, g() = f() 0 Chapter Rational Eponents and Radical Functions

18 . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..C A..E A..A A.7.I Graphing Radical Functions Essential Question How can ou identif the domain and range of a radical function? Identifing Graphs of Radical Functions Work with a partner. Match each function with its graph. Eplain our reasoning. Then identif the domain and range of each function. a. f() = b. f() = c. f() = d. f() = A. B. C. D. Identifing Graphs of Transformations Work with a partner. Match each transformation of f() = with its graph. Eplain our reasoning. Then identif the domain and range of each function. a. g() = + b. g() = c. g() = + d. g() = + A. B. C. D. ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math, ou need to look closel to discern a pattern or structure. Communicate Your Answer. How can ou identif the domain and range of a radical function?. Use the results of Eploration 1 to describe how the domain and range of a radical function are related to the inde of the radical. Section. Graphing Radical Functions 0

19 . Lesson What You Will Learn Core Vocabular radical function, p. 0 Previous transformations parabola circle STUDY TIP A power function has the form = a b, where a is a real number and b is a rational number. Notice that the parent square root function is a power function, where a = 1 and b = 1. Graph radical functions. Write radical functions. Graph parabolas and circles. Graphing Radical Functions A radical function contains a radical epression with the independent variable in the radicand. When the radical is a square root, the function is called a square root function. When the radical is a cube root, the function is called a cube root function. Core Concept Parent Functions for Square Root and Cube Root Functions The parent function for the famil of The parent function for the famil of square root functions is f() =. cube root functions is f() =. (0, 0) f() = (1, 1) Domain: 0, Range: 0 (0, 0) ( 1, 1) f() = (1, 1) Domain and range: All real numbers Graphing Radical Functions ANALYZING MATHEMATICAL RELATIONSHIPS Eample 1(a) uses -values that are multiples of so that the radicand is an integer. Graph each function. Identif the domain and range of each function. a. f() = 1 b. g() = a. Make a table of values and sketch the graph f() = 1 The radicand of a square root must be nonnegative. So, the domain is 0. The range is 0. b. Make a table of values and sketch the graph g() = The radicand of a cube root can be an real number. So, the domain and range are all real numbers. 0 Chapter Rational Eponents and Radical Functions

20 In Eample 1, notice that the graph of f is a horizontal stretch of the graph of the parent square root function. The graph of g is a vertical stretch and a reflection in the -ais of the graph of the parent cube root function. You can transform graphs of radical functions in the same wa ou transformed graphs of functions previousl. Core Concept Transformation f() Notation Eamples Horizontal Translation g() = units right f( h) Graph shifts left or right. g() = + units left Vertical Translation Graph shifts up or down. f() + k g() = + 7 g() = 1 7 units up 1 unit down Reflection Graph flips over - or -ais. f( ) f() g() = g() = in the -ais in the -ais Horizontal Stretch or Shrink Graph stretches awa from or shrinks toward -ais. Vertical Stretch or Shrink Graph stretches awa from or shrinks toward -ais. f(a) a f() g() = shrink b a g() = 1 factor of 1 stretch b a factor of g() = stretch b a factor of g() = 1 shrink b a factor of 1 Transforming Radical Functions ANALYZING MATHEMATICAL RELATIONSHIPS In Eample (b), ou can use the Product Propert of Radicals to write g() =. So, ou can also describe the graph of g as a vertical stretch b a factor of and a reflection in the -ais of the graph of f. Describe the transformation of f represented b g. Then graph each function. a. f() =, g() = + b. f() =, g() = 8 a. Notice that the function is of the form g() = h + k, where h = and k =. So, the graph of g is a translation units right and units up of the graph of f. g f b. Notice that the function is of the form g() = a, where a = 8. So, the graph of g is a horizontal shrink b a factor of 1 8 and a reflection in the -ais of the graph of f. f g Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Graph g() = + 1. Identif the domain and range of the function.. Describe the transformation of f () = represented b g() =. Then graph each function. Section. Graphing Radical Functions 0

