Important Formulas: Exponents and Radicals | Mathematics for Digital SAT PDF Download

 Page 1


Exponents & Radicals
Giventheexpression x
n
, xiscalledthebaseand niscalledtheexponentorpower. Herearethelawsof
exponentsyoushouldknow:
Law Example
x
1
= x 3
1
= 3
x
0
= 1 3
0
= 1
x
m
·x
n
= x
m+n
3
4
·3
5
= 3
9
x
m
x
n
= x
m-n
3
7
3
3
= 3
4
!
x
m
"
n
= x
mn
!
3
2
"
4
= 3
8
(xy)
m
= x
m
y
m
(2 ·3)
3
= 2
3
·3
3
Å
x
y
ã
m
=
x
m
y
m
Å
2
3
ã
3
=
2
3
3
3
x
–m
=
1
x
m
3
–4
=
1
3
4
Page 2


Exponents & Radicals
Giventheexpression x
n
, xiscalledthebaseand niscalledtheexponentorpower. Herearethelawsof
exponentsyoushouldknow:
Law Example
x
1
= x 3
1
= 3
x
0
= 1 3
0
= 1
x
m
·x
n
= x
m+n
3
4
·3
5
= 3
9
x
m
x
n
= x
m-n
3
7
3
3
= 3
4
!
x
m
"
n
= x
mn
!
3
2
"
4
= 3
8
(xy)
m
= x
m
y
m
(2 ·3)
3
= 2
3
·3
3
Å
x
y
ã
m
=
x
m
y
m
Å
2
3
ã
3
=
2
3
3
3
x
–m
=
1
x
m
3
–4
=
1
3
4
Thedifferencebetween
(-3)
2
and-3
2
comesdowntoorderofoperations(PEMDAS),whichdictatesthatparenthesesareprioritized?rst. So,
(-3)
2
= (-3) ·(-3)= 9
Withoutparentheses,exponentstakepriority:
-3
2
=-3 ·3=-9
Noticethatthenegativeisnotapplieduntiltheexponentoperationiscarriedthrough.
Sometimes,theresultturnsouttobethesame,asin:
(-2)
3
and-2
3
Bothyield-8.
EXAMPLE1: Whichofthefollowingisequivalentto-x
2
(-x)
5
?
A)-x
10
B)-x
7
C) x
7
D) x
10
The-x
2
termcannotbesimpli?edanyfurtherbecausethenegativeisnotinsideanyparentheses. The(-x)
5
termhasanoddexponent. Anoddexponentyieldsanoddnumberofnegativesigns,whichinturnyieldsa
negativeresult. So,(-x)
5
simpli?esto-x
5
. Theentireexpressionthenboilsdownto
-x
2
(-x)
5
=-x
2
·-x
5
= x
2+5
= x
7
Answer (C) .
EXAMPLE2: Whichexpressionisequivalentto4
x+1
·3
2x
,where x>0?
A)4(12)
x
B)4(24)
x
C)4(36)
x
D)8(12)
x
Wecanuseourlawsofexponentstosimplifytheexpression. First,thelaw x
m+n
= x
m
·x
n
allowsusto
simplify4
x+1
to4
x
·4
1
. Thenthelaw x
mn
=
!
x
m
"
n
allowsustoturn3
2x
into
!
3
2
"
x
= 9
x
. Afterapplyingthese
laws,weget
4
x+1
·3
2x
= 4
x
·4
1
·9
x
= 4 ·(4
x
·9
x
)= 4(36)
x
Answer (C) . Inthelaststep,weusedthelaw x
m
y
m
= (xy)
m
toget4
x
·9
x
= 36
x
.
Rootsaretheoppositesofexponents. Whereas5
2
willgiveyou25,thesquarerootof25,denotedby
v
25,will
giveyoubackthe5. Similarly,takingthecuberootof6
3
gives
3
#
6
3
= 6andtakingthefourthrootof b
4
gives
4
#
b
4
= b. Noticehowtheoperationscanceleachotherout.
