6.1 Th Roots And Use Rational Exponents
6.2Apply Properties of RationalExponentsGoalYour Notesp Simplify expressions involving rational exponents.VOCABULARYSimplest form of a radicalLike radicalsPROPERTIES OF RATIONAL EXPONENTSLet a and b be real numbers and let m and n berational numbers. The following properties have thesame names as those in Lesson 5.1, but now apply torational exponents.Property1. am p an 5 am 1 n41/2 p 43/2 5 4(1/2 1 3/2)2. (am)n 5 amn(25/2)2 5 2(5/2 p 2)3. (ab)m 5 ambm(16 p 4)1/2 5 161/2 p 41/214. a2m 5 }m, a Þ 012521/2 5 }51/2amaa mam}6. 1 }5,bÞ0b2bm35/2} 5 3(5/2 2 1/2) 531/227 1/3271/3}} 55881/3am 2 n, a Þ 05. }n 5 aCopyright Holt McDougal. All rights reserved.251 2Lesson 6.2 Algebra 2 Notetaking Guide163
6.2Apply Properties of RationalExponentsGoalYour Notesp Simplify expressions involving rational exponents.VOCABULARYSimplest form of a radical A radical with index n isin simplest form if the radicand has no perfect nthpowers as factors and any denominator has beenrationalized.Like radicals Two radical expressions with the sameindex and radicand.PROPERTIES OF RATIONAL EXPONENTSLet a and b be real numbers and let m and n berational numbers. The following properties have thesame names as those in Lesson 5.1, but now apply torational exponents.Property1. am p an 5 am 1 n41/2 p 43/2 5 4(1/2 1 3/2)5 42 5 162. (am)n 5 amn(25/2)2 5 2(5/2 p 2) 5 25 5 323. (ab)m 5 ambm(16 p 4)1/2 5 161/2 p 41/254p25814. a2m 5 }m, a Þ 0112521/2 5 }5 }1/2amaa mam}6. 1 }5,bÞ0b2bm35/2} 5 3(5/2 2 1/2)31/227 1/3271/3}} 55881/3am 2 n, a Þ 05. }n 5 aCopyright Holt McDougal. All rights reserved.251 255 32 5 932}Lesson 6.2 Algebra 2 Notetaking Guide163
Your NotesUse properties of exponentsExample 1Use the properties of rational exponents to simplify theexpression.a. 91/2 p 93/4 5b. (72/3 p 51/6)3 55535/63162/3 4d. }542/3c. }51/312PROPERTIES OF RADICALSProduct Property of RadicalsQuotient Property of RadicalsÎ}ab 5}n}Ïa p b 5n,bÞ0Use properties of radicalsExample 2Use the properties of radicals to simplify theexpression.5}5}a. Ï27 p Ï9 5553}Ï192b. }3} 5Ï35ProductpropertyQuotientproperty5Checkpoint Simplify the expression.1.164( 66Lesson 6.2 Algebra 2 Notetaking Guidep56)21/6}Ï2452. }}Ï5Copyright Holt McDougal. All rights reserved.
Your NotesUse properties of exponentsExample 1Use the properties of rational exponents to simplify theexpression.a. 91/2 p 93/4 5 9(1/2 1 3/4) 5 95/4b. (72/3 p 51/6)3 5 (72/3)3 p (51/6)35 7(2/3 p 3) p 5(1/6 p 3)5 72 p 51/2 5 49 p 51/235/63162/3 4d. }542/3c. }5 3(5/6 2 1/3) 5 33/6 5 31/21/31F12G216 2/3 45 (42/3)4 5 4(2/3 p 4) 5 48/34}PROPERTIES OF RADICALSProduct Property of RadicalsÎn}}n}n}n}Quotient Property of RadicalsÏa p b 5 Ïa p ÏbnÏaa} 5 }n} , b Þ 0bÏbUse properties of radicalsExample 2Use the properties of radicals to simplify theexpression.5}5}5}5}a. Ï27 p Ï9 5 Ï 27 p 9 5 Ï 243 5 33}Ï192b. }3} 5Ï3Î1923}3}3}5 Ï64 5 4ProductpropertyQuotientpropertyCheckpoint Simplify the expression.1.( 66130}164Lesson 6.2 Algebra 2 Notetaking Guidep56)21/6}Ï2452. }}Ï57Copyright Holt McDougal. All rights reserved.
