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u N tr 11 Radical Expressions

List of Objectives

To simplify numerical radical expressions

To simplify variable radical expressions

To add and subtract radical expressions

To multiply radical expressions

To divide radical expressions

To solve an equation containing one or more radical expressions

To solve application problems

UNtT 11 Radical Expressions 38s

SECTION 1 Introduction to Radical Expressions

1.1 Objective To simplify numerical radical expressions

A square root of a positive number x is a number whose square is x.

A square root of 16 is 4 since 42 - -16.

A square root of 16 is -4 since (-4)2: 16.

Every positive number has two square roots, one a positive and one a negative num-ber. The symbol " {", called a radical, is used to indicate the positive or principalsquare root of a number. For example /16 - 4 and \,Ed = 5. The number underthe radical sign is called the radicand.

When the negative square root of a number is to be found, a negative sign is placed infront of the radical. For example -vrl6 - -4 and - vEi = - 5.

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The square of an integer is a perfect square.

An integer which is a perfect square canbe written as the product of prime factors,each of which has an even exponentwhen expressed in exponential form.

To find the square root of a perfect square written in exponential form, remove the

radical sign and multiply the exponent by +

Simplify v6%Write the prime factorization of theradicand in exponential form.

Remove the radical sign and multiply

the exoonent bv 1., '2

Simplify.

/z iit -+J

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122 - 144

49:7'7:72B1 =3'3'3'3-34

144 = 2'2'2'2.3.3 : 2a32

ytr%: rB

=52

-)q

lf a number is not a perfect square, its square root t//=1.4142135...can only be approximated, for example, vtr and t/7. t/7 = Z.G4S7S1T . . .

These numbers are irrational numbers. Therr decimal representations never termi-nate or repeat.

The approximate square roots of the positive integers up to 200 can be found inthe Appendix on page 438. The square roots have been rounded to the nearestthousandth.

A radical expression is in simplest form when the radicand contains no factor whichis a perfect sguare. The Product Property of Square Roots is used to simplify radicalexpressions.

The Product Property of Square Roots

lf a and b are positive real numbers, then v66 = \,tr. \tE

386

Example 1

Simplify 3\6-0.

Solution

3 /00' = 31p:F ,E = 31/!Q. 5) -3\/g\tr:E = 3.3V10-= e/10-

Example 3

Find the decimal approximation of t/252. Use thetable on page 438.

Solution

\tr$ = tP:F:7 = t/F-4rt = 2.3rt =6rt=6(2.646)=15,876

UNIT 11 Radical Expressions

Simplify 1166.

Write the prime factorization of the radicand in expo-nential form.

Write the radicand as a product of a perfect square andfactors which do not contain a perfect square.

Use the Product Property of Square Roots.

Simplify.

Simplify /-a.The square root of a negative number is not a real numbersince the square of a real number is always positive.

Simplify \,tr% Then find the decimal approximation.thousandth.

Write the prrme factorization of the radicand in exponentialform.

Write the radicand as a product of a perfect square andfactors which do not contain a perfect square.

Use the Product Property of Square Roots.

Simplify.

Replace the radical expression by the decimal approxima-tion found on page 438.

= v/F'5

Simplify. = 11.180

Note that in the table on page 438, the decimal approximation of /i25 is 11.180.

Example 2

Simplify -5\/32.

Your solution

Example 4

Find the decimal approximation oltable on page 438.

Your solution

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= 1/Ve.3)

= \/FlE*- 221/24

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real number.

Round to the nearest

\nE = \/*

Use the

= ,/Ft/s

= 5v5

= 5(2.236)

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UNIT 11 Radical Expressrons 387

1.2 Objective To simplify variable radical expressions

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Variable expressions which contain radicals do not always represent real numbers.

The variable expression at the right \/Fdoes not represent a real numberwhen x is a negative number, forexample, -4. \K-T = f 44 Not a real number

For this reason, the variables in this unit will represenl positive numbers unless other-wise stated.

