Download Lesson 1 - Square Roots - Algebra I Keystone Exam Preparation

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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