Page 3


Exponents & Radicals
Giventheexpression x
n
, xiscalledthebaseand niscalledtheexponentorpower. Herearethelawsof
exponentsyoushouldknow:
Law Example
x
1
= x 3
1
= 3
x
0
= 1 3
0
= 1
x
m
·x
n
= x
m+n
3
4
·3
5
= 3
9
x
m
x
n
= x
m-n
3
7
3
3
= 3
4
!
x
m
"
n
= x
mn
!
3
2
"
4
= 3
8
(xy)
m
= x
m
y
m
(2 ·3)
3
= 2
3
·3
3
Å
x
y
ã
m
=
x
m
y
m
Å
2
3
ã
3
=
2
3
3
3
x
–m
=
1
x
m
3
–4
=
1
3
4
Thedifferencebetween
(-3)
2
and-3
2
comesdowntoorderofoperations(PEMDAS),whichdictatesthatparenthesesareprioritized?rst. So,
(-3)
2
= (-3) ·(-3)= 9
Withoutparentheses,exponentstakepriority:
-3
2
=-3 ·3=-9
Noticethatthenegativeisnotapplieduntiltheexponentoperationiscarriedthrough.
Sometimes,theresultturnsouttobethesame,asin:
(-2)
3
and-2
3
Bothyield-8.
EXAMPLE1: Whichofthefollowingisequivalentto-x
2
(-x)
5
?
A)-x
10
B)-x
7
C) x
7
D) x
10
The-x
2
termcannotbesimpli?edanyfurtherbecausethenegativeisnotinsideanyparentheses. The(-x)
5
termhasanoddexponent. Anoddexponentyieldsanoddnumberofnegativesigns,whichinturnyieldsa
negativeresult. So,(-x)
5
simpli?esto-x
5
. Theentireexpressionthenboilsdownto
-x
2
(-x)
5
=-x
2
·-x
5
= x
2+5
= x
7
Answer (C) .
EXAMPLE2: Whichexpressionisequivalentto4
x+1
·3
2x
,where x>0?
A)4(12)
x
B)4(24)
x
C)4(36)
x
D)8(12)
x
Wecanuseourlawsofexponentstosimplifytheexpression. First,thelaw x
m+n
= x
m
·x
n
allowsusto
simplify4
x+1
to4
x
·4
1
. Thenthelaw x
mn
=
!
x
m
"
n
allowsustoturn3
2x
into
!
3
2
"
x
= 9
x
. Afterapplyingthese
laws,weget
4
x+1
·3
2x
= 4
x
·4
1
·9
x
= 4 ·(4
x
·9
x
)= 4(36)
x
Answer (C) . Inthelaststep,weusedthelaw x
m
y
m
= (xy)
m
toget4
x
·9
x
= 36
x
.
Rootsaretheoppositesofexponents. Whereas5
2
willgiveyou25,thesquarerootof25,denotedby
v
25,will
giveyoubackthe5. Similarly,takingthecuberootof6
3
gives
3
#
6
3
= 6andtakingthefourthrootof b
4
gives
4
#
b
4
= b. Noticehowtheoperationscanceleachotherout.
Insteadofusingtheradicalsign, n
v
,wecanalsodenotethe nth-rootusingtheequivalentfractional
exponent
1
n
. Forexample,
v
x= x
1
2
3
v
x= x
1
3
Whatabout x
2
3
? The2ontopmeanstosquare x. The3onthebottommeanstocuberootit:
3
#
x
2
Wecanseethismoreclearlyifwebreakitdownusingourlawsofexponents:
x
2
3
=
Ä
x
2
ä1
3
=
3
#
x
2
Theorderinwhichwedothesquaringandthecube-rootingdoesn’tmatter. Wecould’vebrokenitdownthis
wayaswell:
x
2
3
=
!
x
1
3
"
2
=
!
3
v
x
"
2
You’llsee
3
#
x
2
moreoftenthan
!
3
v
x
"
2
becausetheoutsidecuberootavoidstheneedforparentheses.
EXAMPLE3: Whichofthefollowingisequivalentto
4
#
x
5
?