A variable or a product of variables written in exponentral form is a perfect squarewhen each exponent is an even number.

To find the square root oJ a perfect square, remove the radical sign and multiply eachI

exponent Dy z.

Simplify r,66.

Remove the radical sign and multiply the exponent OV L {F = as

A variable radical expression is in simplest form when the radicand contains no factorwhich is a perfect square.

-simplity vx'.Write x7 as the product of x and a perfect \F : \,eq'xsquare.

Use the Product Property of Square Roots. = rFrRSimplify the perfect square. - xttQ

Simplify 3xttrfrE.Write the prime factorization of the coeffi- Sxt/BFyB = 3xtf*i5y6cient of the radicand in exponential form.

Write the radicand as a product of a per- = 3xy/FPfQxy)fect square and factors which do not con-tain a perfect square.

Use the Product Property of Square = 3xy/FFyot/2xyRoots.

Simplify. - 3x.2xy6{2xy= 6xzyot/2xy

Simplify \/ri6- lFWrite the prime factonzation of the coeffi- \86 -ZF = 1f Sz(x -ficient in exponential form.

Simplify. =5(x-2)= 5x - 10

388 UNIT 11 Radical Expressions

Example 5

Simplify 16-1

Solution

\/F = \/5r:E = 1/Ft. t/6 = brr/6

Example 6

Simplify y!-1

Your solution

Example 7 Example g

Simplify \tr4F Simptify \Ai6rSolution your solution\/24-F = lF.J.P - \EqTz.3x\ -\/F:p\tr:ji - 2x2t/6x

Example 9 gxampte 10

simptify 2atfi6FFd, Simptify sa\/28fl66.

Solution your solutionzatfi6at66 - 26r/2.@:sr. 6o -2at/Fa/Fd1u1 = 2a1/F4.nn\E =2a. 3. r. 6s1/Za - 6"26s ;/2a

Example 11 Example t2Simprify / 6-GTEF. Simptify \tri6 _WSolution your sotution

VT6trTTP = t/FQJ@ - 22(x + 5) :4(x+5)-4x+20

Example 13 Example 14

Simplify Simptify \frT_1IV+ 4s-

Solution your solution

-

t/r2 + 10x + 25 = \ru+ 5F= x + 5

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UNIT 11 Radical Expressions 389

1.1 Exercises

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Simplify.

1. /Td 2. \/64 3. \Ee 4. \n44

5. \/32 6. \60 7. \6' 8. ,,42

9' 6 \n8 10. 4\Ed 11. 5\Fo 12. 2\/28

13. vT5 14' \tr1 15. ,,6 16. ,fr3

17. -srtl 18. 11 vBO 19. ^45

20. \E%

21. vO 22. \/7m 23. 6\4n 24. s\/7n

25. vT05 26. v65 27. \€00 28. \re-00

2s. 5 /180 30. 7 \,tr8 31' \/250 32' \n6

33. /06- 34. vlEo- 35. \FZA 36' \844

Find the decimal approximation. Use the table on page 438.

t=t 37. tE+ol:..:J

41. \trs6

38' \re-oo 39. \,tr6d 40. v600-

42. \@ 43. \,mg 44. 1466

4s. \tr4s 46. \Fn 47. \,AA 48. /365

390 UNIT 11 Radical Expressiors

1.2 Exercises

Simplify.

49' \/F 50' \/nz sl. \/va 52' \/F

ss. \/F 54. \/ai' s5' \/FF 56' \frW

57. \EF s8' \/'v se. \trEF 60' \FW

61' \/FF 62. \/FW 63. \/Fm 64. \/nF

6s. v6OtF 66. \/W 67. \@;w-

6e. \/l6FY? 70- \EFFT 71. \MV

68. \n4TFtr 6Io6o6

72' \frrFP F

73. v6Grbit- 74. \MFV 7s. ztfid$F 76. st/x-FF

77. x\/FF 78' Y\Fp 79. q\/nFF 80. stfrIFF

81. sx1/1zFf 82. +vt/TdFF 83. 2x21/ffif 84. 3y2lfi7f

8s. \/25G;+ry 86. \/6W=W 87. \/46=4

BB. \/96= 2f- 8e. \FlZrTZ eo. {F=e6T1d

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91. \/yr@ s2. \FTA;75 e3. vitz -E T-Td

U N lT 11 Radical Expressions 391

SECTION 2 Addition and Subtraction of RadicalExpressions

2.1 Objective To add and subtract radical expressions

The Distributive Property is used tosimplify the sum or difference ofradical expressions with like radi-cands.