Your NotesWrite radicals in simplest formExample 3Write the expression in simplest form.5}Ï128 5Factor out perfect fifth power.p5Product property5Simplify.Add and subtract like radicals and rootsExample 4Simplify the expression.a. 2(122/3) 1 7(122/3) 54}4}b. Ï48 2 Ï 3 5p25Checkpoint Write the expression in simplest form.Î59}3.4}4}4. 6 Ï6 1 2 Ï 6}3Simplify expressions involving variablesExample 5Simplify the expression. Assume all variables arepositive.a.}Ï32x15 55b. (36m4n10)1/2 5Î5}c.3a9b} 564 742x z5d. }3/2 23 56xCopyright Holt McDougal. All rights reserved.yzLesson 6.2 Algebra 2 Notetaking Guide165
Your NotesWrite radicals in simplest formExample 3Write the expression in simplest form.5}5}Ï128 5 Ï32 p 45}Factor out perfect fifth power.5}5 Ï 32 p Ï 4Product property5 2Ï 4Simplify.5}Add and subtract like radicals and rootsExample 4Simplify the expression.a. 2(122/3) 1 7(122/3) 5 (2 1 7)(122/3) 5 9(122/3)4}4}4}4}4}b. Ï48 2 Ï 3 5 Ï 16 p Ï 3 2 Ï34}4}4}4}5 2Ï 3 2 Ï 3 5 (2 2 1)Ï 3 5 Ï3Checkpoint Write the expression in simplest form.Î59}3.4}4}4. 6 Ï6 1 2 Ï 6}33}Ï154}8Ï 6}3Simplify expressions involving variablesExample 5Simplify the expression. Assume all variables arepositive.}}}}55a.5 Ï p (x 3)5 5 Ï 25 p Ï (x 3)5 5 2x 3b. (36m4n10)1/2 5 361/2(m4)1/2(n10)1/2Ï32x155Î3255 6m(4 p 1/2)n(10 p 1/2) 5 6m2n5}c.5a9}b6}}3a9(a3)3ÏÏa3}5 3} 5 }} 5 }Ïb6 Ï3 (b2)3 b234 742x z5 7x (4 2 3/2)yd. }3/2 23 56xCopyright Holt McDougal. All rights reserved.yz2(23)z(7 2 5)5 7x 5/2y 3z 2Lesson 6.2 Algebra 2 Notetaking Guide165
Your NotesExample 6Write the expression in simplest form. Assume allvariables are positive.You mustmultiplythe originalexpression bya form of 1, inthis caseWrite variable expressions in simplest formÎ}4a2bMultiply to make denominatora perfect fourth power.} 56,when simplifyingso that the newexpression isequivalent.5Simplify.5Quotient property.5Simplify.Example 7Add and subtract expressions involving variablesPerform the indicated operation. Assume all variablesare positive.}}55a. 10 Ïy 2 6Ïy5b. 3a2b1/4 1 4a2b1/4 5Checkpoint Simplify the expression. Assume allvariables are positive.5.Ï3}8x7y3z116.}7 Ï2a53}2 a Ï128a23Homework166Lesson 6.2 Algebra 2 Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesYou mustmultiplythe originalexpression bya form of 1, inÎ4this caseWrite variable expressions in simplest formExample 6}Write the expression in simplest form. Assume allvariables are positive.Î}a24 } 5b6b2}b2 ,5Î ÎÎ}a24 } b6}4}4a2b2b8}b2b2}Multiply to make denominatora perfect fourth power.Simplify.}Ï4 a2b25 }}Ï4 b8}4 2 2a bÏ5 }when simplifyingso that the newexpression isequivalent.Quotient property.Simplify.b2Example 7Add and subtract expressions involving variablesPerform the indicated operation. Assume all variablesare positive.}}}}5555a. 10 Ïy 2 6Ïy 5 (10 2 6)Ïy 5 4Ïyb. 3a2b1/4 1 4a2b1/4 5 (3 1 4)a2b1/4 5 7a2b1/4Checkpoint Simplify the expression. Assume allvariables are positive.5.Ï3}8x7y3z113}2x2yz 3Ï xz26.}7 Ï2a53}2 a Ï128a233}3aÏ 2a2Homework166Lesson 6.2 Algebra 2 Notetaking GuideCopyright Holt McDougal. All rights reserved.