A) x B) x
5
-x
4
C) x
5
4
D) x
4
5
Thefourthrootequatestoafractionalexponentof
1
4
,so
4
#
x
5
=
Ä
x
5
ä1
4
= x
5
4
Answer (C) .
EXAMPLE4: Theexpression
25
2x
Ä
v
5
ä
x
isequivalentto5
kx
,where kisaconstant. Whatisthevalueof k?
Becausemostofthelawsofexponentsinvolvetermswiththesamebase,anoften-usedtacticistoconvert
eachtermtohavethesamebase. Here,we’reaskedtosimplifyto5
kx
,soweshouldconverteverythingto
havethesamebaseof5. Asiscommoninthistypeofquestion,weseethatwehaveaperfectsquare,25,
whichcanbewrittenas5
2
. Inthedenominator,wehave
v
5,whichcanbewrittenas5
1
2
.
25
2x
Ä
v
5
ä
x
=
!
5
2
"
2x
!
5
1
2
"
x
=
5
4x
5
1
2
x
= 5
4x-
1
2
x
= 5
7
2
x
Therefore, k=
7
2
.
Page 4


Exponents & Radicals
Giventheexpression x
n
, xiscalledthebaseand niscalledtheexponentorpower. Herearethelawsof
exponentsyoushouldknow:
Law Example
x
1
= x 3
1
= 3
x
0
= 1 3
0
= 1
x
m
·x
n
= x
m+n
3
4
·3
5
= 3
9
x
m
x
n
= x
m-n
3
7
3
3
= 3
4
!
x
m
"
n
= x
mn
!
3
2
"
4
= 3
8
(xy)
m
= x
m
y
m
(2 ·3)
3
= 2
3
·3
3
Å
x
y
ã
m
=
x
m
y
m
Å
2
3
ã
3
=
2
3
3
3
x
–m
=
1
x
m
3
–4
=
1
3
4
Thedifferencebetween
(-3)
2
and-3
2
comesdowntoorderofoperations(PEMDAS),whichdictatesthatparenthesesareprioritized?rst. So,
(-3)
2
= (-3) ·(-3)= 9
Withoutparentheses,exponentstakepriority:
-3
2
=-3 ·3=-9
Noticethatthenegativeisnotapplieduntiltheexponentoperationiscarriedthrough.
Sometimes,theresultturnsouttobethesame,asin:
(-2)
3
and-2
3
Bothyield-8.
EXAMPLE1: Whichofthefollowingisequivalentto-x
2
(-x)
5
?
A)-x
10
B)-x
7
C) x
7
D) x
10
The-x
2
termcannotbesimpli?edanyfurtherbecausethenegativeisnotinsideanyparentheses. The(-x)
5
termhasanoddexponent. Anoddexponentyieldsanoddnumberofnegativesigns,whichinturnyieldsa
negativeresult. So,(-x)
5
simpli?esto-x
5
. Theentireexpressionthenboilsdownto
-x
2
(-x)
5
=-x
2
·-x
5
= x
2+5
= x
7
Answer (C) .
EXAMPLE2: Whichexpressionisequivalentto4
x+1
·3
2x
,where x>0?
A)4(12)
x
B)4(24)
x
C)4(36)
x
D)8(12)
x
Wecanuseourlawsofexponentstosimplifytheexpression. First,thelaw x
m+n
= x
m
·x
n
allowsusto
simplify4
x+1
to4
x
·4
1
. Thenthelaw x
mn
=
!
x
m
"
n
allowsustoturn3
2x
into
!
3
2
"
x
= 9
x
. Afterapplyingthese
laws,weget
4
x+1
·3
2x
= 4
x
·4
1
·9
x
= 4 ·(4
x
·9
x
)= 4(36)
x
Answer (C) . Inthelaststep,weusedthelaw x
m
y
m
= (xy)
m
toget4
x
·9
x
= 36
x
.
