11.2 Simplifying Radical Expressions

Transcription

1 n a a m Locker LESSON 11. Simplifying Radical Epressions PAGE Name Class Date 11. Simplifying Radical Epressions Essential Question: How can you simplify epressions containing rational eponents or radicals involving nth roots? Common Core Math Standards The student is epected to: N-RN. Rewrite epressions involving radicals and rational eponents using the properties of eponents. Also F-IF.7b Mathematical Practices MP.8 Patterns Language Objective Eplain to a partner the steps for simplifying rational eponents and radical epressions. Eplore Establishing the Properties of Rational Eponents In previous courses, you have used properties of integer eponents to simplify and evaluate epressions, as shown here for a few simple eamples: + 10 ( ) 1 ( ) ( ) Now that you have been introduced to epressions involving rational eponents, you can eplore the properties that apply to simplifying them. Let a, b, m 1, and n. Evaluate each epression by substituting and applying eponents individually, as shown. Resource Locker ENGAGE Essential Question: How can you simplify epressions containing rational eponents or radicals involving nth roots? Possible answer: You can use the same properties of eponents for rational eponents as for integer eponents, apply the properties of square roots to radicals involving nth roots, and translate between radical form and rational eponent form whenever it is helpful. Epression Substitute Simplify Result a m a n (a b) n ( ) (a m ) n a n a m ( a b) n ( 1 ) 1 ( ) PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the volume and surface area of a sphere have a common variable. Then preview the Lesson Performance Task. Module Lesson Name Class Date 11. Simplifying Radical Epressions Essential Question: How can you simplify epressions containing rational eponents or radicals involving nth roots? N-RN. Rewrite epressions involving radicals and rational eponents using the properties of eponents. Also F-IF.7b In previous courses, you have used properties of integer eponents to simplify and evaluate epressions, as shown here for a few simple eamples: + 10 ( ) 1 ( ) ( ) Now that you have been introduced to epressions involving rational eponents, you can eplore the properties that apply to simplifying them. Eplore Establishing the Properties of Rational Eponents Let a, b, m 1, and n. Evaluate each epression by substituting and applying eponents individually, as shown. Epression Substitute Simplify Result a m a n (a b) n ( ) (a m ) n ( b) a n 1 08 ( 1 ) ( ) Resource HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. Module Lesson 87 Lesson 11.

2 Complete the table again. This time, however, apply the rule of eponents that you would use for integer eponents. Epression a m a n 1 + Apply Rule and Substitute Simplify Result PAGE EXPLORE Establishing the Properties of Rational Eponents (a b) n INTEGRATE TECHNOLOGY ( a m ) n Students have the option of completing the Eplore activity either in the book or online. Reflect a n a m ( a b) n 1. Compare your results in Steps A and B. What can you conclude? Applying the same rules as for integer eponents gives the same results as applying the eponents individually. The properties of rational eponents are the same as the corresponding properties of integer eponents.. In Steps A and B, you evaluated a n a two ways. Now evaluate a m m n two ways, using the definition of a negative eponents. Are your results consistent with your previous conclusions about integer and rational eponents? a m n a ; a m 1 n a ; Yes, working with negative rational eponents is consistent with working with negative integer eponents QUESTIONING STRATEGIES If an epression consists of a variable raised to a negative eponent, how can you rewrite the epression with a positive eponent? Rewrite the epression as the reciprocal of the variable raised to the opposite of the eponent. How does that help you write the simplified form of 1 with a positive eponent? You can subtract the eponents, and then apply the rule to the answer ( 1 ) 1 Module Lesson PROFESSIONAL DEVELOPMENT Learning Progressions This lesson etends concepts and properties that students have learned in previous courses and lessons. Students are familiar with the properties of eponents, and have used them to simplify epressions containing integer eponents. They also were introduced to the nth roots and the meaning of rational eponents in the previous lesson. In this lesson, students combine these concepts, etending the properties to epressions containing rational eponents. They also learn about the properties of nth roots. Students will apply these skills in the following lesson, where they will use them to solve radical equations. Simplifying Radical Epressions 88