6.3Perform Function Operationsand CompositionGoalYour Notesp Perform operations with functions.VOCABULARYPower functionCompositionOPERATIONS ON FUNCTIONSLet f and g be any two functions. A new function hcan be defined by performing any of the four basicoperations on f and g.Operation and DefinitionAdditionh(x) 5 f(x) 1 g(x)Subtractionh(x) 5 f(x) 2 g(x)Multiplicationh(x) 5 f(x) p g(x)Example: f(x) 5 3x, g(x) 5 x 1 3h(x) 5 3x 1 (x 1 3)5h(x) 5 3x 2 (x 1 3)5h(x) 5 3x(x 1 3)5Divisionf(x)h(x) 5 }g(x)h(x) 5The domain of h consists of the x-values that are in the. Additionally, the domain of adomains ofquotient does not include x-values for which g(x) 5.Copyright Holt McDougal. All rights reserved.Lesson 6.3 Algebra 2 Notetaking Guide167
6.3Perform Function Operationsand CompositionGoalYour Notesp Perform operations with functions.VOCABULARYPower function A function of the form y 5 ax bwhere a is a real number and b is a rationalnumberComposition The composition of a function g witha function f is h(x) 5 g(f (x)). The domain of h is theset of all x-values such that x is in the domain of fand f(x) is in the domain of g.OPERATIONS ON FUNCTIONSLet f and g be any two functions. A new function hcan be defined by performing any of the four basicoperations on f and g.Operation and DefinitionAdditionh(x) 5 f(x) 1 g(x)Subtractionh(x) 5 f(x) 2 g(x)Multiplicationh(x) 5 f(x) p g(x)Example: f(x) 5 3x, g(x) 5 x 1 3h(x) 5 3x 1 (x 1 3)5 4x 1 3h(x) 5 3x 2 (x 1 3)5 2x 2 3h(x) 5 3x(x 1 3)5 3x 2 1 9xDivisionf(x)h(x) 5 }g(x)3xx13h(x) 5 }The domain of h consists of the x-values that are in thedomains of both f and g . Additionally, the domain of aquotient does not include x-values for which g(x) 5 0 .Copyright Holt McDougal. All rights reserved.Lesson 6.3 Algebra 2 Notetaking Guide167
Your NotesExample 1Add and subtract functionsLet f (x) 5 3x 1/2 and g(x) 5 25x 1/2. Find the following.a. f(x) 1 g(x)b. f(x) 2 g(x)c. the domains of f 1 g and f 2 gSolutiona. f(x) 1 g(x) 5 3x1/2 1 (25x 1/2)5b. f(x) 2 g(x) 5 3x1/2 2 (25x 1/2)5c. The functions f and g each have the same domain:. So, the domains off 1 g and f 2 g also consist of.Example 2Multiply and divide functionsLet f (x) 5 7x and g(x) 5 x 1/6. Find the following.a. f(x) p g(x)f(x)g(x)b. }fc. the domains of f p g and }gSolutiona. f(x) p g(x) 5 (7x)(x1/6) 5f(x)g(x)b. } 5c. The domain of f consists of, andthe domain of g consists of. So, the domain of f p g consists of. Because g(0) 5,fthe domain of }g is restricted to.168Lesson 6.3 Algebra 2 Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesAdd and subtract functionsExample 1Let f (x) 5 3x 1/2 and g(x) 5 25x 1/2. Find the following.a. f(x) 1 g(x)b. f(x) 2 g(x)c. the domains of f 1 g and f 2 gSolutiona. f(x) 1 g(x) 5 3x1/2 1 (25x 1/2)5 [3 1 (25)]x1/2 5 22x 1/2b. f(x) 2 g(x) 5 3x1/2 2 (25x 1/2)5 [3 2 (25)]x1/2 5 8x 1/2c. The functions f and g each have the same domain:all nonnegative real numbers . So, the domains off 1 g and f 2 g also consist of all nonnegative realnumbers .Multiply and divide functionsExample 2Let f (x) 5 7x and g(x) 5 x 1/6. Find the