%VALUATE NTH 2OOTS AND 5SE
2ATIONAL %XPONENTS
'OAL
9OUR .OTES
+ %VALUATE NTH ROOTS AND STUDY RATIONAL EXPONENTS
6/#!"5,!29
NTH ROOT OF A œÀÊ>˜Êˆ˜Ìi}iÀÊ ˜Ê}Ài>ÌiÀÊ̅>˜Ê£]ʈvÊÊ
L˜ÊÊ>]Ê̅i˜ÊLʈÃÊ>˜Ê˜ ̅ÊÀœœÌʜvÊ>°
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)NDEX OF A RADICAL ʘʘ ̅ÊÀœœÌʜvÊ>ʈÃÊÜÀˆÌÌi˜Ê>ÃÊÊÊqÊ>Ê]zÊÊ
܅iÀiʘʈÃÊ̅iʈ˜`iÝʜvÊ̅iÊÀ>`ˆV>°
2%!, NTH 2//43 /& A
,ET N BE AN INTEGER N AND LET A BE A REAL NUMBER
)F N IS AN EVEN INTEGER
)F N IS AN ODD INTEGER
s A .O REAL NTH ROOTS s A /NE REAL NTH ROOT
N]
q A z Ê >£É˜Ê
s A /NE REAL NTH ROOT s A /NE REAL NTH ROOT
N]
N]
q z Ê äÊ
q z Ê äÊ
s A 4WO REAL NTH ROOTS s A /NE REAL NTH ROOT
N]
N]
zq
z A z zÊ >£É˜Ê
q A z Ê >£É˜Ê
%XAMPLE &IND NTH ROOTS
&IND THE INDICATED REAL NTH ROOTS OF A
A N A z
B N A 3OLUTION
A "ECAUSE N IS ODD AND A z Ê Ê HAS
Ê œ˜iÊÀi>ÊVÕLiÊÀœœÌÊ "ECAUSE Ê {Ê z YOU
]
CAN WRITE q zzÊ {Ê OR zÊ {Ê B "ECAUSE N IS EVEN AND A z Ê €Ê HAS
Ê ÌܜÊÀi>ÊÈÝ̅ÊÀœœÌÃÊ "ECAUSE Ê ÎÊ ]
AND Ê ÎÊ YOU CAN WRITE zzq zzÊ ÎÊ OR
z zÊ ÎÊ ,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
#HECKPOINT &IND THE INDICATED REAL NTH ROOTS OF A
N A z
N A Ê n
z {Ê
2!4)/.!, %80/.%.43
,ET A BE A REAL NUMBER AND LET M AND N BE POSITIVE
INTEGERS WITH N ˜]
AMN ANM qÊ>ÊzÊ M
AMN ]
z ]
z
]
Azsz
˜ ] Mz
MN
N M
A
A
qÊ>ÊzÊ %VALUATE AN EXPRESSION WITH RATIONAL EXPONENTS
%XAMPLE %VALUATE 3OLUTION
2ATIONAL %XPONENT &ORM
2ADICAL &ORM
£
Ê Êz
]
{ÉÎ
£
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Ê Êz
{ÉÎ
n
n
£
£
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Ê Êz
Î] {
­ÊÊqÊnÊ®Êz
Ê Êzz
zÊ ]
£ÉÎ {
­n ®
Ê
Ê
Ê
Ê
zÊ ]z
Ê£{ÊÊ Êz
Ó
zÊ ]z
Ê£ÊÊ Ê
£È
%XAMPLE z
Ê
zÊ ]z
Ê£{ÊÊ Ê
Ó
zÊ ]z
Ê£ÊÊ
£È
3OLVE EQUATIONS USING NTH ROOTS
A X X Ê ÇәÊ
È]
B X Î]
X Ê ÊqÊ£ÓÊÊz
Î]
Ê
X Ê zzqÊÇәÊz
X qÊ£ÓÊÊzÊ{Ê
X Ê ÎÊ
X|y|Ê £°Ç£Ê
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
%XAMPLE 5SE NTH ROOTS IN PROBLEM SOLVING
!NIMAL 0OPULATION 4HE POPULATION 0 OF A CERTAIN ANIMAL
SPECIES AFTER T MONTHS CAN BE MODELED BY 0 #T
WHERE # IS THE INITIAL POPULATION &IND THE POPULATION AFTER
MONTHS IF THE INITIAL POPULATION WAS 3OLUTION
0 #T
7RITE MODEL FOR POPULATION
Ê Çx­£°Ó£®£™ÉÎÊ
3UBSTITUTE FOR # AND T
y|Ê Óxä°nÊ
5SE A CALCULATOR
4HE POPULATION OF THE SPECIES IS ABOUT Ê Óx£Ê AFTER
MONTHS
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISES
%VALUATE £
Óx
]ÊÊz
Ê
Ê Ê
3OLVE Y {]
ÊÊÎ|y|È°ÇÈʜÀ
qÊÓääÊz
{]
Ê ÊÊqÊÓääÊÊzÊÎÊyÊä°ÇÈ
:R H
4HE VOLUME OF A CONE IS GIVEN BY 6 ]
z WHERE H
IS THE HEIGHT OF THE CONE AND R IS THE RADIUS &IND THE
RADIUS OF A CONE WHOSE VOLUME IS CUBIC INCHES
AND WHOSE HEIGHT IS INCHES
Ê
Ê £°™™Êˆ˜°
(OMEWORK
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY