Essential Question How can you use properties of exponents to simplify products and quotients of radicals?
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1 . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.7.G Properties of Ratioal Expoets ad Radicals Essetial Questio How ca you use properties of expoets to simplify products ad quotiets of radicals? Reviewig Properties of Expoets Work with a parter. Let a ad b be real umbers. Use the properties of expoets to complete each statemet. The match each completed statemet with the property it illustrates. Statemet Property 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 Expoet e. ( a b ), b 0 E. Zero Expoet f. a, a 0 F. Quotiet of Powers a SELECTING TOOLS To be proficiet i math, you eed to cosider the tools available to help you check your aswers. For istace, the followig calculator scree shows that ad 8 are equivalet. ( ())( ()) (8) g. a 0, a 0 G. Power of a Quotiet Simplifyig Expressios with Ratioal Expoets Work with a parter. Show that you ca apply the properties of iteger expoets to ratioal expoets by simplifyig each expressio. Use a calculator to check your aswers. a. / / b. / / c. ( / ) d. (0 / ) e. 8/ 8 / f. 7/ 7 / Simplifyig Products ad Quotiets of Radicals Work with a parter. Use the properties of expoets to write each expressio as a sigle radical. The evaluate each expressio. Use a calculator to check your aswers. a. b. c. 7 d. 98 e. 0 Commuicate Your Aswer f.. How ca you use properties of expoets to simplify products ad quotiets of radicals?. Simplify each expressio. a. 7 b. 0 c. ( / / ) Sectio. Properties of Ratioal Expoets ad Radicals 9
2 . Lesso What You Will Lear Core Vocabulary simplest form of a radical, p. 97 cojugate, p. 98 like radicals, p. 98 Previous properties of iteger expoets ratioalizig the deomiator absolute value COMMON ERROR Whe you multiply powers, do ot multiply the expoets. For example, 0. Use properties of ratioal expoets to simplify expressios with ratioal expoets. Use properties of radicals to simplify ad write radical expressios i simplest form. Properties of Ratioal Expoets The properties of iteger expoets that you have previously leared ca also be applied to ratioal expoets. Core Cocept Properties of Ratioal Expoets Let a ad b be real umbers ad let m ad be ratioal umbers, such that the quatities i each property are real umbers. Property Name Defiitio Example Product of Powers Power of a Power Power of a Product a m a a m + (a m ) a m (ab) m a m b m / / (/ + /) ( / ) (/ ) Negative Expoet Zero Expoet Quotiet of Powers Power of a Quotiet a m a m, a 0 a 0, a 0 a m a am, a 0 ( a b m, b 0 b) m am ( 9) / / 9 / / 0 / / / (/ /) ( 7 / ) 7/ / 9 Chapter Ratioal Expoets ad Radical Fuctios Usig Properties of Expoets Use the properties of ratioal expoets to simplify each expressio. a. 7 / 7 / 7 (/ + /) 7 / b. ( / / ) ( / ) ( / ) (/ ) (/ ) / / c. ( ) / [( ) ] / ( ) / [ ( /)] d. / ) / e. ( / / ( /) / [ ( / ) Moitorig Progress Simplify the expressio. ] (7 / ) 7 (/ ) 7 /. / /. Help i Eglish ad Spaish at BigIdeasMath.com /. ( 0/ / ). ( / 7 / )
3 Simplifyig Radical Expressios The Power of a Product ad Power of a Quotiet properties ca be expressed usig radical otatio whe m for some iteger greater tha. Core Cocept Properties of Radicals Let a ad b be real umbers ad let be a iteger greater tha. Property Name Defiitio Example Product Property Quotiet Property a b a b a a b, b 0 b 8 8 Usig Properties of Radicals Use the properties of radicals to simplify each expressio. a. 8 8 Product Property of Radicals b Quotiet Property of Radicals A expressio cotaiig a radical with idex is i simplest form whe these three coditios are met. No radicads have perfect th powers as factors other tha. No radicads cotai fractios. No radicals appear i the deomiator of a fractio. To meet the last two coditios, ratioalize the deomiator by multiplyig the expressio by a appropriate form of that elimiates the radical from the deomiator. Write each expressio i simplest form. Writig Radicals i Simplest Form a. b. 7 8 a. 7 Factor out perfect cube. 7 Product Property of Radicals Simplify. 7 7 b Make the radicad i the deomiator a perfect fifth power. Product Property of Radicals Simplify. Sectio. Properties of Ratioal Expoets ad Radicals 97
4 For a deomiator that is a sum or differece ivolvig square roots, multiply both the umerator ad deomiator by the cojugate of the deomiator. The expressios a b + c d ad a b c d are cojugates of each other, where a, b, c, ad d are ratioal umbers. Write i simplest form. + + Writig a Radical Expressio i Simplest Form + The cojugate of + is. ( ) ( ) Sum ad Differece Patter Simplify. Radical expressios with the same idex ad radicad are like radicals. To add or subtract like radicals, use the Distributive Property. Addig ad Subtractig Like Radicals ad Roots Simplify each expressio. a b. (8 / ) + 0(8 / ) c. a ( + 7) b. (8 / ) + 0(8 / ) ( + 0)(8 / ) (8 / ) c. 7 ( ) Moitorig Progress Simplify the expressio Help i Eglish ad Spaish at BigIdeasMath.com (9 / ) + 8(9 / ). + 0 The properties of ratioal expoets ad radicals ca also be applied to expressios ivolvig variables. Because a variable ca be positive, egative, or zero, sometimes absolute value is eeded whe simplifyig a variable expressio. Rule Example 98 Chapter Ratioal Expoets ad Radical Fuctios Whe is odd x x 7 7 ad 7 ( ) 7 Whe is eve x x ad ( ) Absolute value is ot eeded whe all variables are assumed to be positive.
5 Simplifyig Variable Expressios STUDY TIP You do ot eed to take the absolute value of y because y is beig squared. Simplify each expressio. a. x y b. y 8 a. y (y ) (y ) y x x b. y 8 y 8 x ( y ) x y Writig Variable Expressios i Simplest Form COMMON ERROR You must multiply both the umerator ad deomiator of the fractio by y so that the value of the fractio does ot chage. Write each expressio i simplest form. Assume all variables are positive. a. a 8 b c b. a. a 8 b c a a b 0 b c b. x y 8 x y 8 c. xy/ x / z Factor out perfect fifth powers. a b 0 c a b Product Property of Radicals ab c a b Simplify. x y y 8 y x y y 9 x y y c. xy/ x / z 7x ( /) y / z ( ) 7x / y / z Make deomiator a perfect cube. Product Property of Radicals Simplify. Addig ad Subtractig Variable Expressios Perform each idicated operatio. Assume all variables are positive. a. y + y b. z z z a. y + y ( + ) y y b. z z z z z z z (z z) z 9z z Moitorig Progress Sectio. Properties of Ratioal Expoets ad Radicals 99 Help i Eglish ad Spaish at BigIdeasMath.com Simplify the expressio. Assume all variables are positive.. 7q 9. x 0 y. xy / x / y /. 9w w w
6 . Exercises Dyamic Solutios available at BigIdeasMath.com Vocabulary ad Core Cocept Check. WRITING How do you kow whe a radical expressio is i simplest form?. WHICH ONE DOESN T BELONG? Which radical expressio does ot belog with the other three? Explai your reasoig. x 9x Moitorig Progress ad Modelig with Mathematics I Exercises, use the properties of ratioal expoets to simplify the expressio. (See Example.). (9 ) /. ( ) /. /. 7 7 / 7. ( 8 / 0 ) 8. ( 9 / ) 9. ( / / ) 0. ( / / ) /. / / /. 9 /8 9 7/8 7 / I Exercises 0, use the properties of radicals to simplify the expressio. (See Example.). 7. I Exercises 9, write the expressio i simplest form. (See Example.) I Exercises 7, simplify the expressio. (See Example.) ( / ) + 9( / ) 0. (8 / ) (8 / ) I Exercises 8, write the expressio i simplest form. (See Example.) ( / ) ( / ). / + (0 / ) 7. ERROR ANALYSIS Describe ad correct the error i simplifyig the expressio. + ( + ) Chapter Ratioal Expoets ad Radical Fuctios