21 Writing Radical Functions Writing a Transformed Radical Function Let the graph of g be a horizontal shrink b a factor of 1 followed b a translation units to the left of the graph of f() =. Write a rule for g. Step 1 First write a function h that represents the horizontal shrink of f. Check 7 g h f h() = f() Multipl the input b 1 1 =. = Replace with in f(). Step Then write a function g that represents the translation of h. g() = h( + ) Subtract, or add, to the input. = ( + ) Replace with + in h(). = + 18 Distributive Propert The transformed function is g() = Using Technolog to Write a Radical Function Self-Portrait of NASA s Mars Rover Curiosit The table shows the numbers of seconds it takes a dropped object to fall feet on Mars. Write a function that models in terms of. How long does it take a dropped object to fall feet on Mars? Step 1 Use a graphing calculator to create a scatter plot of the data. It appears that a vertical shrink of the parent square root function can be used 10 0 to model the data. 0. Step Create a table of values for the parent square root function = using the -values in the given table. Then compare the -values for each -value X Y1 The -values for Mars are about 0. times the -values of the parent square root function. So, the function = 0. models the data. Evaluate the function when =. = 0. = 0.(8) =. It takes a dropped object about. seconds to fall feet on Mars X= Monitoring Progress Help in English and Spanish at BigIdeasMath.com. In Eample, is the transformed function the same when ou perform the translation followed b the horizontal shrink? Eplain our reasoning.. The table shows the numbers of seconds it takes a dropped object to fall feet on the Moon. Write a function that models in terms of. How long does it take a dropped object to fall feet on the Moon? 0 Chapter Rational Eponents and Radical Functions

22 Graphing Parabolas and Circles To graph parabolas and circles using a graphing calculator, first solve their equations for to obtain radical functions. Then graph the functions. Graphing a Parabola (Horizontal Ais of Smmetr) Use a graphing calculator to graph 1 =. Identif the verte and the direction that the parabola opens. STUDY TIP Notice 1 is a function and is a function, but 1 = is not a function. Step 1 Solve for. 1 = Write the original equation. = Multipl each side b. = ± Step Graph both radical functions. 1 = Take square root of each side. 1 = 10 The verte is (0, 0) and the parabola opens right. Graphing a Circle (Center at the Origin) Use a graphing calculator to graph + = 1. Identif the radius and the intercepts. Step 1 Solve for. + = 1 = 1 = ± 1 Step Graph both radical functions using a square viewing window. 1 = 1 = 1 The radius is units. The -intercepts are ±. The -intercepts are also ±. Write the original equation. Subtract from each side. Take square root of each side Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Use a graphing calculator to graph = + 1. Identif the verte and the direction that the parabola opens.. Use a graphing calculator to graph + =. Identif the radius and the intercepts. Section. Graphing Radical Functions 07

23 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE Square root functions and cube root functions are eamples of functions.. COMPLETE THE SENTENCE When graphing = a h + k, translate the graph of = a h units and k units. Monitoring Progress and Modeling with Mathematics In Eercises 8, match the function with its graph.. f() = +. h() = +. f() =. g() = 7. h() = + 8. f() = + In Eercises 19, describe the transformation of f represented b g. Then graph each function. (See Eample.) 19. f() =, g() = f() =, g() = 1 A. B. 1. f() =, g() = 1. f() =, g() = +. f() = 1/, g() = 1 ( )1/ C. D.. f() = 1/, g() = 1 1/ +. f() =, g() = +. f() =, g() = + E. F. In Eercises 9 18, graph the function. Identif the domain and range of the function. (See Eample 1.) 9. h() = g() = 11. g() = 1. f() = 1. g() = 1 1. f() = f() = () 1/ + 1. g() = ( 1) 1/ 17. h() = 18. h() = 7. ERROR ANALYSIS Describe and correct the error in graphing f() =. 8. ERROR ANALYSIS Describe and correct the error in describing the transformation of the parent square root function represented b g() = 1 +. The graph of g is a horizontal shrink b a factor of 1 and a translation units up of the parent square root function. 08 Chapter Rational Eponents and Radical Functions