Radical expressions which are insimplest form and have unlike radi-cands cannot be simplified by theDistributive Property.

Simplify 4\/6 - 1O\/,

Simplify each term. 4\/8 -

Simplify the expression by using theDistributive Property.

Simplify B\ilBx - 2\hN.Simplify eachterm.

Simplify the expression by usingthe Distributive Property.

5t/2+3t/7=(5+3)rt=grto\tri - 4\E = (6 - q\/trx:2.,,8x

zt/5 + +t/V cannot be simplifiedby the DistributiveProperty.

1O\/2=4\/8-rcrt

I = ^--ri -io.rn-l Do this steP

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- 4.2r/2 - tot/2= BtD - 10vtr

------------r- Do this stepl=rg-1o\.,DL--Y----:-::_1 mentally.

= _2f2

B\nu - 2\/52x = 8t/2:Fi - z\,Ev

,:;wox- --;a ."t, 3:^'ll:'*or i men1allv.

= Bv@\B - 2\E\,8: B'3\//i -r'221/u= 24\/Ex - a{u

'l = ,ro - B\tEi i Do this steP

l-:----:'-::-j mentally.

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Example 1 ExamPle 2

simptify 5\/, - sttr + 12\/, simplify 9r/5+ 3V5- 1Bv6

Solution Your solution

s\/2 - s\/7 + 12\/Z = 1 4\/E

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392 U N lT 11 Radical Expressions

Example 3 '

Simptify s\nZ - s\/n.

Solution

3\n2 - s\E = sy/Fi - 5V35 :s\/Ft/s - s\/g\/5 =3.2\/3 - 5.3r,6:6V5- 15\6= -9\6

Example 4

Simplify 2\FO - 5\8.Your solution

Example 5

Simplify 3\n2xs- - 2xt/3x.

Solution

3r/@- zat/u : 3t/F:3:F - 2xy/!i =g\/F .-FrB - zxt/u - s .2. xt/u - 2xtf3x =6xt/!i-2xt/3x:4x1f3x

Example 6

Simplify vt/N +7\/65F.

Your solution

Example 7

Simplify 2xy/8y - s\trW + z\hmSolution

2xt/8y - 3\trfr + 2th2fr =2xtPS - s\trfr + 2t/2sfr =zxt/Ft/zv - 3\/F\E + 2t/FPt/zy =2x.2t/2y -3.xt/2y +2.22.xt/2y =axt/zy - 3xt/2y a 8x1/2y :9xy/2y

Example 8

Simplify 2\/nF - aat/tze:- + a2t/75a.

Your solution

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UNIT. 11 Radical Expressions393

2.1 Exercises

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Simplify.

1. zt/7 + t/2 '2. S16+ 816 3. -3fi + 2rt

4. 4\/s - j0\6 s. -3VjT - 8\,47 6. -3vA - 5.va

7. 2t/i + a\/, 8. s\/y + Z\/i 9' 8\/i - 1o\/i

10. -5\E + z\Ea 11. -2\/i6 _ e\re6 n. _7 \/5a _ 5\6;

