Name
LESSON
Date
Class
Reteach
11-7 Adding and Subtracting Radical Expressions
You can add and subtract radical expressions just like you add and subtract
expressions with variables.
4x ⫹ 2x ⫽ 6x
2x ⫹ 4y
These are not like
terms. Do not add.
These are like
terms. Add.
Combine radicals only if they are like radicals.
2兹 5 ⫹ 4兹3
These are like
radicals. Add.
These are not like
radicals. Do not add.
Add 8兹10 5兹10 .
Subtract 10兹7x 12兹7x .
8兹 10 ⫹ 5兹 10
10兹 7x ⫺ 12兹7x
These are like
radicals.
These are like
radicals.
⫺2兹7x
13兹10
4_
兹 7 ⫹ 2_
兹 7 ⫽ 6兹 7
Add 4兹2 8兹3 .
4兹 2 ⫹ 8兹3
Add 9兹5 4兹6 .
9兹5 ⫺ 4兹6
These are not like
radicals. Do not add.
These are not like
radicals. Do not
subtract.
State whether the expressions can be added. If yes, find the sum.
1. 3兹 2y ⫹ 8兹2y
2. 2兹5 ⫹ 5兹2
yes; 11兹 2y
no
3. 8 ⫹ 兹 8
4. 5兹11 ⫺ 6兹11
yes; 兹 11
no
Add or subtract.
5. 4兹 13 ⫹ 2兹 13
5兹 2
9. 7兹
x⫹
8. 12兹 3a ⫺ 2兹3a
10兹3a
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a107c11-7_rt.indd 54
兹x
8兹
x
54
7. 5兹5 ⫹ 6兹5
6兹 13
6. 8兹2 ⫺ 3兹2
11兹 5
10. 10兹 6 ⫺ 3兹6
7兹 6
Holt Algebra 1
12/26/05 8:27:36 AM
Process Black
Name
Date
Class
Reteach
LESSON
11-7 Adding and Subtracting Radical Expressions (continued)
Sometimes it is necessary to simplify expressions before adding or subtracting.
Simplify 兹50 兹18 .
兹 50 ⫹ 兹 18
兹 25 ⭈ 2 ⫹ 兹 9 ⭈ 2
Factor the radicands using perfect squares.
兹 25 ⭈ 兹 2 ⫹ 兹 9 ⭈ 兹 2
Product Property
5兹2 ⫹ 3兹2
Simplify.
8兹2
Combine like radicals.
Simplify 兹45a 兹 80a 兹20 .
兹 9 ⭈ 5a ⫹ 兹 16 ⭈ 5a ⫺ 兹 4 ⭈ 5
Factor the radicands using perfect squares.
兹 9 ⭈ 兹 5a ⫹ 兹 16 ⭈ 兹 5a ⫺ 兹 4 ⭈ 兹 5
Product Property
3兹5a ⫹ 4兹 5a ⫺ 2兹5
Simplify.
7兹5a ⫺ 2兹 5
Combine like radicals.
Notice that 兹5a and 兹5 are not like radicals.
Simplify each expression by filling in the boxes below.
11. 兹32 ⫹ 兹2
兹 16
⭈
2
5
兹
2
兹
2
⫹ 兹2
兹 16 兹 2
4
12. 兹27 ⫺ 兹 3
⫹ 兹2
⫹ 兹2
兹
3
4
兹
3
兹
3
兹 25 ⭈
⫹ 兹3
兹9 兹3
3
⭈
9
13. 兹125 ⫹ 兹 5
5
兹 25 兹 5
⫹ 兹3
⫹ 兹3
5
6
兹5
⫹ 兹5
⫹ 兹5
⫹ 兹5
兹5
Simplify.