24 USING TOOLS In Eercises 9, use a graphing calculator to graph the function. Then identif the domain and range of the function. 9. g() = + 0. h() = 1. f() = +. f() =. f() = h() = 1 + ABSTRACT REASONING In Eercises 8, complete the statement with sometimes, alwas, or never.. The domain of the function = a is 0.. The range of the function = a is The domain and range of the function = h + k are (, ). 8. The domain of the function = a ( ) + k is { 0}. In Eercises 9, write a rule for g described b the transformations of the graph of f. (See Eample.) 9. Let g be a vertical stretch b a factor of, followed b a translation units up of the graph of f() = Let g be a reflection in the -ais, followed b a translation 1 unit right of the graph of f() = Let g be a horizontal shrink b a factor of, followed b a translation units left of the graph of f() =.. Let g be a translation 1 unit down and units right, followed b a reflection in the -ais of the graph of 1 f() = + In Eercises and, write a rule for g... g. f() = g f() = In Eercises 8, write a rule for g that represents the indicated transformation of the graph of f.. f() =, g() = f( + ). f() = 1 1, g() = f() f() =, g() = f( + ) 8. f() = + 10, g() = 1 f( ) + 9. MODELING WITH MATHEMATICS The table shows the distances (in miles) an astronaut would be able to see to the horizon at a height of feet above the surface of Mars. Write a function that models in terms of. What is the distance an astronaut would be able to see to the horizon from an height of 10,000 feet above Mars? (See Eample.) Height, Distance, MODELING WITH MATHEMATICS The table shows the speeds of sound waves (in meters per second) in air when the air temperature is kelvins. Write a function that models in terms of. What is the speed of the sound waves in the air when the air temperature is 0 kelvins? Temperature, Speed, In Eercises 1, use a graphing calculator to graph the equation of the parabola. Identif the verte and the direction that the parabola opens. (See Eample.) 1. 1 =. =. 8 + =. =. + 8 = 1. 1 = In Eercises 7, use a graphing calculator to graph the equation of the circle. Identif the radius and the intercepts. (See Eample.) 7. + = = 9. 1 = 0. = 1. =. = 100 Section. Graphing Radical Functions 09

25 . MODELING WITH MATHEMATICS The period of a pendulum is the time the pendulum takes to complete one back-and-forth swing. The period T (in seconds) can be modeled b the function T = 1.11, where is the length (in feet) of the pendulum. Graph the function. Estimate the length of a pendulum with a period of seconds. Eplain our reasoning.. HOW DO YOU SEE IT? Does the graph represent a square root function or a cube root function? Eplain. What are the domain and range of the function? (, ) (, 1). PROBLEM SOLVING For a drag race car with a total weight of 00 pounds, the speed s (in miles per hour) at the end of a race can be modeled b s = 1.8 p, where p is the power (in horsepower). Graph the function. a. Determine the power of a 00-pound car that reaches a speed of 00 miles per hour. b. What is the average rate of change in speed as the power changes from 1000 horsepower to 100 horsepower?. THOUGHT PROVOKING The graph of a radical function f passes through the points (, 1) and (, 0). Write two different functions that could represent f( + ) + 1. Eplain. 7. MULTIPLE REPRESENTATIONS The terminal velocit v t (in feet per second) of a skdiver who weighs 10 pounds is given b v t =.7 10 A where A is the cross-sectional surface area (in square feet) of the skdiver. The table shows the terminal velocities (in feet per second) for various surface areas (in square feet) of a skdiver who weighs 1 pounds. Cross-sectional surface area, A Terminal velocit, v t a. Which skdiver has a greater terminal velocit for each value of A? b. Describe how the different values of A given in the table relate to the possible positions of the falling skdiver. 8. MATHEMATICAL CONNECTIONS The surface area S of a right circular cone with a slant height of 1 unit is given b S = πr + πr, where r is the radius of the cone. r 1 unit a. Use completing the square to show that r = 1 π S + π 1. b. Graph the equation in part (a) using a graphing calculator. Then find the radius of a right circular cone with a slant height of 1 unit and a surface area of π square units. Maintaining Mathematical Proficienc Solve the equation. Check our solutions. (Section 1.) Reviewing what ou learned in previous grades and lessons 9. + = = = = + Solve the inequalit. (Section.) < > Chapter Rational Eponents and Radical Functions