13. 3xt/7 - xt/2 14. 2vt/5 - gv\/3 ls. 2ay/ia _ sat/k

16. -sb\/ji - 2b\/jI 17. 3\8 - B\N 18. -4\/xy + 6\/xy

21. 2t/7 + 3t/ale. t/+s + \mi 20. \/s/ _ \,66

22. 4\/128 - 3\hZ 23. 5V18-- z\ns 24. s\/7s - 2\fl8

25. 5\/4x - 3\M 26. _s\//il + 8\/4W 27. 3\/sF _ s\EF

28. -2\/sF + s\/jz7 2e. 2xt/if _ syytf, s0. aay/b%- _ sb\/Fo

31. sxlh2x - s\EF g2. zat/soa + 7\ltrzF 33. qy\/8F-7\nBF

34. 2at/BaF - zb\trF 35. b'?\EFE .r3a2y/a6s- 36. y'z\/W + xt/FF


Simplify.

s7. 4\/, - stq + a\E 38. 316+ B\6- 16\6

3e. 5\/i - ltF + gt/i 40. tfr-tt/x+at/i

41. 8\// - 3\fr - 8\/, 42. 8\6 - St/T - st/5

43. 8w - 4\E - e\60- 44. 2\fr2 - 4{n + ,/ts

45. -2\/i + 5\E - 4\Fs 46. -2\/8 - g\E + 3r,60 --r

3

47. 4\nd + 3\R - \,69 48. 2\fr5 - 5\/N + 2\/45 =p@

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4e. \/N - l/sx + 1frax so. \Ei - fiTo-t - \M ;zoo

51. 31/u + \m - 8\/75x 52. 5\F + 2\/45x - g/60t

53. 2at/tsb - at/zob + aa1/ffi 54. zb\nsa - sb\m + 2b\/2va

s5. ,t/V - 2vt/t* + xy\/t 56. atFF + 3bt/Ta7F - ab\/3

57. g\/N + aat/ab - 5b\/4ai 58. s\/FE + ay/ffi - 3\/48d6

s9. sat/T# - JFg + abt/#b 60. z\FFF - sat/e# + abt@o


SECTION 3 Muttiplication and Division of RadicalExpressions

3.1 Obiective To multiply radical expressions

The Product Property of Square Roots can also be \trr\/W = yf2x:3y: \/6xyused to multiply varrable radical expressions

Simplify ({4'Multiply the radicands.

Simplify.

Note: For a ) 0, (\ftr2 - aE = a

Simplify \,ee''PZxl.Use the Product ProPertY ofSquare Roots.Multiply the radicands.Simplify.

Simplify t/zx(x + {u).Use the Distributive ytri1xProperty to removeparentheses.

Simplify 6ltr - 3x)6/2 + 4.Use the FOILmethod to removeparentheses.

T--------l. -,^ r rl r- |(\/x)':VxVxi-Vx.xl' L-------l

_ ,/F=^

--------------l\/fxz \/gIF | = \,trx'-EF I

L-- --------l

= \/64x?= \/FF: \EFt/i_ 2sx3t/i- s*r&

----------+ t/n>t=_!3!9_I-8-q)

= x;/2x + t/+F= x;f2x + t/PF- xt/2x + 2x

Do this stepmentally.

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The expressions a + b and a - b, which are the sum and difference of two terms, are

called conjugates of each other. Conjugates differ only in the sign of the secondterm.

SimplifyQ+{DQ-\,ry)The product of conlugates of the form(a + bXa - b) = a2 - b2.

SimprifyG+{DG-{v>The product o{ conjugates of the form(a+b)(a-b)=a2-b2.

6/2 - 3x)6'D. o

: f:i;:-';,,rf'*""

Q + \fr)Q - tn

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_ rlv,-3

-a2=9-y

(3+r,ay)(3-{v> ,/7


Example 1 Example 2

Simprify \/gFveW\/W Simplify Vselm%Tv-3bl


\EF \trW \/6 x-y' : tf!6 x7 y3 =\trgFvt = lfFFFf \/xy :,. 3rsyl/w = 6#yf xy

Example 3 Example 4

Simplify lulft/u + .urgol. Simplify ,fsrfr/s, - lFzsn.


t/e;6f\/ge + V-gO = \h46 + tf*aF =\Fet/o + y@Ft/ia :3ayE + sb\,€e

Example 5 Example 6

Simplify (\/a - ,/6X.r[- + \/6). Simplify (zt/i + 7)(2\/i - 7).