7 8. MULTIPLE REPRESENTATIONS Which radical expressios are like radicals? A ( /9 ) / B C E D ( ) F I Exercises 9, simplify the expressio. (See Example.) 9. 8y 8 0. r t m 0. k. z g. h h p 7 p. ERROR ANALYSIS Describe ad correct the error i simplifyig the expressio. h g h g (h ) g h g I Exercises 70, perform the idicated operatio. Assume all variables are positive. (See Example 8.). y + 9 y. z z 7. x 7/ x 7/ 8. 7 m 7 + m 7/ 9. w 0 + w w 70. (p / p / ) p MATHEMATICAL CONNECTIONS I Exercises 7 ad 7, fid simplified expressios for the perimeter ad area of the give figure. 7. x / x 7. x / x / 7. MODELING WITH MATHEMATICS The optimum diameter d (i millimeters) of the pihole i a pihole camera ca be modeled by d.9[(. 0 ) ] /, where is the legth (i millimeters) of the camera box. Fid the optimum pihole diameter for a camera box with a legth of 0 cetimeters. pihole film. OPEN-ENDED Write two variable expressios ivolvig radicals, oe that eeds absolute value i simplifyig ad oe that does ot eed absolute value. Justify your aswers. I Exercises 7, write the expressio i simplest form. Assume all variables are positive. (See Example 7.) 7. 8a 7 b c 9 8. r s 9 t 7 0m 9. 0x 7 0. y x y.. w w w. v 7 v 8w / v / 7w / v /. 7x / y / z / x / y / tree 7. MODELING WITH MATHEMATICS The surface area S (i square cetimeters) of a mammal ca be modeled by S km /, where m is the mass (i grams) of the mammal ad k is a costat. The table shows the values of k for differet mammals. Mammal Rabbit Huma Bat Value of k a. Fid the surface area of a bat whose mass is grams. b. Fid the surface area of a rabbit whose mass is. kilograms (. 0 grams). c. Fid the surface area of a huma whose mass is 9 kilograms. Sectio. Properties of Ratioal Expoets ad Radicals 0
8 7. MAKING AN ARGUMENT Your fried claims it is ot possible to simplify the expressio 7 9 because it does ot cotai like radicals. Is your fried correct? Explai your reasoig. 7. PROBLEM SOLVING The apparet magitude of a star is a umber that idicates how fait the star is i relatio to other stars. The expressio.m tells m. how may times faiter a star with apparet magitude m is tha a star with apparet magitude m. Star Apparet magitude Costellatio Vega 0.0 Lyra Altair 0.77 Aquila Deeb. Cygus a. How may times faiter is Altair tha Vega? b. How may times faiter is Deeb tha Altair? c. How may times faiter is Deeb tha Vega? Deeb Cygus Altair Vega Lyra Aquila 77. CRITICAL THINKING Fid a radical expressio for the perimeter of the triagle iscribed i the square show. Simplify the expressio. 78. HOW DO YOU SEE IT? Without fidig poits, match the fuctios f(x) x ad g(x) x with their graphs. Explai your reasoig. A. 8 y x B. 8 y x 79. REWRITING A FORMULA You have filled two roud balloos with water. Oe balloo cotais twice as much water as the other balloo. a. Solve the formula for the volume of a sphere, V πr, for r. b. Substitute the expressio for r from part (a) ito the formula for the surface area of a sphere, S πr. Simplify to show that S (π) / (V) /. c. Compare the surface areas of the two water balloos usig the formula i part (b). 80. THOUGHT PROVOKING Determie whether the expressios (x ) / ad (x / ) are equivalet for all values of x. 8. DRAWING CONCLUSIONS Substitute differet combiatios of odd ad eve positive itegers for m ad i the expressio x m. Whe you caot assume x is positive, explai whe absolute value is eeded i simplifyig the expressio. 8 Maitaiig Mathematical Proficiecy Reviewig what you leared i previous grades ad lessos Idetify the focus, directrix, ad axis of symmetry of the parabola. The graph the equatio. (Sectio.) 8. y x 8. y x 8. y x Write a rule for g. Describe the graph of g as a trasformatio of the graph of f. (Sectio.7) 8. f(x) x x x, g(x) f(x) 8. f(x) x x, g(x) f(x) 87. f(x) x, g(x) f(x ) 88. f(x) x + x x, g(x) f(x) 0 Chapter Ratioal Expoets ad Radical Fuctios
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