Rootsaretheoppositesofexponents. Whereas5
2
willgiveyou25,thesquarerootof25,denotedby
v
25,will
giveyoubackthe5. Similarly,takingthecuberootof6
3
gives
3
#
6
3
= 6andtakingthefourthrootof b
4
gives
4
#
b
4
= b. Noticehowtheoperationscanceleachotherout.
Insteadofusingtheradicalsign, n
v
,wecanalsodenotethe nth-rootusingtheequivalentfractional
exponent
1
n
. Forexample,
v
x= x
1
2
3
v
x= x
1
3
Whatabout x
2
3
? The2ontopmeanstosquare x. The3onthebottommeanstocuberootit:
3
#
x
2
Wecanseethismoreclearlyifwebreakitdownusingourlawsofexponents:
x
2
3
=
Ä
x
2
ä1
3
=
3
#
x
2
Theorderinwhichwedothesquaringandthecube-rootingdoesn’tmatter. Wecould’vebrokenitdownthis
wayaswell:
x
2
3
=
!
x
1
3
"
2
=
!
3
v
x
"
2
You’llsee
3
#
x
2
moreoftenthan
!
3
v
x
"
2
becausetheoutsidecuberootavoidstheneedforparentheses.
EXAMPLE3: Whichofthefollowingisequivalentto
4
#
x
5
?
A) x B) x
5
-x
4
C) x
5
4
D) x
4
5
Thefourthrootequatestoafractionalexponentof
1
4
,so
4
#
x
5
=
Ä
x
5
ä1
4
= x
5
4
Answer (C) .
EXAMPLE4: Theexpression
25
2x
Ä
v
5
ä
x
isequivalentto5
kx
,where kisaconstant. Whatisthevalueof k?
Becausemostofthelawsofexponentsinvolvetermswiththesamebase,anoften-usedtacticistoconvert
eachtermtohavethesamebase. Here,we’reaskedtosimplifyto5
kx
,soweshouldconverteverythingto
havethesamebaseof5. Asiscommoninthistypeofquestion,weseethatwehaveaperfectsquare,25,
whichcanbewrittenas5
2
. Inthedenominator,wehave
v
5,whichcanbewrittenas5
1
2
.
25
2x
Ä
v
5
ä
x
=
!
5
2
"
2x
!
5
1
2
"
x
=
5
4x
5
1
2
x
= 5
4x-
1
2
x
= 5
7
2
x
Therefore, k=
7
2
.
EXAMPLE5: Ifthesquareof aisequaltothecubeof b,where a= 0and b= 0,forwhatvalueof xis
v
a
x
equalto b?
Essentially,weneedtosolve
v
a
x
= b
for xgiventhat a
2
= b
3
. Let’s?rstcuberootbothsidesof a
2
= b
3
toisolate b:
3
#
a
2
=
3
#
b
3
a
2
3
= b
Substituting a
2
3
for bintheequationabove,weget
v
a
x
= a
2
3
a
1
2
x
= a
2
3
Sincebothsideshavethesamebaseof a,wecanjustequatetheexponentsandsolvefor x:
1
2
x=
2
3
x=
4
3
TheSATwillalsotestyouonsimplifyingsquareroots(alsocalled“surds”). Tosimplifyasquareroot,factor
thenumberinsidethesquarerootandtakeoutanypairs:
v
48=
v
2 ·2 ·2 ·2 ·3=
$
2 ·2 · 2 ·2 ·3= 2 ·2
v
3= 4
v
3
Intheexampleabove,wetakea2outforthe?rst 2 ·2 . Thenwetakeanother2outforthesecondpair 2 ·2 .
Finally,wemultiplythetwo2’soutsidethesquareroottoget4andleavethe3inside. Ofcourse,aquicker
routewouldhavebeentobreakdown48intobiggerfactorpairs:
v
48=
$
4 ·4 ·3= 4
v
3
Here’sanotherexample:
v
72=
$
2 ·2 · 3 ·3 ·2= 2 ·3
v
2= 6
v
2
Togobackwards,takethenumberoutsideandputitbackunderthesquarerootasapair:
6
v
2=
v
6 ·6 ·2=
v
72
Tosimplifyacuberootsuchas
3
v
16,takeoutanytriplets:
3
v
16=
3
$
2 ·2 ·2 ·2= 2
3
v
2
Page 5


Exponents & Radicals
Giventheexpression x
n
, xiscalledthebaseand niscalledtheexponentorpower. Herearethelawsof
exponentsyoushouldknow:
Law Example
x
1
= x 3
1
= 3
x
0
= 1 3
0
= 1
x
m
·x
n
= x
m+n
3
4
·3
5
= 3
9
x
m
x
n
= x
m-n
3
7
3
3
= 3
4
!
x
m
"
n
= x
mn
!
