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Name:_________________________________________________________Period:______Date:___________________________ AlgebraIKeystoneExamPreparationProgramoMasteredOn:_____________________ Square Roots AnchorsAddressed A1.1.1.1.2–Simplifysquareroots(e.g. 24 = 2 6). LessonPretest Solvethefollowingproblems.Showyourworkinthespacebelowandfillinthebubblethatrepresentsthe correctanswer. 1.Anexpressionisshownbelow. 21𝑥 Forwhichvalueofxshouldtheexpressionbe furthersimplified? A. 5 2.Anexpressionisshownbelow. 2 43𝑥 Which value of 𝑥 makes the expression equivalentto14 43? A. 7 B. 6 B. 12 C. 11 C. 49 D. 55 D. 98 1. 2. A B C D A B C D LessonKeywords CompositeNumber,CubeRoot,Integer,PerfectSquare,PrimeNumber,RadicalExpression,SquareRoot Concepts Square roots are the reverse operation of squaring a value. Numbers whose square roots are Integers are calledperfectsquares.Forexample: 4 = 2,so4isaperfectsquare.Thesquarerootsymbol( )iscalled a radical. You can use a calculator to determine the approximate value of a square root or simplify square roots,asshownbelow: CalculatingtheValueofaSquareRoot Toestimatethevalueofasquareroot,useacalculator.Somecalculatorsrequireyoutoenterthevaluefirst andthenpressthesquarerootkey(usually % b)andothersrequireyoutopressthesquarerootkeyand thenenterthevalue.Youwillneedtodeterminewhichmethodyourcalculatoruses. Example1:Findthesquarerootof 18.Roundtheanswertothenearesttenth. 4.242640687 Solution:Press% bandtype18,thenpresse . Roundingtheanswertothenearesttenth,yieldsafinalsolutionof4.2. ? TestTakingTip:Useacalculatorthatyouknowhowtouse.Understandhowthefeaturesofthe calculator work, including how to enter expressions into the calculator. Make sure your calculatorhasfreshbatteriesandisworkingproperlypriortothedayofthetest. SimplifyingSquareRoots Squarerootsthatarenotperfectsquarescansometimesbesimplified.Tosimplifyasquareroot,factorthe numberinsidetheradicalintoprimefactors.Eachpairofthesamefactorscanbemovedtotheoutsideofthe radical.Simplifythefactorsontheoutsideoftheradicalbymultiplying,andthendothesametotheinside. Example2:Simplify 160. Solution: STEP1:Factor160insidetheradical: 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 5. STEP2:Sincetherearetwopairsoftwo,one2fromeachpaircanberemovedfromthe radical(Thinkofitas2times2is4andthesquarerootof4is2,soa2isplaced outside). STEP3:Repeatthisstepforallpairs,asshown:2 ∙ 2 2 ∙ 5. STEP4:Finally,multiplythenumbersoutsidetheradicalandthenthoseinsidethe radical,asshown:4 10. The following examples demonstrate two additional methods of determining your understanding of simplifyingsquareroots. Example3:Anexpressionisshownbelow. 14𝑥 Forwhichvalueofxshouldtheexpressionbefurthersimplified? A. 15 B. 23 C. 35 D. 37 Solution: STEP1:Factorthecoefficientofx,insidetheradical: 2 ∙ 7 ∙ 𝑥. STEP2:Factoreachofthevaluesintheanswerchoices. A. 15 = 3 ∙ 5 B. 23 = 1 ∙ 23(Prime) C. 35 = 5 ∙ 7 D. 37 = 1 ∙ 37(Prime) STEP3:ComparethefactorsandmatchthefactorsfromSTEPONEandSTEPTWO.Onlyone oftheanswerchoicesshouldhaveoneormorematchingfactors.Thisanswerchoice isthecorrectanswer.Inthisexample,boththenumberintheradicalandchoiceC haveafactorof7.Therefore,Cisthecorrectanswer. STEP 4: Check your work by replacing 𝑥 with the value you selected and simplifying the squareroot,asshownbelow. 14𝑥 = 14(35) = 490 = 7 10 Since 490canbesimplifiedto7 10,theansweriscorrect. Example4:Anexpressionisshownbelow. 2 23𝑥 Whichvalueof𝑥makestheexpressionequivalentto10 23? A. 5 B. 10 C. 25 D. 50 Solution: STEP1:Dividethecoefficientoftheequivalentexpressionbythecoefficientoftheoriginal expression. 10 ÷ 2 = 5 STEP2:Recallthatwhenwemoveaperfectsquarefromtheradical,wetakethesquareroot oftheperfectsquare.Therefore,squarethevaluefromSTEPONE. 53 = 25 Therefore,thecorrectansweris25,whichischoiceC. STEP 3: Check your work by replacing 𝑥 with the value you selected and simplifying the squareroot,asshownbelow. 2 23𝑥 = 2 23(25) = 2 575 = 10 23 Exercises A.Estimatethevaluesofthefollowingsquarerootstothenearesttenth. 1. 11 3. 92 5. 14 2. 38 4. 18 6. 56 B.Simplifythefollowingsquareroots.Useacalculatortoverifyyouranswersandmakecorrectionsas needed. 7. 24 12. 98 17. 63 8. 10 13. 20 18. 125 9. 32 14. 17 19. 45 10. 18 15. 112 20. 12 11. 54 16. 132 21. 28 C.Findavalueofxthatallowseachexpressiontobefurthersimplified.Donotuseperfectsquares. 22. 22𝑥 26. 110𝑥 23. 35𝑥 27. 60𝑥 24. 42𝑥 28. 33𝑥 25. 30𝑥 29. 70𝑥 D.Findthevalueofxthatmakeseachexpressiontrue. 30.2 14𝑥 = 12 14 34.6 29𝑥 = 12 29 31.3 51𝑥 = 21 51 35.3 22𝑥 = 18 22 32.5 27𝑥 = 25 27 36.2 71𝑥 = 18 71 33.4 33𝑥 = 40 33 37.7 120𝑥 = 42 120 E.Answerthefollowingquestions. 38.Simplifythevalueof 32.Explaineachstep. LessonPostTest Solvethefollowingproblems.Showyourworkandfillinthebubblethatrepresentsthecorrectanswer. 1.Anexpressionisshownbelow. 55𝑥 Forwhichvalueofxshouldtheexpressionbe furthersimplified? A. 7 A. 3 B. 12 B. 9 C. 13 C. 12 D. 15 D. 18 1. 2. 2.Anexpressionisshownbelow. 6 63𝑥 Which value of 𝑥 makes the expression equivalentto18 63? A B C D A B C D