14. 兹12 ⫹ 兹300
4兹 3
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a107c11-7_rt.indd 55
18. 兹63x ⫹ 兹 28x ⫺ 兹7x
4兹 7x
55
4兹 7 兹14
兹3
17. 兹75 ⫹ 兹12 ⫺ 兹27
16. 兹112 ⫹ 兹 14
12兹 3
15. 兹48 ⫺ 兹 27
19. 兹160y ⫺ 兹 90y ⫺ 兹 40y
兹10y
Holt Algebra 1
12/26/05 8:27:36 AM
Process Black
*À>V̈ViÊ
--"
££‡Ç !DDINGAND3UBTRACTING2ADICAL%XPRESSIONS
*À>V̈ViÊ
--"
££‡Ç !DDINGAND3UBTRACTING2ADICAL%XPRESSIONS
!DDORSUBTRACT
^
!DDORSUBTRACT
^
qq n
qq Ó q
^
^
q
qq £Î q ^
^
^
^
^
^
q q Êr ÇÊÊ ÊÈÊr £ÎÊÊ
^
^
^
^
^
Ê{ÊrB
qB qBq B ÈÊr ÓBÊ
^
q^
Mq^
M
^
£Î q M ^
^
^
^
qq q {Êr £äÊÊ
3IMPLIFYEACHEXPRESSION
^
^
^
^
^
q
qqq
x q { q
™ q ^
^
^
^
^
^
^
ÎÊr xÊÊʙÊr ÓÊÊ
^
M
M
^
^
^
^
^
^
^
^
^
^
^
^
^
^
?
Ê
ÊÊ
ÊÚÚÚÚÚÚ
Ê{xÊrÓÊ££ÊÊ
#
?
Êr ÇÓÊÊ՘ˆÌÃ
C 7RITEASIMPLIFIEDRADICALEXPRESSIONFORTHE
?
?
COMBINEDLENGTHOF!"AND#$
^
^
qq ÞiÃÆÊ££Êr ÓY
˜œ
^
^
^
qq
!
^
$
X
ÈÊr£ÎÊÊ
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
^
^
qq œÌʏ}iLÀ>Ê£
^
q AqA
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
87
^
xÊr ÓÊÊ
^
^
£äÊrÎA
^
^
qq ^
ÞiÃÆÊr ££ÊÊ
!DDORSUBTRACT
"
^
˜œ
££Êr ÓÊÊ՘ˆÌÃ
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
^
^
^
AK4up.indd 87
^
4HESEARENOTLIKE
RADICALS$ONOT
SUBTRACT
3TATEWHETHERTHEEXPRESSIONSCANBEADDED)FYESFINDTHESUM
4HESEARENOTLIKE
RADICALS$ONOTADD
^
^
!DDr r ^
^
q q
q
^
4HESEARELIKE
RADICALS
q X
^
qYq Y
Y
Êr xäÊÊ՘ˆÌÃ
B &INDTHELENGTHOF#$
^
^
^
£ÎÊr xZÊÊÊr^
Z
^
^
qXq X
^
^
?
^
^
^
qZq^
Z qZ
A &INDTHELENGTHOF!"
(INT5SETHE0YTHAGOREAN4HEOREM
^
3UBTRACTrXrX
ÈÊr ÎABÊ
Ê{ÊrAB
4HESEARENOTLIKE
RADICALS$ONOTADD
4
HESEARELIKE
RADICALS
!DDr r ^
^
q q
^
4HECOORDINATEPLANESHOWS!"AND#$
^
Î
^
ÓÎÊr ÓT
?
^
^
q q ^
q
^
qAB
qABqAB
^
^
^
qq ????
q
^
^
^
^
^
^
^
^
ÇÊÊr xXYZ
^
^
qq
££ÊrÊ ÓÊÊ
Ê
ÊÊ
ÊÚÚÚÚÚ
^
^
££Êr ÓBÊÊ{Êr B
^
Ó
qTq T
^
^
qXYZ
qXYZ
^
!DDr r
£{ÊrxÊÊ
q
????