3 EXPLAIN 1 Simplifying Rational-Eponent Epressions PAGE Eplain 1 Simplifying Rational-Eponent Epressions Rational eponents have the same properties as integer eponents. Properties of Rational Eponents For all nonzero real numbers a and b and rational numbers m and n Words Numbers Algebra AVOID COMMON ERRORS Students may, in error, multiply or divide the common bases in a product or quotient of powers. Help them to see that the simplified product (or quotient) represents factors of the common base. Use a numerical eample with integer eponents, such as, to help students see why this is so. Product of Powers Property To multiply powers with the same base, add the eponents. Quotient of Powers Property To divide powers with the same base, subtract the eponents. Power of a Power Property To raise one power to another, multiply the eponents m 1 1a a n a m+n m a 1 a a m - n n (8 ) 8 m 8 (a ) n a m n QUESTIONING STRATEGIES How do you multiply powers with the same base when the eponents are rational? Add the eponents, and write the result as a power of the common base. Power of a Product Property To find a power of a product, distribute the eponent. Power of a Quotient Property To find the power of a qoutient, distribute the eponent (1 ) 1 0 (ab) m a m m b 1 1 ( 81) a ( b) m a m b m How do you divide powers with the same base when the eponents are rational? Subtract the eponents, and write the result as a power of the common base. INTEGRATE TECHNOLOGY Students can use a graphing calculator to check their work. Review the correct use of parentheses when entering epressions containing rational eponents. Also, encourage them to use parentheses around the numerator and the denominator of a quotient of epressions. Eample 1 Simplify the epression. Assume that all variables are positive. Eponents in simplified form should all be positive. a. 7 b Product of Powers Prop. 8 - Quotient of Powes Prop. Simplify Simplify. Definition of neg. power Simplify. Module Lesson COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs. Provide each pair with several fairly comple epressions to simplify. Instruct one student in each pair to simplify one of the epressions while the other gives verbal instructions for each step. Then have the student who simplified the epression write an eplanation net to each step, describing what was done. Have students switch roles and repeat the eercise using a different epression. 89 Lesson 11.

4 a. ( y 1y ) ( y - ) 1 ( ) y 1 ( y ) 1 y b. ( 7 Simplify. 7 Power of a power Prop. 1 y Simplify. Your Turn ( ) ( ) Power of a Product Prop. ). Power of a Power Prop. Prop. Simplify. Simplify the epression. Assume that all variables are positive. Eponents in simplified form should all be positive. 1. ( 1 1 ) ( ). y ( 1 1 ) ( 1 + ) (1 ) ( 1 ) 1 Eplain Simplifying Radical Epressions Using the Properties of Eponents When you are working with radical epressions involving nth roots, you can rewrite the epressions using rational eponents and then simplify them using the properties of eponents. Eample ( y )( y ) Quotient of Powers Prop. 7 Power of a Quotient Simplify the epression by writing it using rational eponents and then using the properties of rational eponents. Assume that all variables are positive. Eponents in simplified form should all be positive. (y) 1 ( y ) 1 Write using rational eponents. (y y 1 ) Power of a Product Property (8 y 1 ) Product of Powers Property ( y) Power of a Product Property y Product of Powers Property 9 1 y 1 y - 1 y y - y y PAGE EXPLAIN Simplifying Radical Epressions Using the Properties of Eponents QUESTIONING STRATEGIES How does knowing the relationship between roots and epressions containing rational eponents help you to simplify epressions containing radicals? Convert each radical to an epression containing a rational eponent, and then apply the rules for eponents. How do the properties of eponents help you to multiply two radicals that have the same radicands but different indices? Convert each radical to an epression containing a rational eponent, add the eponents, and then write the result as a power of the common radicand. INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP. Enhance students understanding of the properties and relationships used to simplify these epressions by having them eplain each step, identifying the properties being applied. You may want to provide students with additional eamples, then have them eplain the steps of their own, and possibly each other s, work. Module Lesson DIFFERENTIATE INSTRUCTION Communicating Math For many students, the descriptive sentences in the first column of each table of properties will be the most helpful when applying the various properties. Have students read these out loud, and use the eamples in the second column to check for understanding. Encourage students to memorize the sentences, rewording them in their own words, as necessary, to clarify the instruction. Then show them how repeating the sentence that applies, at each step of the simplification process, can provide the guidance they need to correctly apply each property. Simplifying Radical Epressions 90

5 EXPLAIN Simplifying Radical Epressions Using the Properties of nth Roots PAGE 7 B y y ( y) 1 (y) Write using rational eponents. (y) Quotient of Powers Property 1 (y) Simplify. QUESTIONING STRATEGIES How do you rationalize a denominator that contains an nth root? Multiply the numerator and denominator of the fraction by the nth root of enough factors of the radicand to create a perfect nth root. AVOID COMMON ERRORS After learning the product and quotient properties for nth roots, students may assume there are similar properties for sums and differences. Show students, by numerical eample, that n n a + b a n + b and n n a - b a n - b for a, b > 0. Power of a Product Property Simplify. Your Turn Simplify the epression by writing it using rational eponents and then using the properties of rational eponents.. 1 y 1 y 1 ( ) 1 ( ) 1 Eplain ( ) 1 ( ) ( ) 7 18 Simplifying Radical Epressions Using the Properties of nth Roots From working with square roots, you know, for eample, that and 8 8. The corresponding properties also apply to nth roots. Properties of nth Roots For a > 0 and b > 0 Words Numbers Algebra Product Property of Roots The nth root of a product is equal to the product of the nth roots. Quotient Property of Roots The nth root of a Quotient is equal to the Quotient of the nth roots n ab n a n b n a n a b n b Module Lesson 91 Lesson 11.

6 Eample y 7 y 7 Simplify the epression using the properties of nth roots. Assume that all variables are positive. Rationalize any irrational denominators. 8 y 7 Write as a power. y y Product Property of Roots y y Factor out perfect cubes. y y Simplify PAGE 8 INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Have each student work with a partner to use the relationship between radicals and rational eponents to derive the quotient property of roots, n a b n a n b. Have students justify each step of their derivations. 81 Quotient Property of Roots Simplify. Rationalize the denominator. Product Property of Roots Simplify. Reflect 7. In Part B, why was used when rationalizing the denominator? What factor would you use to rationalize a denominator of y? It was chosen to make the product of the radicands be a perfect fourth power so that the fourth root could be taken; 8 y. Module 11 9 Lesson Simplifying Radical Epressions 9

7 EXPLAIN Rewriting a Radical-Function Model QUESTIONING STRATEGIES How can you check that the simplified form of the epression is equivalent to the original epression? You could graph both epressions as functions on a graphing calculator and make sure their graphs are the same. You could also evaluate both epressions for several values and make sure the resulting values are the same. PAGE 9 Your Turn Simplify the epression using the properties of nth roots. Assume that all variables are positive y y 1 1 y 1 y Eplain 1 1 Rewriting a Radical-Function Model When you find or apply a function model involving rational powers or radicals, you can use the properties in this lesson to help you find a simpler epression for the model Eample Gregor Schuster/Corbis Manufacturing A can that is twice as tall as its radius has the minimum surface area for the volume it contains. The formula S π ( π) V epresses the surface area of a can with this shape in terms of its volume. a. Use the properties of rational eponents to simplify the epression for the surface area. Then write the approimate model with the coefficient rounded to the nearest hundredth. b. Graph the model using a graphing calculator. What is the surface area in square centimeters for a can with a volume of 0 c m? a. S π V ( π) V Power of a Quotient Property π (π) (π) Group Powers of π. V (π) Quotient of Powers Property (π) 1 - V Simplify. (π) 1 V Use a calculator.. V A simplified model is S (π) 1 V, which gives S. V. Module 11 9 Lesson 9 Lesson 11.