(\/a - {oXr/u + {D = t/F - \/F - a - b

Example 7 Example 8

simptify (2\/i - t/*D5tfr - ztfn. simplify (3\/x - t/iXsr/V - z{D.


e\,q - {D6t/x - zt/lt) =1o\/F - 4\/xy - s\/xy + 2rF =lOx-g{xy+2y

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3.2 Objective To divide radical expressions

The Quotient Property ol Square Roots

The square root of a quotient isequal to the quotient of the squareroots.

simplify vry,Rewrite the radical expression as thequotient of the square roots.

Simplify.

lf a and b are positive real numbers andl; .,8 ,E labl0, then l-=-:-and-i-= l-.' vb \/b \/b vb

r-------ll4x2 ', \/4x2 |

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2x

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Simolifv lry-' ' ,,,1 sx,y.

Simplify the radicand.

Rewrite the radical expression asthe quotient of the square roots.

Simplify.

Multiply the numerator and denominator_f;

bv v". which eouals 1.' /^ 'V.1

The radicand in the denominator is a per-

fect square.

Simplify.The radical has been removed from thedenominator.

Simplify "; -V-l/ -r u

Multiply the numerator and de-nominator OV lA - 3, the con-jugate of t/y + S.

Simplify.

i-;vt 1 Do this step

i_:E_j mentarry

1 - 1 .vF=st0+s ,,0+s tfi-z

F----------

i- t/v-z I DothissteP

i-@-3'? I mentallY.L_*____-_-!

_./y_3y-9

l24x3y7 lgvs| 3x'v' - 'J xo

l---------ri - t/evu I

i ,/f ili

t/2"ru.,F\/Ff\/N:+ .,F I

_ 2yrt/g- ---',--

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Simplify ry\/ xY

Use the Quotient Property of Square Roots.

Simplify the radicand.

Simplify the radical exPression.

A radical expression is nol considered to be in simplest form if a radical remains in the

denominator. The procedure used to remove a radical from the denominator is called

rationalizing the denominator.

tzPv.,1 ,y

=\R= \/Ft/V= 2\/x

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simolify ft2 - z .yBv.r v.r vr

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Simptifv t/Z + yFF/^vz

Divide each term in the numerator by the denomi- t// + tfldF _ ,/z , \nBFnator. vT -: r1*-iUse the Quotient Property of Square Roots.

Simplify.-1+ IIY'v 2

:t + tfgF= I + ,/1T'z-1+3y

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Example 9

Simplify ry\/3x"Y

Solution

Example 10

Simptifv @.t/3xryt

Your solution

Example 11

Simplify -:g-t/2 - t/x

Solution

\,tr _ \/, . t/1+ t/x _J2-\/i \f2-\n \/r+fi-

Example 12

Simplify -!-t/Y+s

Your solution

2+t/x2-x

Example 13

simotifv Qo - ztfinvb

Solution

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Example 14

Simolifv t/nxs--s\/OxV3x

Your solution

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17t-'J+ = \/4 - 2\f25 :-2\/g-2-2.s:2-jo--B 6ori

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399

3.1 Exercises

Simplify.

1. \/i.\/i

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2. \/n. \nT 3. \/s-. \rc

4. \/2. \/d 5. Vx' t/x 6. \/y. rfr

7. \/F.\/n 8. {alFF. 1;fi e. \tra$. \/d;F

10. \/W.\/TW 11. \/de3.e.\trC%- 12. \/d;F. \frTFS

t/Zft/Z - vO

t/b(t/a - \/6)

14. s(\/TZ - \,O ls. tRftfr _ \O)

17. v6(/i0'_ \/i) 18. \/d6/y - r/Te--;

le. t/e-fttr _ v/o 20.. t/to-cr/N _ G) 21. (t/i - s1,

22. (2G _ D2 23. tMrt/u- _ v66l 24. tqrtnO; _ \/i)

25. \trac . \/sab--. \frdc6 26. \/W. \/w.\/ry

27. @\fr - 2h|6t/i _ qy) 28. (sv&+ zt?)@tfr _ t/-v>

2e. (\/i - tElt{i + \O) 30. (\/* + D6/u- _ v)

13.