?????
q
xÊrÊÈÊÊ
Ê
ÊÊ
ÊÚÚÚÚ
^
4HESEARENOTLIKE
TERMS$ONOTADD
4HESEARELIKE
RADICALS!DD
^
^
nÊÊ£{Êr ÎÊÊ
qBqB
^
^
qq q ????
q
????
XY
q
q
q
?
?
xÊr£xÊÊ
^
^
q^
??
??
q
^
ÎÊrÎÊÊ
^
^
œÌʏ}iLÀ>Ê£
^
^
qq
q
^
#OMBINERADICALSONLYIFTHEYARELIKERADICALS
^
£nÊrÓÊÊ
^
^
'b^
4HESEARELIKE
TERMS!DD
^
^
^
qq ^
^
^
Êr ÎÊÊÊÊr xÊÊ
(b^
' ,b^
XXX
^
qq
{ÊrÓÊÊ
qq '-b^
££Êr ££ÊÊÊÓÊr xÊÊ
qq
ÓÈÊrÓÊÊ
^
9OUCANADDANDSUBTRACTRADICALEXPRESSIONSJUSTLIKEYOUADDANDSUBTRACT
EXPRESSIONSWITHVARIABLES
^
^
qq q
xÊr T
^
qq
^
,iÌi>V…
--"
££‡Ç !DDINGAND3UBTRACTING2ADICAL%XPRESSIONS
^
^
£ÇÊrÎÊÊ
^
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
3IMPLIFYEACHEXPRESSION
^
^
£ÓÊr ÈKÊÊ£ÓÊr xÊÊ
^
œÌʏ}iLÀ>Ê£
q T q T
^
^
££Êr £ÎXÊÊÓÊr £ÎÊÊ
^
^
qKq xÊÊnÊr ÇÊʓˆiÃ
qq
^
£äÊr ÈT
^
!DDORSUBTRACT
^
^
qq
^
Êr^
ZÊÊ{ÊrÓZ
^
^
^
qXq Xq
4HEMAPATRIGHTSHOWSTHEPATHTRAVELEDBYA
DELIVERYPERSONONHISAFTERNOONROUTE7RITETHE
TOTALDISTANCETRAVELEDASASIMPLIFIEDRADICAL
EXPRESSION
*À>V̈ViÊ
--"
££‡Ç !DDINGAND3UBTRACTING2ADICAL%XPRESSIONS
ÈÊrÎÊÊ
^
££Êr ÓM
^
&&'b^
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
^
^
qZq Zq^
Z
{Êr ÓÊÊ]ÊÊrxäÊÊ]ÊÎÊr nÊÊ
^
^
Êr ÎB
^
INORDERFROMLEASTTOGREATEST
£{Êr ÓÊʓ
£äÊrÎÊÊ
^
^
^
7RITETHENUMBERSqqANDq
&INDTHEPERIMETEROFTHERECTANGLESHOWNATRIGHT'IVE
YOURANSWERASARADICALEXPRESSIONINSIMPLESTFORM
^
£ÈÊrÎÊÊ
^
^
Ó£Êr^
Mʈ˜°
^
^
^
q
BqB
^
^
^
^
&INDTHEPERIMETEROFANEQUILATERALTRIANGLEWHOSESIDES
EACHMEASUREq^
MINCHES'IVEYOURANSWERASARADICAL
EXPRESSIONINSIMPLESTFORM
^
ÎÊr{ÓÊÊ
^
^
qq
qTq T
^
qq £ÎÊrÎÊÊ
ÈÊr ÎM
^
^
£ÈÊrÎT
^
^
xÊr £ÇÊÊÊ£ÇÊr xÊÊ
^
^
^
qTq Tq T
£ÓÊr ÎX
^
ÓÎÊrÓÊÊ
^
qq ^
nÊr ÈÊÊ
^
xÊrÇX
^
^
qq qMqMqM qMqM
^
^
^
^
^
^
^
nÊr £ÎÊÊ
£ÓÊrÓÊÊ
^
qq
^
^
Êr ÓÊÊÊÎÊr ÓX
^
qXqX
^
^
qq
™ÊrxÊÊ
^
£ÓÊrÇÊÊ
^
^
qq
^
^
qq
£äÊr ÎÊÊ
^
^
^
^
qXqX
™Êr X
^
^
^
££ÊrÎÊÊ
^
^
^
^
^
qXqX
^
^
^
^
^
^
q q
qq
^
£äÊrÓÊÊÊxÊr ÎÊÊ
^
nÊr ÎB
^
qqq ^
^
qq
^
xÊr ££ÊÊ
^
^
{ÊrxÊÊÊ£äÊr ÓÊÊ
qq ^
^
qq Xq ^
^
^
qq
^
^
^
™ÊrÓÊÊÊ{Êr ÎÊÊ
ÈÊr £xÊÊ
^
^
q q
^
^
^
Ó q { q x È q x ÇÊr ÎÊÊ
qq ^
^
^
^
^
^
^
^
^
^
^
qq ^
^
q { q q £È q x ^
^
^
nÊr ÇÊÊ
^
^
^
^
qBqB
qq q
™ q q È{ q
^
qq ^
£äÊr Y
qq q ^
q { q £È x ££ q ^
^
^
^
Î q n q ^
^
q q q ™ q È{ ^
^
^
^
^
q q q
q ^
^
^
q^
Y q^
Y ^
qq n q ^
3IMPLIFYEACHEXPRESSION
^
q q ^
^
^
^
qX qX
££Êr xÊÊ
^
nÊr^
X
^
qq ^
q q
^
ÇÊrÈÊÊ
œÌʏ}iLÀ>Ê£
Holt Algebra 1
12/26/05 8:01:39 AM
,iÌi>V…
--"
££‡Ç !DDINGAND3UBTRACTING2ADICAL%XPRESSIONSCONTINUED
…>i˜}i
--"
££‡Ç 2ADICAL#ONCENTRATION
3OMETIMESITISNECESSARYTOSIMPLIFYEXPRESSIONSBEFOREADDINGORSUBTRACTING
4HISGAMEISFORPLAYERS%ACHSHOULDHAVEPAPERANDAPENCIL
^
^
3IMPLIFYr r ^
0REPARATION
^
q q
^
^
q q ^
^
^
^
q q q q ^
q q
3IMPLIFY
^
q 'AME0LAY
#OMBINELIKERADICALS
^
s 0UTALLOFTHEGAMECARDSUPSIDEDOWNONATABLEANDMIXTHEMUP
+EEPINGTHECARDSUPSIDEDOWNARRANGETHEMINTOABYGRID
0RODUCT0ROPERTY
^
s #UTOUTTHEGAMECARDSBELOW9OUCANMAKETHECARDSSTURDIERAND
NOTSEETHROUGHBYMOUNTINGTHEMONCARDBOARDWITHGLUEORTAPE
&ACTORTHERADICANDSUSINGPERFECTSQUARES
s 0LAYERPICKSANYTWOCARDSANDTURNSTHEMRIGHTSIDEUP4HEPLAYER
IDENTIFIESWHETHERTHETWOEXPRESSIONSCANSIMPLIFYTOBELIKERADICALS
^
3IMPLIFYr Ar Ar
^
^
^
^
qA
q
q A
^
^
^
^
^
^
^
^
^
q Aq Aq 0RODUCT0ROPERTY
s )FTHEEXPRESSIONSARENOTLIKERADICALSOR0LAYERADDSINCORRECTLY
THECARDSREMAINANDARETURNEDUPSIDEDOWNAGAIN
3IMPLIFY
q Aq s )FTHEEXPRESSIONSCANBELIKERADICALSTHEN0LAYERADDSTHEM
0LAYERCHECKSTHEOPPONENTSADDITION)FTHESUMISCORRECT
0LAYERREMOVESANDKEEPSTHETWOCARDS
&ACTORTHERADICANDSUSINGPERFECTSQUARES
^
^
q q Aq q Aq q #OMBINELIKERADICALS
^
^
.OTICETHATqAANDq ARENOTLIKERADICALS
s 0LAYERSALTERNATEREPEATINGTHEPROCESSDESCRIBEDABOVE4OIMPROVE