3
2
"
4
= 3
8
(xy)
m
= x
m
y
m
(2 ·3)
3
= 2
3
·3
3
Å
x
y
ã
m
=
x
m
y
m
Å
2
3
ã
3
=
2
3
3
3
x
–m
=
1
x
m
3
–4
=
1
3
4
Thedifferencebetween
(-3)
2
and-3
2
comesdowntoorderofoperations(PEMDAS),whichdictatesthatparenthesesareprioritized?rst. So,
(-3)
2
= (-3) ·(-3)= 9
Withoutparentheses,exponentstakepriority:
-3
2
=-3 ·3=-9
Noticethatthenegativeisnotapplieduntiltheexponentoperationiscarriedthrough.
Sometimes,theresultturnsouttobethesame,asin:
(-2)
3
and-2
3
Bothyield-8.
EXAMPLE1: Whichofthefollowingisequivalentto-x
2
(-x)
5
?
A)-x
10
B)-x
7
C) x
7
D) x
10
The-x
2
termcannotbesimpli?edanyfurtherbecausethenegativeisnotinsideanyparentheses. The(-x)
5
termhasanoddexponent. Anoddexponentyieldsanoddnumberofnegativesigns,whichinturnyieldsa
negativeresult. So,(-x)
5
simpli?esto-x
5
. Theentireexpressionthenboilsdownto
-x
2
(-x)
5
=-x
2
·-x
5
= x
2+5
= x
7
Answer (C) .
EXAMPLE2: Whichexpressionisequivalentto4
x+1
·3
2x
,where x>0?
A)4(12)
x
B)4(24)
x
C)4(36)
x
D)8(12)
x
Wecanuseourlawsofexponentstosimplifytheexpression. First,thelaw x
m+n
= x
m
·x
n
allowsusto
simplify4
x+1
to4
x
·4
1
. Thenthelaw x
mn
=
!
x
m
"
n
allowsustoturn3
2x
into
!
3
2
"
x
= 9
x
. Afterapplyingthese
laws,weget
4
x+1
·3
2x
= 4
x
·4
1
·9
x
= 4 ·(4
x
·9
x
)= 4(36)
x
Answer (C) . Inthelaststep,weusedthelaw x
m
y
m
= (xy)
m
toget4
x
·9
x
= 36
x
.
Rootsaretheoppositesofexponents. Whereas5
2
willgiveyou25,thesquarerootof25,denotedby
v
25,will
giveyoubackthe5. Similarly,takingthecuberootof6
3
gives
3
#
6
3
= 6andtakingthefourthrootof b
4
gives
4
#
b
4
= b. Noticehowtheoperationscanceleachotherout.
Insteadofusingtheradicalsign, n
v
,wecanalsodenotethe nth-rootusingtheequivalentfractional
exponent
1
n
. Forexample,
v
x= x
1
2
3
v
x= x
1
3
Whatabout x
2
3
? The2ontopmeanstosquare x. The3onthebottommeanstocuberootit:
3
#
x
2
Wecanseethismoreclearlyifwebreakitdownusingourlawsofexponents:
x
2
3
=
Ä
x
2
ä1
3
=
3
#
x
2
Theorderinwhichwedothesquaringandthecube-rootingdoesn’tmatter. Wecould’vebrokenitdownthis
wayaswell:
x
2
3
=
!
x
1
3
"
2
=
!
3
v
x
"
2
You’llsee
3
#
x
2
moreoftenthan
!