26 .1. What Did You Learn? Core Vocabular nth root of a, p. 90 inde of a radical, p. 90 simplest form of a radical, p. 97 conjugate, p. 98 like radicals, p. 98 radical function, p. 0 Core Concepts Section.1 Real nth Roots of a, p. 90 Rational Eponents, p. 91 Section. Properties of Rational Eponents, p. 9 Properties of Radicals, p. 97 Section. Parent Functions for Square Root and Cube Root Functions, p. 0 Transformations of Radical Functions, p. 0 Mathematical Thinking 1. How can ou use definitions to eplain our reasoning in Eercises 1 on page 9?. How did ou use structure to solve Eercise 7 on page 0?. How can ou check that our model is a good fit in Eercise 9 on page 09?. How can ou make sense of the terms of the surface area formula given in Eercise 8 on page 10? Stud Skills Analzing Your Errors Application Errors What Happens: You can do numerical problems, but ou struggle with problems that have contet. How to Avoid This Error: Do not just mimic the steps of solving an application problem. Eplain out loud what the question is asking and wh ou are doing each step. After solving the problem, ask ourself, Does m solution make sense? 11

27 .1. Quiz Find the indicated real nth root(s) of a. (Section.1) 1. n =, a = 81. n =, a = 10. Evaluate (a) 1 / and (b) 1 / without using a calculator. Eplain our reasoning. (Section.1) Find the real solution(s) of the equation. Round our answer to two decimal places. (Section.1). = 18. ( + ) = 8 Simplif the epression. (Section.). ( 81/ 1/ ) Simplif z 1. (Section.) Write the epression in simplest form. Assume all variables are positive. (Section.) 11. 1p 9 1. m 1. Graph f() = + 1. Identif the domain and range of the function. (Section.) 1. n q + 7n q Describe the transformation of the parent function represented b the graph of g. Then write a rule for g. (Section.) g g g 18. Use a graphing calculator to graph =. Identif the verte and direction the parabola opens. (Section.) 19. A jeweler is setting a stone cut in the shape of a regular octahedron. A regular octahedron is a solid with eight equilateral triangles as faces, as shown. The formula for the volume of the stone is V = 0.7s, where s is the side length (in millimeters) of an edge of the stone. The volume of the stone is 11 cubic millimeters. Find the length of an edge of the stone. (Section.1) 0. An investigator can determine how fast a car was traveling just prior to an accident using the model s = d, where s is the speed (in miles per hour) of the car and d is the length (in feet) of the skid marks. Graph the model. The length of the skid marks of a car is 90 feet. Was the car traveling at the posted speed limit prior to the accident? Eplain our reasoning. (Section.) s SPEED LIMIT 1 Chapter Rational Eponents and Radical Functions