following.a. f(x) p g(x)f(x)g(x)b. }fc. the domains of f p g and }gSolutiona. f(x) p g(x) 5 (7x)(x1/6) 5 7x (1 1 1/6) 5 7x7/6f(x)7xb. } 5 }5 7x (1 2 1/6) 5 7x 5/61/6g(x)xc. The domain of f consists of all real numbers , andthe domain of g consists of all nonnegative realnumbers . So, the domain of f p g consists of allnonnegative real numbers . Because g(0) 5 0 ,fthe domain of }g is restricted to all positive realnumbers .168Lesson 6.3 Algebra 2 Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesCheckpoint Complete the following exercise.1. Let f(x) 5 5x 3/2 and g(x) 5 22x 3/2. Find (a) f 1 g,f(b) f 2 g, (c) f p g, (d) }g, and (e) the domains.COMPOSITION OF FUNCTIONSThe composition of afunction g with a functionf is h(x) 5.The domain of h is theset of all x-values suchthat x is in the domainand f(x) is in theofdomain of.Copyright Holt McDougal. All rights reserved.Domain of fRange of fInputof fOutputof fxf (x)g(f (x))Inputof gOutputof gDomain of gRange of gLesson 6.3 Algebra 2 Notetaking Guide169
Your NotesCheckpoint Complete the following exercise.1. Let f(x) 5 5x 3/2 and g(x) 5 22x 3/2. Find (a) f 1 g,f(b) f 2 g, (c) f p g, (d) }g, and (e) the domains.a. 3x 3/2 b. 7x 3/2c. 210x 352d. 2}e. The domain of f 1 g, f 2 g, and f p g is allfnonnegative real numbers. The domain of }gis all positive real numbers.COMPOSITION OF FUNCTIONSThe composition of afunction g with a functionf is h(x) 5 g(f (x)) .The domain of h is theset of all x-values suchthat x is in the domainof f and f(x) is in thedomain of g .Copyright Holt McDougal. All rights reserved.Domain of fRange of fInputof fOutputof fxf (x)g(f (x))Inputof gOutputof gDomain of gRange of gLesson 6.3 Algebra 2 Notetaking Guide169
Your NotesExample 3Find compositions of functionsLet f (x) 5 6x 21 and g(x) 5 3x 1 5. Find the following.a. f (g(x))b. g(f(x))c. f(f(x))d. the domain of each compositionSolutiona. f (g(x)) 5 f(3x 1 5) 5b. g(f(x)) 5 g(6x21)5c. f (f (x)) 5 f (6x21) 5d. The domain of f(g(x)) consists ofexcept x 5theisconsist ofbecausebecause g12 5 0 is not in. (Note that f(0) 5, which.) The domains of g(f(x)) and f(f(x))except x 5, again.Checkpoint Complete the following exercise.2. Let f(x) 5 5x 2 4 and g(x) 5 3x21. Find (a) f (g(x)),(b) g(f(x)), (c) f(f(x)), and (d) the domain of eachcomposition.Homework170Lesson 6.3 Algebra 2 Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesExample 3Find compositions of functionsLet f (x) 5 6x 21 and g(x) 5 3x 1 5. Find the following.a. f (g(x))b. g(f(x))c. f(f(x))d. the domain of each compositionSolution6a. f (g(x)) 5 f(3x 1 5) 5 6(3x 1 5)21 5 }3x 1 5b. g(f(x)) 5 g(6x21)18155 3(6x21) 1 5 5 18x21 1 5 5 }xc. f (f (x)) 5 f (6x21) 5 6(6x21)21 5 6(621x) 5 60x 5 xd. The domain of f(g(x)) consists of all real numbers53153except x 5 2} because g 2}2 5 0 is not in60the domain of f . (Note that f(0) 5 } , whichis undefined .) The domains of g(f(x)) and f(f(x))consist of all real numbers except x 5 0 , againbecause 0 is not in the domain of f .Checkpoint Complete the following exercise.2. Let f(x) 5 5x 2 4 and g(x) 5 3x21. Find (a) f (g(x)),(b) g(f(x)), (c) f(f(x)), and (d) the domain of eachcomposition.15a. }x 243b. }5x 2 4c. 25x 2 24d. The domain of f (g(x)) and f (f (x)) is all realnumbers except x 5 0. The domain of g(f(x))4.is all real numbers except x 5 }5Homework170Lesson 6.3 Algebra 2 Notetaking GuideCopyright Holt McDougal. All rights reserved.