8 b. The surface area is about 0 c m. Commercial fishing The buoyancy of a fishing float in water depends on the volume of air it contains. The radius of a spherical float as a function of its volume is given by r V π. a. Use the properties of roots to rewrite the epression for the radius as the product of a coefficient term and a variable term. Then write the approimate formula with the coefficient rounded to the nearest hundredth. b. What should the radius be for a float that needs to contain. f t of air to have the proper buoyancy? a. r V π Rewrite radicand. π V Product Property of Roots π V Use a calculator 0. V The rewritten formula is r, which gives r. π V 0. V b. Substitute. for V. PAGE 0 Reflect r The radius is about 1.0 feet. 10. Discussion What are some reasons you might want to rewrite an epression involving radicals into an epression involving rational eponents? By rewriting radical epressions, especially complicated ones and those involving nth roots and powers in the radicand, rewriting using rational eponents lets you use the properties of rational eponents to make simplification easier. Also, rational eponents make it easier to enter an epression into a calculator for evaluation or graphing. Module 11 9 Lesson Simplifying Radical Epressions 9

9 ELABORATE QUESTIONING STRATEGIES Can you use the properties of rational eponents to simplify a b? Eplain. No; in rational eponent form, the epression is a 1 b 1. Because the bases are different, the product of powers property does not apply, and the epression cannot be simplified. SUMMARIZE THE LESSON How can the properties of eponents be applied to the simplification of epressions containing rational eponents and to those containing radicals? For epressions containing rational eponents, the properties of eponents can be applied directly. For radical epressions, convert the radical epressions to eponent form using the fact that n a m a m n, and then apply the properties. Your Turn 11. The surface area as a function of volume for a bo with a square base and a height that is twice the side length of the base is S 10 ( V ). Use the properties of rational eponents to simplify the epression for the surface area so that no fractions are involved. Then write the approimate model with the coefficient rounded to the nearest hundredth. 10 V S 10 ( V ) Elaborate V ( 1 - ) V 1 V.0 V The epression is S 1 V, which gives S.0 V. 1. In problems with a radical in the denominator, you rationalized the denominator to remove the radical. What can you do to remove a rational eponent from the denominator? Eplain by giving an eample. You can multiply the epression by a form of 1 so that the denominator of the resulting epression has an eponent that is an integer. For eample, for, multiply the numerator and denominator by : Show why n a n is equal to a for all natural numbers a and n using the definition of nth roots and using rational eponents. By definition, the nth root of a number b is the number whose nth power is b. So the nth root of a n is the number whose nth power is a n, or a. Using rational eponents, the nth root is indicated by the eponent n 1, so n a n ( a n 1 ) n a n n 1 a n n a 1 a. 1. Show that the Product Property of Roots is true using rational eponents. The nth root is indicated by the eponent n 1, so n 1 ab (ab) n a 1 n b 1 n n a n b. 1. Essential Question Check-In Describe the difference between applying the Power of a Power Property and applying the Power of a Product Property for rational eponents using an eample that involves both properties. Possible answer: Consider the epression ( of and. The result is the product of the ). This is the power of each factor: ( epression contains the power of the power and the Simplifying then gives ( ) ( ) 1 1 power of the product ) ( ). This power of the power. Module 11 9 Lesson LANGUAGE SUPPORT Communicate Math Have students work in pairs. The first student eplains the steps for simplifying rational eponents to the second student, including the properties involved. The second student takes notes and writes down the steps, and repeats them back using his or her own words. Students switch roles and repeat the procedure for radical epressions involving nth roots. 9 Lesson 11.