28 . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..F A..G A..B A.7.H Solving Radical Equations and Inequalities Essential Question How can ou solve a radical equation? Solving Radical Equations Work with a partner. Match each radical equation with the graph of its related radical function. Eplain our reasoning. Then use the graph to solve the equation, if possible. Check our solutions. a. 1 1 = 0 b. + + = 0 c. 9 = 0 d. + = 0 e. + = 0 f. + 1 = 0 A. B. C. D. E. F. ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math, ou need to look closel to discern a pattern or structure. Solving Radical Equations Work with a partner. Look back at the radical equations in Eploration 1. Suppose that ou did not know how to solve the equations using a graphical approach. a. Show how ou could use a numerical approach to solve one of the equations. For instance, ou might use a spreadsheet to create a table of values. b. Show how ou could use an analtical approach to solve one of the equations. For instance, look at the similarities between the equations in Eploration 1. What first step ma be necessar so ou could square each side to eliminate the radical(s)? How would ou proceed to find the solution? Communicate Your Answer. How can ou solve a radical equation?. Would ou prefer to use a graphical, numerical, or analtical approach to solve the given equation? Eplain our reasoning. Then solve the equation. + = 1 Section. Solving Radical Equations and Inequalities 1

29 . Lesson What You Will Learn Core Vocabular radical equation, p. 1 Previous rational eponents radical epressions etraneous solutions solving quadratic equations Solve equations containing radicals and rational eponents. Solve radical inequalities. Solving Equations Equations with radicals that have variables in their radicands are called radical equations. An eample of a radical equation is + 1 =. Core Concept Solving Radical Equations To solve a radical equation, follow these steps: Step 1 Isolate the radical on one side of the equation, if necessar. Step Raise each side of the equation to the same eponent to eliminate the radical and obtain a linear, quadratic, or other polnomial equation. Step Solve the resulting equation using techniques ou learned in previous chapters. Check our solution. Solving Radical Equations Solve (a) + 1 = and (b) 9 1 =. a. + 1 = Write the original equation. Check + 1 =? =? = + 1 = Divide each side b. ( + 1 ) = Square each side to eliminate the radical. + 1 = Simplif. = Subtract 1 from each side. b. The solution is =. 9 1 = 9 = Write the original equation. Add 1 to each side. Check (18) 9 1 =? 7 1 =? = ( 9 ) = Cube each side to eliminate the radical. 9 = 7 Simplif. = Add 9 to each side. = 18 Divide each side b. The solution is = 18. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. Check our solution =. + =. = 1 Chapter Rational Eponents and Radical Functions

30 Solving a Real-Life Problem In a hurricane, the mean sustained wind velocit v (in meters per second) can be modeled b v( p) =. 101 p, where p is the air pressure (in millibars) at the center of the hurricane. Estimate the air pressure at the center of the hurricane when the mean sustained wind velocit is. meters per second. v( p) =. 101 p Write the original function.. =. 101 p Substitute. for v( p) p Divide each side b.. 8. ( 101 p ) Square each side p Simplif. EXPLAINING MATHEMATICAL IDEAS To understand how etraneous solutions can be introduced, consider the equation =. This equation has no real solution; however, ou obtain = 9 after squaring each side. 98. p Subtract 101 from each side. 98. p Divide each side b 1. The air pressure at the center of the hurricane is about 98 millibars. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? Estimate the air pressure at the center of the hurricane when the mean sustained wind velocit is 8. meters per second. Raising each side of an equation to the same eponent ma introduce etraneous solutions. When ou use this procedure, ou should alwas check each apparent solution in the original equation. Solve + 1 = Solving an Equation with an Etraneous Solution + 1 = Write the original equation. ( + 1) = ( ) Square each side = = 0 Epand left side and simplif right side. Write in standard form. ( 7)( + ) = 0 Factor. 7 = 0 or + = 0 Zero-Product Propert = 7 or = Solve for. Check =? 7(7) =? 7( ) =? 1 =? 1 8 = The apparent solution = is etraneous. So, the onl solution is = 7. Section. Solving Radical Equations and Inequalities 1