THEIRCHANCESOFWINNINGBOTHPLAYERSSHOULDREMEMBERWHAT
EXPRESSIONSHAVEBEENREVEALEDANDWHERETHEYARELOCATED
3IMPLIFYEACHEXPRESSIONBYFILLINGINTHEBOXESBELOW
^
^
^
qq q £È
^
q £È q Ó
{
x
^
q
q
^
^
q Ó q
^
q Ó ^
^
™
^
Î
{
^
^
^
qÎ
^
^
^
^
x
È
^
^
q
^
q Óx q x q
Î
q Î q
^
q q Óx x ^
^
q q
s !FTERALLCARDSAREREMOVEDTHEWINNERISTHEPLAYERWITHTHEMOSTCARDS
^
q q
^
q q ™ Î Ó
^
^
q q ^
^
q
q
^
^
q
q
^
q x q
^
qx
^
^
q
q
^
^
q
q
3IMPLIFY
^
^
^
q
q ^
^
q q ^
q q
^
^
^
£ÓÊrÎÊÊ
^
^
^
^
^
^
q
^
^
r £äY
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
^
q
^
^
{Êr ÇX
^
q
q YqYqY
^
^
{ÊrÎÊÊ
^
q
{Êr ÇÊÊÊÊr £{ÊÊ
^
q Xq XqX
qq q ^
q
^
^
Êr ÎÊÊ
œÌʏ}iLÀ>Ê£
q
^
^
q
q
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
œÌʏ}iLÀ>Ê£
*ÀœLi“Ê-œÛˆ˜}
--"
££‡Ç !DDINGAND3UBTRACTING2ADICAL%XPRESSIONS
,i>`ˆ˜}Ê-ÌÀ>Ìi}ˆiÃ
--"
££‡Ç #ONNECTING#ONCEPTS
7RITETHECORRECTANSWERASARADICALEXPRESSIONINSIMPLESTFORM
4HEPROCESSFORADDINGANDSUBTRACTINGRADICALEXPRESSIONSISSIMILARTO
THEPROCESSFORSIMPLIFYINGALGEBRAICEXPRESSIONS,OOKATTHECONNECTIONS
BELOW
!RECTANGULARLAUNDRYROOMHASALENGTH
^
^
OFqFEETANDAWIDTHOFqFEET
&INDTHEPERIMETEROFTHELAUNDRYROOM
4HEPARKSDEPARTMENTISINSTALLINGA
FENCEALONGASCENICOVERLOOK4HEAREA
^
TOBEFENCEDHASSIDESMEASURINGq ^
^
FEETq FEETANDqFEET&INDTHE
TOTALAMOUNTOFFENCINGTHATNEEDSTOBE
INSTALLED
^
4RAVONISHANGINGTHESAMEWALLPAPER
BORDERINTWOROOMS/NEROOMISA
PERFECTSQUAREWITHANAREAOF
SQUAREFEET4HEOTHERROOMARECTANGLE
^
HASAWIDTHOFq FEETANDALENGTHOF
FEET(OWMUCHWALLPAPERBORDERDOES
4RAVONNEEDTOGOAROUNDTHEPERIMETER
OFTHETWOROOMS
^
-R,ANSBERRYBOUGHTFOURWATERMELONS
FORAFAMILYREUNION4HEWATERMELONS
^
^
WEIGHEDq POUNDSqPOUNDS
^
^
q POUNDSANDq POUNDS(OW
MANYPOUNDSOFWATERMELONDID
-R,ANSBERRYBRINGTOTHEREUNION
^
^
^
^
q qq
X ANDX ^
^
q ANDq
3TEP2EARRANGESOLIKETERMSLIKERADICALSARETOGETHER