16.

31. (zt/i + ,fr>tst/i + 4\0) 32. (s\fr _ zt'lrs\/i _ 4\o)


3.2 Exercises

3s. #

38. #

-frc34. vi:vb

s7. lmVga

36' #

46. Lt/2x

4e. @t/3xv3

Simplify.

33. #

39. \nsFYt/3xy

42. ,/^Wt/sxtY

51. -:Lvz-J

40. Iryt/5xY

55. J* -t/x - t/Y

44. l:t/e

4s. +

53. --j--=b+vb52. =5l-vt - J

41. \/ZFg\/1Bab4

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47. ft'"' E- it"4-2 bv-r\ t I

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50. ,/Tryt/Bxty

56' ffi

45' +

48. \/w\,trix

54. J/^YZ - |

sr. 3\'tr -_B\/, sg. 5\/3 -3_\/j 59. zytr t:_\/2\// zvtr ,,tr,

60. ,fi#tr! 61. S*tlE 62. "E#tfr


SECTION 4 Solving Equations Containing RadicalExpressions

4.1 Objective To solve an equation containing one or more radical expressions

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An equation that contains a variable ex- rfr : 4 I naOicat

f;:""..r'".inaradicandisaradical equa- \/qTZ: yfx _t j Equations

The following property of equality is used to solve radical equations.

Property of Squaring Both Sides ol an Equation

lf two numbers are equal, then the lf a and b are real numbers and a = b,squares of the numbers are equal. then a2 : b2.

Solve \&-2 -7 =0.Rewrite the equation with the radical \/q - 2 - 7 - O

on one side of the equation and the ,/82 : Iconstant on the other side.

Square both sides of the equation. 1yF - Z1z = lzSolvethe resulting equation. x -2:49

x=5'1The solution is 51 .

Checkthesolution. Check; \/q-2-7:OWhen squaring both sides of an equa- \,qf - Z - 7 : 0tion, the resulting equation may have u * _ 7 - O

solution which is not a solution of the lp _ 7 = Ooriginal equation. 7_7=O

0:0 Atrueequation

Example 1 Example 2

Solve: 16i * 2 :5 Solve: y/4y + 3 =7

Solution Your solutiony/u+2-s check: l/u+2-s

t/3x:3 .,6'g +2-5(yEiyz=32 J*+z-s

3x:9 3+2-5X=3 5=5

The solution is 3.

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Example 3

Solve: 0=3-\pr-

SolutlonO:3-\tri=3

-3 - - \tri=3(-3)2=(-t/Tx-S)z

9-2x-312-2x6=X

Tne solution is 6.

Example 4

Solve: y/1x-Z-s

Your solutionCheck:0=3-0=3-0:3-0:3-0=3-0=3-0=0

\tri -3\E:{-\nz -3v0\/g3

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4.2 Objective To solve application problems

Example 5

How far would a submarine periscope have to beabove the water to locate a ship 4 mi away? Theequation for the distance in miles that the lookoutcan see is d = 1.4\n, where h is the height in

feet above the surface of the water. Bound to thenearest hundredth.

Strategy

To find the height above water, replace d in theequation with the given value and solve for h.

Solution

1.4\/6 = d1.4\f n = 4

rAVn =aa

(r/-hy = (+)'I ^- 16

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h = 8.16

The perisrcope nnust be 8.16 ft above ihe water

Example 6

Find the length of a pendulum that makes oneswing in 2.5 s. The equation for the time for one

n-swing is T = 2n.f n, where f is the time in sec-

onds and L is the length in feet. Use 3.14 lor n.

Round to the nearest hundredth.

Your strategy

Your solution

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4.1 Exercises

UNIT 11 Radical Expressrons 403

Solve and check.

