Algebra II Notes Unit Seven: Powers, Roots, and Radicals

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1 Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical. a is called the radicad. th Root: If Note: The square root has a implied ide of. E: 9, because th root of a. This is writte a 9 b. is called the ide of the E: 8, because 8 Multiple Roots E: Fid the real th root(s) of a if a = 6 ad =. 6, because 6 ad Roots as Ratioal Epoets: The 6. th root, a, ca be writte as a epoet a. a m a m E: Evaluate 8. 8, because E: Evaluate., because E: Evaluate E: Evaluate. 8. Note: This could also be writte as 8.. Page of 7 McDougal Littell:

2 Fidig Roots o the Calculator E: Use the graphig calculator to approimate. We will rewrite with a ratioal epoet:. Type this ito your calculator usig the carrot key for the epoet. Be sure to put the epoet i paretheses. Questio: What epressio is your calculator evaluatig if you do ot use paretheses? Solvig Equatios with th roots: To solve a equatio i the form sides. a a a m m both sides to the reciprocal power: am a E: Solve the equatio. a, take the th root of both. Note: To solve a equatio i the form m m a, raise Step Oe: Rewrite i the form a. Step Two: Take the th ( rd ) root of both sides. 7 7 Page of 7 McDougal Littell:

3 E: Solve the equatio Step Oe: Take the th ( th ) root of both sides. Cautio: 8 has real th roots! Step Two: Solve for. E: Solve the equatio Step Oe: Rewrite i the form. a Step Two: Raise both sides to the reciprocal power. Because there is o real square root of, this equatio has o real solutio. You Try: Solve the equatio. Verify your aswer o the graphig calculator. QOD: Ca you evaluate a radical if the radicad is egative ad the ide is odd? Eplai. Ca you evaluate a radical if the radicad is egative ad the ide is eve? Eplai. Sample CCSD Commo Eam Practice Questio(s): Write the radical epressio 7 i epoetial otatio. A. B. C. D Page of 7 McDougal Littell:

4 Sample SAT Questio(s): Take from College Board olie practice problems.. If ad y are real umbers ad the square of y is equal to the square root of, which of the followig must be true? I. y II. 0 III. y 0 (A) I oly (B) I ad II oly (C) I ad III oly (D) II ad III oly (E) I, II, ad III. If ad 7 (A) (B) (C) (D) (E) m, what is the value of m? Page of 7 McDougal Littell:

5 Syllabus Objectives: 7. The studet will simplify radical epressios by applyig properties of radicals. 7. The studet will use properties of ratioal epoets to simplify ad evaluate epressios. Review: Properties of Epoets (Allow studets to come up with these o their ow.) We will ow eted these properties for use with ratioal epoets. Let a ad b be real umbers, ad let m ad be itegers. Product of Powers Property Quotiet of Powers Property m m a a a m m a m a a or, a 0 m a a a m Power of a Power Property a a m ab a b Power of a Product Property m m m m a a Power of a Quotiet Property, b 0 m b b m Negative Epoet Property Zero Epoet Property m a b a m or, a 0 a b a a 0, a 0 Note: The product ad quotiet properties for epoets ca be eteded to radicals, as we ow kow that a radical is simply a ratioal epoet. Product Property: ab a b Quotiet Property: a b a b. E: Use the properties of epoets to simplify the epressio Power of a Product: Power of a Power: Page of 7 McDougal Littell:

6 E: Use the properties of epoets to simplify the epressio. Power of a Quotiet: Rewrite with Ratioal Epoets: Power of a Power: Simplest Form of a Radical: all perfect th powers are removed ad all deomiators are ratioalized E: Simplify the epressio 0. Step Oe: Factor out the perfect cube ( rd root). Step Two: Rewrite usig the product property. Step Three: Simplify by takig the cube root of the perfect cube. E: Simplify the epressio 8. Step Oe: Multiply the umerator ad deomiator by a root that will make the deomiator s radicad a perfect th root. (We ca multiply by 7 for a product of 8, which is a perfect th root. Step Two: Simplify by takig the th root of the deomiator Like Radicals: radical epressios that have the same ide ad same radicad Note: Oly like radicals ca be added or subtracted. Page 6 of 7 McDougal Littell:

7 E: Simplify the followig epressio by addig/subtractig like radicals. 6 7 Step Oe: Simplify each radical by etractig ay perfect cube roots. Step Two: Add/subtract the like radicals (cube roots of ) Simplifyig Variable Epressios Note: For the followig eercises, we must assume all variables are positive. E: Simplify the epressio 8 y. Step Oe: Rewrite the radicad etractig perfect th roots. y y. Note: Powers of are perfect th roots Step Two: Take the th root of ay perfect roots. y y Step Three: Simplify. y E: Simplify the epressio 8 0. Step Oe: Multiplicatio (umerator). Step Two: Rewrite the radicads etractig perfect square roots. Step Three: Take the square root of ay perfect roots. Step Four: Multiply ad simplify Step Five: Ratioalize the deomiator. Page 7 of 7 McDougal Littell:

8 Studets should discover a shorter way to etract perfect th roots? Have studets share their ideas. (E: They ca divide the variable powers i the radicad by the ide. The remaider is the power of the variable left i the radicad. Note This ca be eplaied by illustratig usig ratioal epoets.) You Try: Simplify the epressio. Assume the variable is positive. 6y y 8 QOD: Whe is, ad whe is? Sample CCSD Commo Eam Practice Questio(s): What is the value of the epressio 6 6? A. B. C. 8 D. 6 Page 8 of 7 McDougal Littell:

9 Syllabus Objectives: 7. The studet will perform arithmetic operatios of fuctios. 7. The studet will fid a compositio of fuctios. Fuctio Operatios: Let f ad g be two fuctios, ad h be the resultig fuctio after performig the followig operatios. The domai of h will the itersectio of the domais of f ad g (all -values that are i the domai of BOTH f ad g). Additio: h f g E: Let f ad g. Fid h f g h h ad state its domai. The domai of f ad g is All Real Numbers, so the domai of h is All Real Numbers. Subtractio: h f g E: Let f ad g. Fid h f g h h h ad state its domai. The domai of f ad g are 0, so the domai of h is 0. Multiplicatio: h f g E: Let f ad h h g. Fid h f g ad state its domai. The domai of f is All Real Numbers, ad the domai of g 0, so the domai of h is 0. Page 9 of 7 McDougal Littell:

10 Divisio: f Note: I the domai of g h h, g 0. E: Let f ad g. Fid h f ad state its domai. g h The domai of f is, ad the domai of g is All Real Numbers. Because g is i the deomiator, the domai of h must be restricted so that g 0. So the domai of h is,. Compositio of Two Fuctios: the compositio h of the fuctio f with the fuctio g is. The domai of h is the set of all -values such that is i the domai of g ad g h f g is i the domai of f. E: Let f ad g. Fid f g ad state the domai. Step Oe: Substitute g i for i the fuctio f. f g Step Two: Determie the domai. The domai of g is All Real Numbers. The domai of f g is,. E: Usig the fuctios above, fid g f ad state its domai. Step Oe: Substitute f i for i the fuctio g. g f Step Two: Determie the domai. The domai of. f is, 0 The domai of g f is, 0. Note: f g g f Page 0 of 7 McDougal Littell:

11 E: Let f. Fid f f ad state the domai. Step Oe: Substitute f i for i the fuctio f. f f f f Step Two: Determie the domai. The domai of. f is, 0 The domai of f f is, 0. Note: The fuctio y is restricted because the domai of f f f has a domai of All Real Numbers. However, the domai of was restricted. You Try: Let f ad g. Fid the followig fuctios ad state their domais.. f g. f g. f g. g f QOD: Give a eample of two distict fuctios f ad g such that f g g f. Sample CCSD Commo Eam Practice Questio(s):. Let f ad f g? A. B. C. D. g. What is the differece of the two fuctios, Page of 7 McDougal Littell:

12 . If f ad g, what is the product of f ad A. B. C. 0 D. 8 g?. Let f ad g. What epressio is equal to A. B. C. D. 6 f g? Sample SAT Questio(s): Take from College Board olie practice problems.. The graphs of the fuctios f ad g i the iterval from ad are show above. Which of the followig could epress g i terms of f? (A) g f (B) g f (C) g f (D) g f (E) g f. Let the fuctio f be defied by f a, where a is a costat. If f f what is the value of a? (A) (B) 0 (C) (D) 0 (E) 0 Page of 7 McDougal Littell: ,