6.4Use Inverse FunctionsGoalYour Notesp Find inverse functions.VOCABULARYInverse relationInverse functionExample 1Find an inverse relationFind an equation for the inverse of the relationy 5 7x 2 4.y 5 7x 2 4Write original equation.Switch x and y.Addto each side.Solve for y. This is the inverse relation.INVERSE FUNCTIONSFunctions f and g are inverses of each other provided:f(g(x)) 5g(f(x)) 5andThe function g is denoted by f 21, read as “f inverse.”Example 2Verify that functions are inverses14Verify that f(x) 5 7x 2 4 and f 21(x) 5 } x 1 } are77inverses.Show that f (f 21 (x)) 5 x.f (f 21 (x)) 5 f 1 }x1}7721Copyright Holt McDougal. All rights reserved.4Show that f 21 (f(x)) 5 x.f 21 (f(x)) 5 f 21 (7x 2 4)555555Lesson 6.4 Algebra 2 Notetaking Guide171
6.4Use Inverse FunctionsGoalYour Notesp Find inverse functions.VOCABULARYInverse relation A relation that interchanges theinput and output values of the original relationInverse function The original relation and its inverserelation whenever both relations are functionsExample 1Find an inverse relationFind an equation for the inverse of the relationy 5 7x 2 4.y 5 7x 2 4x 5 7y 2 4x 1 4 5 7y17Write original equation.Switch x and y.Add 4 to each side.47}x 1 } 5 ySolve for y. This is the inverse relation.INVERSE FUNCTIONSFunctions f and g are inverses of each other provided:f(g(x)) 5 xg(f(x)) 5 xandThe function g is denoted by f 21, read as “f inverse.”Example 2Verify that functions are inverses14Verify that f(x) 5 7x 2 4 and f 21(x) 5 } x 1 } are77inverses.Show that f 21 (f(x)) 5 x.Show that f (f 21 (x)) 5 x.f (f 21 (x)) 5 f 1 }x1}77214145 71 }x1}224145 } (7x 2 4) 1 }5 x1424445 x2}1}7775 xCopyright Holt McDougal. All rights reserved.f 21 (f(x)) 5 f 21 (7x 2 4)7775 xLesson 6.4 Algebra 2 Notetaking Guide171
Your NotesCheckpoint Find the inverse of the function. Then verifythat your result and the original function are inverses.1. f (x) 5 23x 1 5Find the inverse of a power functionExample 3Find the inverse of f(x) 5 4x 2, x 0. Then graph fand f 21.f (x) 5 4x 2Write original function.y 5 4x 2Replace f(x) with y.Switch x and y.Divide each side by 4.You can check thesolution by notingthat the graph of1 }f 21(x) 5 2}2 Ï x isthe reflection ofthe graph off(x) 5 4x 2, x 0,in the line y 5 x.Take square roots of each side.yThe domain of f is restricted tonegative values of x. So, the rangeof f 21 must also be restricted tonegative values, and therefore theinverse isf 21 (x)51. (If the1xdomain were restricted to x 0, youwould choose f 21 (x) 5.)HORIZONTAL LINE TESTThe inverse of a function f is also a function if and onlyif no horizontal line intersects the graph of f.FunctionNot a functionyyx172Lesson 6.4 Algebra 2 Notetaking GuidexCopyright Holt McDougal. All rights reserved.
Your NotesCheckpoint Find the inverse of the function. Then verifythat your result and the original function are inverses.1. f (x) 5 23x 1 551f 21(x) 5 2}x1}33Find the inverse of a power functionExample 3Find the inverse of f(x) 5 4x 2, x 0. Then graph fand f 21.f (x) 5 4x 2Write original function.y 5 4x 2Replace f(x) with y.x 5 4y 2Switch x and y.14}x 5 y 2You can check thesolution by notingthat the graph of1 }f 21(x) 5 2}2 Ï x isthe reflection ofthe graph off(x) 5 4x 2, x 0,in the line y 5 x.12Divide each side by 4.}6}Ï x 5 yTake square roots of each side.yThe domain of f is restricted tonegative values of x. So, the rangeof f 21 must also be restricted tonegative values, and therefore theinverse isf 21 (x)y 5 4x 2, x # 011 }5 2}Ï x . (If the2x1y5domain were restricted to x 0, you122x}1would choose f 21 (x) 5 }Ï x .)2HORIZONTAL LINE TESTThe inverse of a function f is also a function if and onlyif no horizontal line intersects the graph of f morethan once .FunctionNot a functionyyx172Lesson 6.4 Algebra 2 Notetaking GuidexCopyright Holt McDougal. All rights reserved.
Your NotesExample 4Find the inverse of a cubic function1Consider the function f(x) 5 } x 3 1 3. Determine4whether the inverse of f is a function. Then find theinverse.ySolutionGraph the function f. Notice that nointersects the graphmore than once. So, the inverse of f is. To find an equationitself a21for f , complete the following steps.1f(x) 5 } x 3 1 341y 5 }x3 1 3411xWrite original function.Replace f(x) with y.Switch x and y.Subtractfrom each side.Multiply each side by.Take cube root of each side.The inverse of f is f 21(x) 5.Checkpoint Find the inverse of the function.2. f (x) 5 2x 4 1 113. g(x) 5 }x532HomeworkCopyright Holt McDougal. All rights reserved.Lesson 6.4 Algebra 2 Notetaking Guide173
Your NotesExample 4Find the inverse of a cubic function1Consider the function f(x) 5 } x 3 1 3. Determine4whether the inverse of f is a function. Then find theinverse.ySolutionGraph the function f. Notice that nohorizontal line intersects the graphmore than once. So, the inverse of f isitself a function . To find an equationfor f 21, complete the following steps.1f(x) 5 } x 3 1 341y 5 }x3 1 341 3x5}y 1341 3x235}y44x 2 12 5 y 3y51 3x41311xWrite original function.Replace f(x) with y.Switch x and y.Subtract 3 from each side.Multiply each side by 4 .3}Ï4x 2 12 5 yTake cube root of each side.3}The inverse of f is f 21(x) 5 Ï 4x 2 12 .Checkpoint Find the inverse of the function.12. f (x) 5 2x 4 1 1f 21(x) 5Î12 x 2 12}4}}3. g(x) 5 }x5325}g 21(x) 5 2Ï xHomeworkCopyright Holt McDougal. All rights reserved.Lesson 6.4 Algebra 2 Notetaking Guide173
6.5Graph Square Root and CubeRoot FunctionsGoalYour Notesp Graph square root and cube root functions.VOCABULARYRadical functionPARENT FUNCTIONS FOR SQUARE ROOT AND CUBEROOT FUNCTIONS The parent function forthe family of square root}functions is f(x) 5 Ï x . The domain is x, andthe range is y. The parent function forthe family of cube root3}functions is g(x) 5 Ï x . The domain and rangeare.Example 1Graph a square root function}Graph y 5 2Ï x , and state the domain and range.}Compare the graph with the graph of y 5 Ï x .SolutionMake a table of values and sketchthe graph.x0123y411yxThe radicand of a square root is always nonnegative. So,the domain is x0. The range is y0.}The graph of y }5 2Ï x is a verticalgraph of y 5 Ï x .174Lesson 6.5 Algebra 2 Notetaking Guideof the parentCopyright Holt McDougal. All rights reserved.