1. lV=S 2. ly :7 3. \fa = 12

4' t/a :9

7. t/+x = a 8. Jax:g 9. \/rr-4=O

10. 3- \,6t=0 11. t/qx+5=2 12. t/u+9=4

13. \/3x-2=4 14. /Sxao-t 15. t/zxll=t

16. 1/Sx1+-S 17. O:2 - \/3 - x 18. 0 = 5 - \frdTt

19. \/5x+2=O 20. 1/sx-t =O 21. ,/g^-6:-4

22. y/Sx + B = 23 23. O:1fix-S-O 24. O=1/Zxat-Z

25. \/5x-1= 1fra9 26. lix+4=\nZx-w

27. \/ii-- = \f4x - 28. \tri-g=\/2x*3

6.

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o

Eou!CocoCoO

2s. \F=ZrTA = \/F - sv q s 30. \R=2iJ4 = y/P a 5y - 12


4.2 ApplicationProblems

Solve.

1. Four added to the square root of

lhe product of f ive and a number is

equal to nine. Find the number.

How far would a submarine Peri-scope have to be above the waterto locate a ship 5 mi awaY? Theequaiion for the distance in milesthat the lookout can see is

d - 1.4Vh, where h is the heightin feet above the surface of thewater. Round to the nearest hun-dredth.

An object is dropped from a highbuilding. Find the distance theobject has fallen when the sPeed

reaches 64 ftls. The equation forthe distance is v = /-6+d, where vis the speed of the object and d isthe distance.

7. A stone is dropped from a bridgeand hits the water 1 .5 s later. How

high is the bridge? The equationlor the distance an object falls in f

1.1seconds is given bY I = ./iC,where d is the distance in feet.

Round to the nearest hundredth.

Fl 9. Find the length of a pendulum thatUiJ makes one swing in 2 s. The

equation for the time of one swing

of a pendulum is given bYlt

T - 2n /1, where f is the time in'--"tlsz'seconds and L is the length in

feet. Use 3.14 for a'. Round to thenearest hundredth.

2. The product of a number and thesquare root of two is equal to thesquare root of eighteen. Find thenumber.

How far would a submarine peri-scope have to be above the waterto locate a ship 6 mi away? Theequation for the distance in milesthat the lookout can see is

d - 1.4r,1-h wf,ere h is the heightin feet above the surface of thewater. Round to the nearest hun-dredth.

An object is droPPed from a

plane. Find the distance the ob-ject has fallen when the sPeed

reaches 512 fI/ s. The equationfor the distance is v = y/64d,where v is the speed of the objectand d is the distance.

A stone is dropped into a mine

shaft and hits the bottom 3 s later,

How deep is the mine shaft? The

equation ior the distance an objectfalls in f seconds is given bY

- frtT = J#, where d is the distance

in feet. Round to the nearest hun-dredth.

Find the length of a pendulum thatmakes one swing in 1.5 s. Theequation for the time of oneswing of a pendulum is given bY

TTT = 2n /*, where f is the time

\JIin seconds and L is the lengthin feet. Use 3.14 for n. Round tothe nearest hundredth.

4.3.

6.5.

oO

Io@oo6E6I!cdccoo

8.

10.


Review/Test

SECTION 1 t.ta Simplify f5. 1.1b Simplity /-5.

1.1c Find the decimal approximation 1.2a Simplify \\XFF.ot t/125. Use the table on page438.

1.2b Simplify \nu'y'?- 1,2c Simplify \/4rFStooI@P@

E,!cd

E SECTION 2 2.1a Simplifyalv-s\/y. 2.1b Simplify5/8-3/-50.coo

2.1c Simplify 2.1d Simplify3\/W - 2\fr?2x + s\nly 2xt/5$ - zvlTlfr - 3xyt/xy.

SECTION 3 3.1a Simplify\/W\/1uy4-. 3.1b Simplify\FWttdVOx.


Review /Test3.1c Simplify G6/" - t/-o>. 3.1d Simplify (\/y - eXl/y + s).