31 Solve =. Solving an Equation with Two Radicals = Write the original equation. ( ) = ( ) Square each side = Epand left side and simplif right side. + = Isolate radical epression. + = Divide each side b. ( + ) = ( ) Square each side. ANOTHER WAY You can also graph each side of the equation and find the -value where the graphs intersect. + = Simplif. 0 = Write in standard form. 0 = ( )( + 1) Factor. = 0 or + 1 = 0 Zero-Product Propert = or = 1 Solve for. Intersection X=-1 Y= Check =? =? ( 1) + 1 =? =? 1 = The apparent solution = is etraneous. So, the onl solution is = 1. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. Check our solution(s) = +. + = = When an equation contains a power with a rational eponent, ou can solve the equation using a procedure similar to the one for solving radical equations. In this case, ou first isolate the power and then raise each side of the equation to the reciprocal of the rational eponent. Solve () / + = 10. Solving an Equation with a Rational Eponent () / + = 10 () / = 8 [() / ] / = 8 / Write the original equation. Subtract from each side. Raise each side to the four-thirds. Check = 1 Simplif. = 8 Divide each side b. The solution is = 8. ( 8) / + =? 10 1 / + =? = 10 1 Chapter Rational Eponents and Radical Functions

32 Solve ( + 0) 1/ =. Solving an Equation with a Rational Eponent Check ( + 0) 1/ =? 1/ =? = ( + 0) 1/ =? 1/ =? ( + 0) 1/ = Write the original equation. [( + 0) 1/ ] = Square each side. + 0 = Simplif. 0 = 0 Write in standard form. 0 = ( )( + ) Factor. = 0 or + = 0 Zero-Product Propert = or = Solve for. The apparent solution = is etraneous. So, the onl solution is =. Monitoring Progress Solve the equation. Check our solution(s). Help in English and Spanish at BigIdeasMath.com 8. () 1/ = 9. ( + ) 1/ = 10. ( + ) / = 8 Solving Radical Inequalities To solve a simple radical inequalit of the form n u < d, where u is an algebraic epression and d is a nonnegative number, raise each side to the eponent n. This procedure also works for >,, and. Be sure to consider the possible values of the radicand. Solve 1 1. Step 1 Solve for. Solving a Radical Inequalit 1 1 Write the original inequalit. Check 0 = 1 Intersection X=17 Y=1 8 = 1 1 Divide each side b. 1 1 Square each side. 17 Add 1 to each side. Step Consider the radicand. 1 0 The radicand cannot be negative. 1 Add 1 to each side. So, the solution is Monitoring Progress Help in English and Spanish at BigIdeasMath.com 11. Solve (a) and (b) + 1 < 8. Section. Solving Radical Equations and Inequalities 17

33 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY Is the equation = a radical equation? Eplain our reasoning.. WRITING Eplain the steps ou should use to solve + 10 < 1. Monitoring Progress and Modeling with Mathematics In Eercises 1, solve the equation. Check our solution. (See Eample 1.). + 1 = = 8 In Eercises 1, solve the equation. Check our solution(s). (See Eamples and.) 1. = = 9. 1 =. 10 = = = = = = = = = = = = = = 0 + = = + 1. MODELING WITH MATHEMATICS Biologists have discovered that the shoulder height h (in centimeters) of a male Asian elephant can be modeled b h =. t + 7.8, where t is the age (in ears) of the elephant. Determine the age of an elephant with a shoulder height of 0 centimeters. (See Eample.). + = In Eercises 7, solve the equation. Check our solution(s). (See Eamples and.) 7. / = 8 8. / = 9. 1/ + = 0 0. / 1 = 0 1. ( + ) 1/ =. ( ) 1/ = 0 h. ( + 11) 1/ = +. ( ) 1/ = ERROR ANALYSIS In Eercises and, describe and correct the error in solving the equation. 1. MODELING WITH MATHEMATICS In an amusement park ride, a rider suspended b cables swings back and forth from a tower. The maimum speed v (in meters per second) of the rider can be approimated b v = gh, where h is the height (in meters) at the top of each swing and g is the acceleration due to gravit (g 9.8 m/sec ). Determine the height at the top of the swing of a rider whose maimum speed is 1 meters per second.. 8 = ( 8 ) = 8 = = 1 =. 8 / = ( / ) / = 1000 / 8 = 100 = 18 Chapter Rational Eponents and Radical Functions