6.5Graph Square Root and CubeRoot FunctionsGoalYour Notesp Graph square root and cube root functions.VOCABULARYRadical functionA function containing a radical}such as y 5 Ï xPARENT FUNCTIONS FOR SQUARE ROOT AND CUBEROOT FUNCTIONS The parent function forthe family of square root}functions is f(x) 5 Ï x . The domain is x 0 , andthe range is y 0 . The parent function forthe family of cube root3}functions is g(x) 5 Ï x . The domain and rangeare all real numbers .Example 1Graph a square root function}Graph y 5 2Ï x , and state the domain and range.}Compare the graph with the graph of y 5 Ï x .SolutionMake a table of values and sketchthe graph.yy52 xx01234y022.833.464y511xxThe radicand of a square root is always nonnegative. So,the domain is x 0. The range is y 0.}The graph of y }5 2Ï x is a vertical stretch of the parentgraph of y 5 Ï x .174Lesson 6.5 Algebra 2 Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesGraph a cube root functionExample 213}Graph y 5 2}Ï x , and state the domain and range.23}Compare the graph with the graph of y 5 Ï x .SolutionMake a table of values and sketch the graph.x2221y0y11x1x2yThe domain and range are1.3}The graph of y 5 2}Ï x is a vertical2of the3}parent graph of y 5 Ï x by a factor ofreflection in the x-axis.followed by aCheckpoint Graph the function. Then state the domainand range.3}}1. y 5 2 Ï x2. y 5 22Ï xyy111Copyright Holt McDougal. All rights reserved.x1xLesson 6.5 Algebra 2 Notetaking Guide175
Your NotesSolve an equation with a rational exponentExample 2(3x 1 4)2/3 5 16Original equationRaise each side to the53power }2 .5Apply properties ofexponents.5Simplify.5Subtract5Divide each side by. Check this in the original equation.The solution isSolve an equation with an extraneous solutionExample 3}x 2 2 5 Ï x 1 1050CHECKCheck x 5Original equation5Square each side.5Expand left side andsimplify right side.50Write in standardform.50Factor.orx5or.}x 2 2 5 Ï x 1 1000The only solution isextraneous.)Copyright Holt McDougal. All rights reserved.from each side.50x5Zero product propertySolve for x.Check x 5 2.}x 2 2 5 Ï x 1 1000. (The apparent solutionLesson 6.6 Algebra 2 Notetaking Guideis179
Your NotesSolve an equation with a rational exponentExample 2(3x 1 4)2/3 5 16Original equationRaise each side to the[(3x 1 4)2/3]3/2 5 163/23power }2 .3x 1 4 5 (161/2)3Apply properties ofexponents.3x 1 4 5 64Simplify.3x 5 60Subtract 4 from each side.x 5 20Divide each side by 3 .The solution is 20 . Check this in the original equation.Solve an equation with an extraneous solutionExample 3}x 2 2 5 Ï x 1 10Original equation}(x 2 2)2 5 (Ï x 1 10 )2Square each side.x 2 2 4x 1 4 5 x 1 10Expand left side andsimplify right side.x 2 2 5x 2 6 5 0Write in standardform.(x 2 6)(x 1 1) 5 0Factor.x26 50Zero product propertyor x 1 1 5 0x 5 6 orCHECKCheck x 5 6 .}x 2 2 5 Ï x 1 10}6 2 2 0 Ï6 1 10}4 0 Ï164 5 4x 5 21Solve for x.Check x 5 2 1 .}x 2 2 5 Ï x 1 10}21 2 2 0 Ï 21 1 10}23 0 Ï 923 Þ 3The only solution is 4 . (The apparent solution 21 isextraneous.)Copyright Holt McDougal. All rights reserved.Lesson 6.6 Algebra 2 Notetaking Guide179