3.2a simplify #

3.2c Simplify G-

g.2b Simolifv -)ry.t/2a"b'

4.1b Solve: 1/di+3=18

ooEIo@o@

oE

I!coEoE

C)SECTION 4 4.1a Sotve: y/sx-6=t

4.2a The square rool of the sum of two 4.2b Find the length of a pendulumconsecutive odd integers is equal that makes one swing in 3 s. Theto 10. Find the larger integer. equation for the time of one

swing of a pendulum is given byrT-

T : 2n JS, where f is the time

in seconds and L is the length infeet. Use 3.'l 4 tor n. Round tothe nearest hundredth.


Review /TestSECTION 1

SECTION 2

1.1a Simplify 126a) 4rtc) 2\/7

1.1c Find the decimal approximatlonot 16e Use the table on page438.a) 12.124 b) 7.484c) 7.937 d) 7.398

1.2b Simplify \/6Wa) 4x2tf|yb) 4x2yEyc) 16x2t/3yd) axt/jfr

2.1a Simplify g\/a - g\,6a) -6̂ -o) -6va,c) -12\/ad) -6\/6

2.1c Simplifyz\E;-s\/4%+B\/ Baa) 38 \ree - 35 \6b) 3\ree6c) 2ByEe - 3s\,6d) 38\ree - 5\@6

3.1a Simplify \,IZF;t{A;F.a) ,6"+5a labb) 4"t6+ lZabc) 6"26+ lf1abd) otz6+ yfdab

1.1b Simplify vtOda) 6\6c) 36va

b) 7vZd) 3\,6-

b) 7vEd) 3v5

1.2a Simplifya) 3aac) 3a2

\/lTF.b) e-\/Fd) 9a2

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o

EoI!cG

Cooo

1.2c Simplify \ndZFsna) tx3y2tf3fyb) 8x3y2 1/2xyc) Bx6y2tf3xyd) 7 x6y2 tf2xy

2.1b Simptify 3\hZ - 2\,4na) -4vtrb) -48\/,c) 12\/Z - 16 V3d) 28\/z

2.1d Simplify2aytraF + b \/EFd - sab t/ab,a) gabt/abb) -zaby66c) aaby66 - Sa2b2

d) aabt/2ab - sabv66

3.1 b Simplify \tra%- \88;F ttra.a) -7

us6z yf 2b) , Ous6z yEac) 14a2b2 nEad) rrz5z{2a

fr

SECTION 3

408 UNIT 11 Radicat Expressions

Review/Test

3.1c Simplify v6(\6-- {Vr)a) zt/s - x\/tb) 3t/z - xt/5c) s - xt/id) 3\/, - s\/i

3.2a Simplify $v5

Simolifv 3 -.' ' a -/Ea - vo

^\ z+r/5d)7

b)

=#trc) -u lut€d) -O - 3\6

4.1a Solve: 16x-2 -4=0a)2 b)4c)8 d)6

4.2a The square root of the sum of twoconsecutive integers is equal to9. Find the smaller integer.a) 20 b) 80c) 81 d) 40

3.1d Simplify(\/a-r/rXG+\/2)a) a2-4a+4b) a2-4c) a-2d) a2+4

g.2b simorifv @\/3a4b

a) 6a3bo

b) 6aby/ac) 6a3btfabd) 6a3b2

3.2d Simolifv \frZF -- \E,/g

a) 2x-3b) 4i3^\ 2xvE - sw)3

d) 4+f

4.1b Solve: 3-B-a) 11 b)

a) 8\,6b) 5wc) 64d)8

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6EouN

coc:oO

SECTION 4 -VbX.1

zzc)5

A stone is dropped from a build-ing and hits the ground 5 s later.How high is the building? Theequation for the distance an ob-ject falls in f seconds is given by

fA, = J *, where d is the distance

in feet. Round to the nearesthundredtha) 200 ftc) 10 it

b) 400 ftd) 300 ft

1

$r',.i

l

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