10 Simplify the epression. Assume that all variables are positive. Eponents in simplified form should all be positive. 1. ( ( 1. 1) - ) ( ( 1 1 ) - ) ( ) 1. y y Simplify the epression by writing it using rational eponents and then using the properties of rational eponents. Assume that all variables are positive. Eponents in simplified form should all be positive ( ) Evaluate: Homework and Practice ( 1 1 ) y. ( y 1 ) y y ( 1 y ) 1 y ( ) (-) 9 1 ( 1 ) - 1 y 1 y ( ) 1 1 () y y 9 ( 1 y ) y y y y y y Online Homework Hints and Help Etra Practice 9 ( 1 9 ) 1 y - ( y 1 y y ) 1 y 1 y ( - ) 1 ( y ) 1 ( ) 1 y 1 + y - 1 y 1 PAGE 1 EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore Establishing the Properties of Rational Eponents Eample 1 Simplifying Rational-Eponent Epressions Eample Simplifying Radical Epressions Using the Properties of Eponents Eample Simplifying Radical Epressions Using the Properties of nth Roots Eample Rewriting a Radical-Function Model QUESTIONING STRATEGIES Practice Eercises 1 Eercises 7 1 Eercises 1 1 Eercises When do you add the eponents on two epressions? when the epressions have the same base and they are being multiplied When do you apply the power of a power property? when an epression containing an eponent is being raised to another eponent Module 11 9 Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 1 1 Recall of Information MP. Logic 1 1 Recall of Information MP. Using Tools 17 0 Skills/Concepts MP. Modeling INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Discuss with students that there is often more than one way to go about simplifying these types of epressions. Encourage students to be aware of this, and to use one method to check their results found using a different method. 1 Skills/Concepts MP. Logic Strategic Thinking MP. Logic Simplifying Radical Epressions 9

11 VISUAL CUES For epressions that involve applying the power of a product property or the power of a quotient property, suggest that students use arrows to show the eponent being applied to each factor in the product or to each factor in the numerator and denominator in the quotient. In this way, students may avoid errors such as forgetting to apply the eponent to a numerical coefficient or to a variable that does not contain an eponent. AVOID COMMON ERRORS Students may make errors in rationalizing denominators when the inde is greater than. For eample, when trying to rationalize an epression such as, they may multiply by, instead of, as if the inde were instead of. Help them to see that this choice does not make the radicand a perfect fifth, which is the goal of rationalizing this denominator. Reinforce that the resulting eponent must be a multiple of the inde. PAGE David R. Frazier/Science Photo Library 11. s t 9 st Simplify the epression using the properties of nth roots. Assume that all variables are positive. Rationalize any irrational denominators s t 9 st 8 y y ( s t 9 ) 1 (st) 1 s 1 - t 1 - s 1 t 7 s t s 1 t y y 8 - y - 1 y y y 8 1 w v 17. Weather The volume of a sphere as a function of its surface area is given by V π S (. π) a. Use the properties of roots to rewrite the epression for the volume as the product of a simplified coefficient term (with positive eponents) and a variable term. Then write the approimate formula with the coefficient rounded to the nearest thousandth. b. A spherical weather balloon has a surface area of 00 ft. What is the approimate volume of the balloon? S (π) (π) 1 - S 1 1 (π) 1 π a. V ( S π ) π S b. V 0.09 (00) 100 ft ( ) y 8 8 y 8 8 y 8 y 1 v w v w w ( w ) v w w v w w w w v w v w w w (π) 1 - S S 1 π 1 S w v w Module Lesson 97 Lesson 11.

12 18. Amusement parks An amusement park has a ride with a free fall of 18 feet. The formula t d g gives the time t in seconds it takes the ride to fall a distance of d feet. The formula v gd gives the velocity v in feet per second after the ride has fallen d feet. The letter g represents the gravitational constant. a. Rewrite each formula so that the variable d is isolated. Then simplify each formula using the fact that g ft/ s. a. t d g g d d 1 1 d b. Find the time it takes the ride to fall halfway and its velocity at that time. Then find the time and velocity for the 1 full drop. b. halfway: d ft, so t s and v 8 ft/s; 1 1 full drop: d 18 ft, so t 18 () c. What is the ratio of the time it takes for the whole drop to the time it takes for the first half? What is the ratio of the velocity after the second half of the drop to the velocity after the first half? What do you notice? whole time c. 1st half time 1.1; velocity at end velocity halfway 19. Which choice(s) is/are equivalent to? A. ( 8 ) B. - C. ( D. - E. 1 d ; v v d 1 ) gd g The formulas are t d () d d 1 g d, or t d, and v g d, or v 8 d. () ft/s, or about 91 ft/s 1.1; the ratios are the same s, or about.8 s, and David R. Frazier/Science Photo Library TECHNOLOGY Encourage students to use a graphing calculator to check their work. To check problems involving the simplification of variable epressions, suggest that students assign values to the variables and use the calculator to check that the value of the original epression and the value of the simplified epression are the same when evaluated for the same values of the variables. Module Lesson Simplifying Radical Epressions 98