13 . Let the fuctio f be defied by f 7 0 ad value of t? Grid-I f t 0. What is oe possible Page of 7 McDougal Littell:

14 Syllabus Objective: 7. The studet will derive ad verify iverses of fuctios. Iverse Relatio: a mappig of the output values of a relatio to its iput values E: Fid the iverse relatio of the relatio. 0 y 7 To fid the iverse relatio, we will switch the iput () values ad the output (y) values. 7 y 0 y 0 Note: Lookig at the graph of the relatio (solid poits) ad its iverse relatio (ope poits), we ca see that the iverse relatio icludes all of the poits reflected over the lie y Fidig the Equatio of a Iverse Relatio: recall that the iverse of a relatio is its reflectio over the lie y. Therefore, to fid the equatio of a iverse relatio, we will reverse the ad y variables ad solve for y. Note: If both the relatio ad its iverse relatio are fuctios, the the two relatios are called iverse fuctios. Notatio for Iverse Fuctios: The iverse of a fuctio f is deoted f. Cautio: This is ot to be cofused with the epoet!! E: Fid the equatio of the iverse fuctio of f. Step Oe: Switch the ad y variables. Note: Step Two: Solve for y. Step Three: Write i iverse otatio. f y. y y y f Page of 7 McDougal Littell:

15 We ca verify our aswer usig the graph of the ordered pairs, as i our first eample. However, a fuctio ad its iverse have aother special relatioship. Verifyig Iverse Fuctios: To verify that two fuctios are iverses, we must show that f f f f. The fuctio y is the idetity fuctio, so the compositio of a fuctio ad its iverse is the idetity. E: Show algebraically ad graphically that the fuctios f ad f are iverses. Step Oe: Show that Step Two: Show that. f f. f f f f f f f f f f f f f f Step Three: Graph the fuctios ad show that they are a reflectio over the lie y I this graph, the lie y is bold. We have show that these fuctios are iverses.. E: Fid the iverse of the fuctio y for 0. Step Oe: Switch the ad y. y Step Two: Solve for y. y Because 0, we oly eed the positive square root. y Step Three: Rewrite i iverse otatio. y Page of 7 McDougal Littell:

16 Let s take a look at the graphs of these two fuctios. They are reflectios of each other over the lie y. Calculator Note: To graph y o its restricted domai, use paretheses after the fuctio. What if the domai of y was ot restricted? Let s take a look at the graphs. You ca see that the iverse of y is ot a fuctio. By lookig at a fuctio s graph, we ca see if it has a iverse fuctio usig the Horizotal Lie Test. Horizotal Lie Test: If a horizotal lie itersects the graph of a fuctio f ot more tha oce, the the iverse of f is a fuctio. E: Determie whether y is a fuctio. If it is, determie if it has a iverse fuctio. If it does, fid the iverse fuctio ad graph to verify. Step Oe: Look at the graph of to verify it is a fuctio. y ad use the vertical lie test y 0 It passes the vertical lie test, so it is a fuctio. Step Two: Look at the graph of y ad use the horizotal lie test to verify it has a iverse fuctio. It passes the horizotal lie test, so it has a iverse fuctio Step Three: Fid the iverse fuctio. y y y y Step Four: Graph the two fuctios. The iverse is a fuctio, ad it is the reflectio of the origial fuctio over the lie y. Page 6 of 7 McDougal Littell:

17 Power Fuctio: a fuctio of the form b y a, where a is a real umber ad b is a ratioal umber Note: As show i the eamples above, the iverse of a power fuctio is a radical fuctio, which will be discussed i the et sectio. Graphig Iverses o the Graphig Calculator Your calculator caot fid the iverse of a fuctio, but it ca draw the iverse. E: Use a graphig calculator to graph the iverse of y. Step Oe: Eter the fuctio ito Y. (For absolute value, use abs i the MATH meu.) Step Two: O the home scree choose 8:DrawIv from the Draw meu, the choose Y from the VARS meu. Press Eter, ad the calculator will automatically draw the iverse. Is the iverse a fuctio? Did you kow before the calculator graphed it? You Try: Fid the iverse of the fuctio f, 0. Verify that these are iverses both algebraically ad graphically. Is the iverse a fuctio? QOD: What is the differece betwee the vertical ad horizotal lie tests? Eplai why each test is used i each case. Sample CCSD Commo Eam Practice Questio(s): Which is the iverse of the fuctio y? A. B. C. D. y, where 0 y, where 0 9 y, where 0 y, where 0 Page 7 of 7 McDougal Littell:

18 Syllabus Objectives: 7.6 The studet will graph square root ad cube root equatios. 7.7 The studet will fid the domai ad rage of square root ad cube root equatios. Radical Fuctio: a fuctio of the form y a h k I the previous sectio, we graphed square root ad cube root fuctios. (These are two types of radical fuctios.) The square root fuctio is the iverse of a quadratic fuctio, ad the cube root fuctio is the iverse of a cubic fuctio. (These are both power fuctios.) We will use the paret fuctios y ad y. Graph of y : Domai: 0 ; Rage: y 0 Graph of y Domai: all real umbers; Rage: all real umbers Eplore: Graph the followig o the graphig calculator ad make ote of the chages to the appropriate paret fuctio. y y y y y y y y Do the same activity with cube roots i place of the square roots. E: Usig your fidigs from the eploratio, predict how the graph of would compare to the graph of y calculator to verify your cojecture.. Use the graphig y 8 Sample aswer: It would be reflected over the -ais, stretched vertically, ad shifted to the left ad up 8. Page 8 of 7 McDougal Littell:

19 Summary of the Trasformatios o Square Root ad Cube Root Fuctios: y a h k y a h k a: If a 0, the graph is reflected over the -ais. If a, the graph is stretched vertically. If a, the graph is stretched horizotally. h: The graph is shifted h uits horizotally. k: The graph is shifted k uits vertically. E: Describe how the graph of y compares to y 0 y. The sketch the graph ad state its domai ad rage. h ad k, so the graph will be shifted left ad up. a, so the graph will ot be stretched. Note: It helps to sketch the graph of the paret fuctio first. The graph of y is the bold graph Domai: ; Rage: y (Note: These ca be foud by lookig at the graph.) You Try: Sketch the graph the fuctio i the form y a h k first.) y ad state its domai ad rage. (Hit: Write it QOD: If f? f, what would the graph of f look like? What is the domai ad rage of Page 9 of 7 McDougal Littell:

20 Sample CCSD Commo Eam Practice Questio(s): What is the graph of y? Page 0 of 7 McDougal Littell:

21 Syllabus Objective: 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. Solvig a Radical Equatio Step Oe: Isolate the radical. Step Two: Raise both sides to the th power, where is the ide of the radical. Step Three: Isolate the variable. Step Four: Check your solutio i the origial equatio. This is crucial, as you may obtai etraeous solutios. E: Solve the equatio 8. Step Oe: Step Two: 7 Step Three: 7 Step Four: The solutio works i the origial equatio, so 7 E: Solve the equatio Step Oe: Step Two: Step Three: 6 9 Step Four: The solutio does ot work i the origial equatio. Therefore, it is a etraeous solutio, ad this equatio has NO SOLUTION. Questio: Could you have determied earlier i the process of solvig that this equatio had o solutio? Eplai. Page of 7 McDougal Littell:

22 E: Solve the equatio. Step Oe: Doe (radical is isolated) Step Two: Step Three: Because this is a quadratic equatio, you may use oe of the methods for solvig quadratic equatios (quadratic formula, factorig, or completig the square) , This ca be factored, so we will solve usig this method. Step Four: 0 0 Solutio Set:, Check: : : Equatios with Two Radicals: To solve, we will move the radicals to opposite sides, the raise both sides to the th power, where is the ide of the radical. E: Solve the equatio 0 0. Step Oe: Move oe of the radicals to the other side. 0 Step Two: Raise both sides to the th power Step Three: Solve for. 0 7 Step Four: These are ot perfect th roots, so we will check o the calculator. Page of 7 McDougal Littell:

23 The solutio is. 7 Solvig Equatios with Ratioal Epoets Step Oe: Isolate the epressio with the ratioal epoet. Step Two: Raise both sides to the reciprocal power. Step Three: Isolate the variable. Step Four: Check your solutio i the origial equatio. This is crucial, as you may obtai etraeous solutios. E: Solve the equatio 8. 8 Step Oe: 6 Step Two: 6 Step Three: 7 Step Four: The solutio is 7. Solvig a Radical Equatio o the Graphig Calculator: We will solve equatios by graphig. You may either graph both sides of the equatio as two fuctios ad fid the -coordiate of the poit of itersectio, or set the equatio equal to zero ad fid the -itercept of the resultig fuctio. E: Solve the equatio by graphig. Method : Graph both sides of the equatio. The solutio is 8. Questio: What does the y-value represet i the poit of itersectio? Method : Set the equatio equal to zero. 0 The solutio is 8. Page of 7 McDougal Littell:

24 You Try: Solve the equatio 6. Verify your aswer o the graphig calculator. QOD: Eplai usig ratioal epoets why raisig a radical to the th power, where is the ide of the radical, will elimiate the radical. Page of 7 McDougal Littell:

25 Sample CCSD Commo Eam Practice Questio(s): What is the value of i the equatio 6? A. = 0 B. = 8 C. = D. = 80 Page of 7 McDougal Littell:

26 Syllabus Objective: 7.9 The studet will solve applicatio problems usig roots, ratioal epoets, power fuctios, fuctio operatios, ad radical equatios. Powers, roots, ad radicals are foud i may real-life situatios. We will eplore several. Note: Ofte, real-life applicatios do ot yield ice umbers. We will use our calculator to approimate our results i these problems. Solvig Equatios Usig th Roots E: A basketball has a volume of about.6 cubic iches. The formula for the volume of a basketball is V r. Fid the radius of the basketball. Substitute the kow value(s) ito the formula. V.6.6 r Solve for the remaiig variable..6 r.7 r.7.77 r r Calculator Note: Be sure ot to roud too soo i the problem. Type i the epressio.7 Do ot use a approimatio (i.e..) for π.. Be sure to aswer the questio ad use appropriate uits. The radius is.77 iches. Operatios with Fuctios E: A professor performs a eperimet o bacteria ad fids that the growth rate G of the bacteria ca be modeled by Gt t, ad that the death rate D is Dt t is the time i hours. Fid a epressio for the umber N of bacterial livig at a time t. The epressio for the umber of bacteria livig will equal the umber of bacteria growig mius the umber of bacteria dyig. Nt () Gt () Dt (), where t Usig the epressios give, we have Nt ( ) 8t 0.8t or Nt ( ) 7.t 0. Iverse Fuctios E: The formula to covert temperatures from degrees Fahreheit to degrees Celsius is C F. Write the iverse of the fuctio ad eplai how it ca be used. 9 Page 6 of 7 McDougal Littell:

27 To fid the iverse, we will solve for F. 9 C F 9 F C This ew equatio will covert temperatures from degrees Celsius to degrees Fahreheit. Graphig Radical Fuctios ad Solvig Radical Equatios E: The legth of a whale ca be modeled by L.0 W, where L is the legth i feet, ad W is the weight i tos. Graph the model, the use the graph to fid the weight of a whale that is 60 ft log. Graph the model. Graph the fuctio L 60 i Y ad fid the poit of itersectio to solve the problem. The weight of the whale is approimately.9 tos. You Try: The strigs of guitars ad piaos are uder tesio. The speed v of a wave o the strig depeds o the force (tesio) F o the strig ad the mass M per uit legth L accordig to the formula F L v. A wave travels through a strig with a mass of 0. kilograms at a speed of 9 meters per M secod. It is stretched by a force of 9.6 Newtos. Fid the legth of the strig. Sample CCSD Commo Eam Practice Questio(s): A aimal populatio ca be modeled over time by P t t 0, where t is measured i weeks. After how may weeks will the populatio be 8 aimals? A. 8 B. C. D. 7 Page 7 of 7 McDougal Littell:

n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a

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