3
v
x
"
2
becausetheoutsidecuberootavoidstheneedforparentheses.
EXAMPLE3: Whichofthefollowingisequivalentto
4
#
x
5
?
A) x B) x
5
-x
4
C) x
5
4
D) x
4
5
Thefourthrootequatestoafractionalexponentof
1
4
,so
4
#
x
5
=
Ä
x
5
ä1
4
= x
5
4
Answer (C) .
EXAMPLE4: Theexpression
25
2x
Ä
v
5
ä
x
isequivalentto5
kx
,where kisaconstant. Whatisthevalueof k?
Becausemostofthelawsofexponentsinvolvetermswiththesamebase,anoften-usedtacticistoconvert
eachtermtohavethesamebase. Here,we’reaskedtosimplifyto5
kx
,soweshouldconverteverythingto
havethesamebaseof5. Asiscommoninthistypeofquestion,weseethatwehaveaperfectsquare,25,
whichcanbewrittenas5
2
. Inthedenominator,wehave
v
5,whichcanbewrittenas5
1
2
.
25
2x
Ä
v
5
ä
x
=
!
5
2
"
2x
!
5
1
2
"
x
=
5
4x
5
1
2
x
= 5
4x-
1
2
x
= 5
7
2
x
Therefore, k=
7
2
.
EXAMPLE5: Ifthesquareof aisequaltothecubeof b,where a= 0and b= 0,forwhatvalueof xis
v
a
x
equalto b?
Essentially,weneedtosolve
v
a
x
= b
for xgiventhat a
2
= b
3
. Let’s?rstcuberootbothsidesof a
2
= b
3
toisolate b:
3
#
a
2
=
3
#
b
3
a
2
3
= b
Substituting a
2
3
for bintheequationabove,weget
v
a
x
= a
2
3
a
1
2
x
= a
2
3
Sincebothsideshavethesamebaseof a,wecanjustequatetheexponentsandsolvefor x:
1
2
x=
2
3
x=
4
3
TheSATwillalsotestyouonsimplifyingsquareroots(alsocalled“surds”). Tosimplifyasquareroot,factor
thenumberinsidethesquarerootandtakeoutanypairs:
v
48=
v
2 ·2 ·2 ·2 ·3=
$
2 ·2 · 2 ·2 ·3= 2 ·2
v
3= 4
v
3
Intheexampleabove,wetakea2outforthe?rst 2 ·2 . Thenwetakeanother2outforthesecondpair 2 ·2 .
Finally,wemultiplythetwo2’soutsidethesquareroottoget4andleavethe3inside. Ofcourse,aquicker
routewouldhavebeentobreakdown48intobiggerfactorpairs:
v
48=
$
4 ·4 ·3= 4
v
3
Here’sanotherexample:
v
72=
$
2 ·2 · 3 ·3 ·2= 2 ·3
v
2= 6
v
2
Togobackwards,takethenumberoutsideandputitbackunderthesquarerootasapair:
6
v
2=
v
6 ·6 ·2=
v
72
Tosimplifyacuberootsuchas
3
v
16,takeoutanytriplets:
3
v
16=
3
$
2 ·2 ·2 ·2= 2
3
v
2
EXAMPLE6: Whichofthefollowingisequivalentto
Ä
x
2
ä3
4
,where x>0?
A)
v
x B) x
v
x C)
3
#
x
2
D)
4
v
x
Solution1:
Ä
x
2
ä3
4
= x
(2·
3
4
)
= x
3
2
=
#
x
3
=
»
x ·x ·x= x
v
x
Answer (B) .
Solution2: Since
Ä
x
2
ä3
4
= x
(2·
3
4
)
= x
3
2
,wecancomparethisexponentof
3
2
totheexponentof xineachofthe
answerchoices.
ChoiceA:
v
x= x
1
2
ChoiceB: x
v
x= x
1
·x
1
2
= x
(1+
1
2
)
= x
3
2
ChoiceC:
3
#
x
2
= x
2
3
ChoiceD:
4
v
x= x
1
4
Theseresultscon?rmthattheansweris (B) .