13 JOURNAL Have students describe how the properties of nth roots are similar to the corresponding properties of rational eponents. PAGE 0. Home Heating A propane storage tank for a home is shaped like a cylinder with hemispherical ends, and a cylindrical portion length that is times the radius. V The formula S 1π ( 1π ) epresses the surface area of a tank with this shape in terms of its volume. a. Use the properties of rational eponents to rewrite the epression for the surface area so that the variable V is isolated. Then write the approimate model with the coefficient rounded to the nearest hundredth. a. S 1π ( V 1π ) 1π ( 1π ) V.7 V b. Graph the model using a graphing calculator. What is the surface area in square feet for a tank with a volume of 10 ft? b. The surface area is about 1 ft. H.O.T. Focus on Higher Order Thinking 1. Critique Reasoning Aaron s work in simplifying an epression is shown. What mistake(s) did Aaron make? Show the correct simplification. Aaron incorrectly applied the Quotient of Powers Property. He should have subtracted the eponents (- ) Critical Thinking Use the definition of nth root to show that the Product Property of Roots is true, that is, that n ab n a n b. (Hint: Begin by letting be the nth root of a and letting y be the nth root of b.) ) (- -1 (- ) 1 Let be the nth root of a and let y be the nth root of b. Then by the definition of nth root, a n and b y n. So, ab n y n n (y). This means by definition that y is the nth root of ab, or n ab y. But y n a n b, so n ab n a n b.. Critical Thinking For what real values of a is a greater than a? For what real values of a is a greater than a? for 0 < a < 1; for a < -1 or 0 < a < 1 Module Lesson 99 Lesson 11.

14 Lesson Performance Task You ve been asked to help decorate for a school dance, and the theme chosen is The Solar System. The plan is to have a bunch of papier-mâché spheres serve as models of the planets, and your job is to paint them. All you re told are the volumes of the individual spheres, but you need to know their surface areas so you can get be sure to get enough paint. How can you write a simplified equation using rational eponents for the surface area of a sphere in terms of its volume? r PAGE AVOID COMMON ERRORS When dividing one term by another, some students may divide the eponents instead of subtracting. π Eplain that to simplify, you subtract the (π) eponents: 1-1. The simplified term would 1 then be (π). (The formula for the volume of a sphere is V π r and the formula for the surface area of a sphere is A π r.) Solve the volume formula for r. V π r V π r V π r ( V 1 π ) r Substitute the value for r into the formula for surface area and simplify. A π r π 1 V Substitute. π π V π π (V) (π) Multiply eponents. Distribute the eponent. (π) 1 (V) Subtract eponents. The surface area in terms of the volume is equal to A (π) 1 (V). Volume: V πr INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Have students discuss why the final eponent of V turns out to be. Have them relate the numbers in the rational eponent to the geometry of the situation. Have them discuss whether the eponent would be if the shape were something other than a sphere. Module Lesson EXTENSION ACTIVITY Have students research the equations for the surface area and volume for a truncated icosidodecahedron, a -face solid similar to a soccer ball. Have students derive an equation for the surface area in terms of the volume, and then have them discuss whether they will need more or less paint to cover this shape, compared to painting a sphere. Scoring Rubric points: Student correctly solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Simplifying Radical Epressions 00