XXX
^
^
XX
^
^
^
# qIN
^
$ qIN
& S
^
' qS
(
^
qS
^
q q
^
^
^
' qCM
/…iÞʅ>ÛiÊ̅iÊÃ>“iÊÀ>`ˆV>˜`]ÊÈ°
^
^
$ESCRIBEHOWTOCOMBINETHERADICALSq ANDq
^
ii«ÊÊr ÈÊÊ>˜`ÊÃÕLÌÀ>VÌÊxÊÊΰ
^
7ÀˆÌiÊÓÊqx{ÊÊ>ÃÊÈÊr ÈÊÊ°
3IMPLIFYTHERADICALEXPRESSIONGIVENINPROBLEM
^
nÊrÈÊÊ
3IMPLIFYEACHEXPRESSION
^
^
^
^
AK4up.indd 88
88
^
qYq Y
ÓäÊrÓÊÊ
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
^
ÈÊr ÎXÊÊÓÊr £äXÊ
^
qq q œÌʏ}iLÀ>Ê£
^
^
^
^
^
q Xq XqX
ÎÊr xÊÊ
^
* CM
^
qqq
^
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
^
^
^
^
7HATEXTRASTEPWOULDNEEDTOBEDONEFIRSTIFTHEPROBLEMWASq qq * S
!TRIANGULARPENNANTHASTWOSIDESTHAT
^
MEASUREq CENTIMETERSANDATHIRD
^
SIDETHATMEASURESqCENTIMETERS
-RS+WANISSEWINGROWSOFGOLD
RIBBONAROUNDTHEPERIMETEROFTHE
PENNANT(OWMUCHRIBBONDOESSHE
NEED
^
^
& qCM
( q CM
" qq IN
^
^
# q q IN
^
^
$ q q IN
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
^
q qq
3TEP#OMBINELIKETERMSLIKERADICALS
7HATMAKESqANDq LIKERADICALS
*ACK!ISLINN-ERCEDESAND$AEARE
ATEAMPARTICIPATINGINARELAYATAFALL
^
FESTIVAL*ACKSTIMEWASq SECONDS
^
!ISLINNSWASqSECONDS-ERCEDESS
^
WASqSECONDSAND$AESWAS
^
q SECONDS7HATWASTHETEAMSTOTAL
TIME
,ILYHASTWOPICTUREFRAMESSHEIS
REPLACING4HEFIRSTFRAMEISSHAPEDLIKE
AREGULAROCTAGONANDALLSIDESMEASURE
^
q IN4HESECONDFRAMEISSHAPEDLIKE
ARECTANGLETHELENGTHANDWIDTHARE
^
^
q INANDq INRESPECTIVELY(OW
MUCHTOTALFRAMINGWILLSHENEED
^
^
! qq IN
^
X XX ^
^
3IMPLIFYr r r
#OMPLETETHEFOLLOWINGBASEDONTHEEXAMPLESABOVE
!CAFETERIATRAYISSHAPEDLIKEAN
ISOSCELESTRAPEZOID4HEBASESMEASURE
^
^
INANDq IN4HELEGS
q ^
MEASUREqIN&INDTHEPERIMETEROF
THETRAY
^
^
3TEP)DENTIFYLIKETERMSLIKERADICALS
3ELECTTHEBESTANSWER
! q
IN
^
IN
" q
^
£äÊr ÎäÊÊÊ£ÈÊvÌ
^
££ÊrÎÊÊʙÊrxÊʏLÃ
3IMPLIFYXXX
ÈÊr ÈÊÊÊnÊr ÎÊÊvÌ
£ÓÊr xÊÊvÌ
^
^
Ó£ÊrÇY
œÌʏ}iLÀ>Ê£
Holt Algebra 1
12/26/05 8:02:13 AM