Your NotesExample 4Solve an equation with two radicals}}Solve Ïx 1 6 1 2 5 Ï 10 2 3x .}}Ïx 1 6 1 2 5 Ï10 2 3x55550x5CHECK5Divide eachside by 4.5Square eachside again.5Simplify.05Write instandard form.05Factor.50orx5orCheck x 5Write originalequation.Square eachside.Expand leftside andsimplify rightside.Isolate radicalexpression.Zero productpropertySolve for x.005Check x 5 2.005The only solution isis extraneous.)180Lesson 6.6 Algebra 2 Notetaking Guide. (The apparent solutionCopyright Holt McDougal. All rights reserved.
6.1 Evaluate nth Roots and Use Rational Exponents Goal p Evaluate nth roots and study rational exponents. VOCABULARY nth root of a For an integer n greater than 1, if bn 5 a, then b is an nth root of a. Index of a radical An nth root of a is written as Ïn a , where n is the index of the radical. REAL nth ROOTS OF a Let n be an integer (n 1) and let a be a real number.
Roots of complex numbers Every number has two square roots. The square roots of 16 are: The square roots of 24 are: The square roots of -81 are: The square roots of -75 are: Likewise, every number has three cube roots, four fourth roots, etc. (over the complex number system.) So if we want to find the four fo
Sum of Roots : _ Product of Roots : _ 4. Use the sum and product rule to determine if the two given values are the roots of the quadratic equation. a. Are 2 and -2 the roots of 3x 2x – 5 0 b. Are -1 6 and the roots of 3x2 2x – 5 0 c. Are and
Tree Roots: Facts and Fallacies Thomas O. Perry A proper understanding of the structure and function of roots can help people become better gardeners. Plant roots can grow anywhere-in the soil, on the surface of the soil, in the water, and even in the air.Except for the first formed roots that respond positively to gravity, most roots do not grow toward anything
List of medical roots, suffixes and prefixes 1 List of medical roots, suffixes and prefixes This is a list of roots, suffixes, and prefixes used in medical terminology, their meanings, and their etymology. There are a few rules when using medical roots.
Simplifying Cube Roots For any real number a, 3a3 a. Example 7: Simplify each of the following. a. 38x3 b. 27x3 c. 31000x3 Even and Odd nth Roots Radical expressions can have roots other than square roots and cube roots. The radical expression na means the nth root of a. The number n is called the index, and a is called the radicand. In general,
Category A: Estimating Square Roots and Cube Roots . Between what two consecutive integers . do the following real numbers lie between? 5 38 53 99 326 3214 227 77 171 194 380 147 3999 3119 380. Category B: Square Roots and Cube
Jul 10, 2018 · The Roots air blowers described in this manual represent both the basic type of Roots Anti-friction System (RAS) rotary . lobe arrangement and the proprietary Roots RAS WHISPAIR design, with gear diameters ranging from 10 to 20 inches. All units are equipped with an effective splash oil lubrication system. The Roots RAS Blower
Installation, Operation and Maintenance Manual Roots 800 RCS Series Blowers ISRB-2003 800 RCS series blowers.indd 1 17/09/2015 12:50. ISRB-2003 800 RCS series blowers Page 2 of 24 Contents . Do these things to get the most from your Roots Blower Check shipment for damage. If found, file claim with carrier and notify Howden Roots.
