Mathematics Formulas, Formulas and forms for Mathematics. Purbanchal University
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Mathematics Formulas, Formulas and forms for Mathematics. Purbanchal University

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1300 Mathematics Formulas for all kinds of Mathematics Problems.
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1300 Math Formulas

1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

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Preface = = = = qÜáë= Ü~åÇÄççâ= áë= ~= ÅçãéäÉíÉ= ÇÉëâíçé= êÉÑÉêÉåÅÉ= Ñçê= ëíì- ÇÉåíë= ~åÇ= ÉåÖáåÉÉêëK= fí= Ü~ë= ÉîÉêóíÜáåÖ= Ñêçã= ÜáÖÜ= ëÅÜççä= ã~íÜ=íç=ã~íÜ=Ñçê=~Çî~åÅÉÇ=ìåÇÉêÖê~Çì~íÉë=áå=ÉåÖáåÉÉêáåÖI= ÉÅçåçãáÅëI=éÜóëáÅ~ä=ëÅáÉåÅÉëI=~åÇ=ã~íÜÉã~íáÅëK=qÜÉ=ÉÄççâ= Åçåí~áåë= ÜìåÇêÉÇë= çÑ= Ñçêãìä~ëI= í~ÄäÉëI= ~åÇ= ÑáÖìêÉë= Ñêçã= kìãÄÉê= pÉíëI= ^äÖÉÄê~I= dÉçãÉíêóI= qêáÖçåçãÉíêóI= j~íêáÅÉë= ~åÇ= aÉíÉêãáå~åíëI= sÉÅíçêëI= ^å~äóíáÅ= dÉçãÉíêóI= `~äÅìäìëI= aáÑÑÉêÉåíá~ä=bèì~íáçåëI=pÉêáÉëI=~åÇ=mêçÄ~Äáäáíó=qÜÉçêóK== qÜÉ= ëíêìÅíìêÉÇ= í~ÄäÉ= çÑ= ÅçåíÉåíëI= äáåâëI= ~åÇ= ä~óçìí= ã~âÉ= ÑáåÇáåÖ= íÜÉ= êÉäÉî~åí= áåÑçêã~íáçå= èìáÅâ= ~åÇ= é~áåäÉëëI= ëç= áí= Å~å=ÄÉ=ìëÉÇ=~ë=~å=ÉîÉêóÇ~ó=çåäáåÉ=êÉÑÉêÉåÅÉ=ÖìáÇÉK===

= =

iii

Contents = = = =

1 krj_bo=pbqp= NKN= pÉí=fÇÉåíáíáÉë==1= NKO= pÉíë=çÑ=kìãÄÉêë==5= NKP= _~ëáÅ=fÇÉåíáíáÉë==7= NKQ= `çãéäÉñ=kìãÄÉêë==8= = 2 ^idb_o^= OKN= c~ÅíçêáåÖ=cçêãìä~ë==12= OKO= mêçÇìÅí=cçêãìä~ë==13= OKP= mçïÉêë==14= OKQ= oççíë==15= OKR= içÖ~êáíÜãë==16= OKS= bèì~íáçåë==18= OKT= fåÉèì~äáíáÉë==19= OKU= `çãéçìåÇ=fåíÉêÉëí=cçêãìä~ë==22= = 3 dbljbqov= PKN= oáÖÜí=qêá~åÖäÉ==24= PKO= fëçëÅÉäÉë=qêá~åÖäÉ==27= PKP= bèìáä~íÉê~ä=qêá~åÖäÉ==28= PKQ= pÅ~äÉåÉ=qêá~åÖäÉ==29= PKR= pèì~êÉ==33= PKS= oÉÅí~åÖäÉ==34= PKT= m~ê~ääÉäçÖê~ã==35= PKU= oÜçãÄìë==36= PKV= qê~éÉòçáÇ==37= PKNM= fëçëÅÉäÉë=qê~éÉòçáÇ==38= PKNN= fëçëÅÉäÉë=qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==40= PKNO= qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==41=

iv

PKNP= háíÉ==42= PKNQ= `óÅäáÅ=nì~Çêáä~íÉê~ä==43= PKNR= q~åÖÉåíá~ä=nì~Çêáä~íÉê~ä==45= PKNS= dÉåÉê~ä=nì~Çêáä~íÉê~ä==46= PKNT= oÉÖìä~ê=eÉñ~Öçå==47= PKNU= oÉÖìä~ê=mçäóÖçå==48= PKNV= `áêÅäÉ==50= PKOM= pÉÅíçê=çÑ=~=`áêÅäÉ==53= PKON= pÉÖãÉåí=çÑ=~=`áêÅäÉ==54= PKOO= `ìÄÉ==55= PKOP= oÉÅí~åÖìä~ê=m~ê~ääÉäÉéáéÉÇ==56= PKOQ= mêáëã==57= PKOR= oÉÖìä~ê=qÉíê~ÜÉÇêçå==58= PKOS= oÉÖìä~ê=móê~ãáÇ==59= PKOT= cêìëíìã=çÑ=~=oÉÖìä~ê=móê~ãáÇ==61= PKOU= oÉÅí~åÖìä~ê=oáÖÜí=tÉÇÖÉ==62= PKOV= mä~íçåáÅ=pçäáÇë==63= PKPM= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê==66= PKPN= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê=ïáíÜ=~å=lÄäáèìÉ=mä~åÉ=c~ÅÉ==68= PKPO= oáÖÜí=`áêÅìä~ê=`çåÉ==69= PKPP= cêìëíìã=çÑ=~=oáÖÜí=`áêÅìä~ê=`çåÉ==70= PKPQ= péÜÉêÉ==72= PKPR= péÜÉêáÅ~ä=`~é==72= PKPS= péÜÉêáÅ~ä=pÉÅíçê==73= PKPT= péÜÉêáÅ~ä=pÉÖãÉåí==74= PKPU= péÜÉêáÅ~ä=tÉÇÖÉ==75= PKPV= bääáéëçáÇ==76= PKQM= `áêÅìä~ê=qçêìë==78= = = 4 qofdlkljbqov= QKN= o~Çá~å=~åÇ=aÉÖêÉÉ=jÉ~ëìêÉë=çÑ=^åÖäÉë==80= QKO= aÉÑáåáíáçåë=~åÇ=dê~éÜë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==81= QKP= páÖåë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==86= QKQ= qêáÖçåçãÉíêáÅ=cìåÅíáçåë=çÑ=`çããçå=^åÖäÉë==87= QKR= jçëí=fãéçêí~åí=cçêãìä~ë==88=

v

QKS= oÉÇìÅíáçå=cçêãìä~ë==89= QKT= mÉêáçÇáÅáíó=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKU= oÉä~íáçåë=ÄÉíïÉÉå=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKV= ^ÇÇáíáçå=~åÇ=pìÄíê~Åíáçå=cçêãìä~ë==91= QKNM= açìÄäÉ=^åÖäÉ=cçêãìä~ë==92= QKNN= jìäíáéäÉ=^åÖäÉ=cçêãìä~ë==93= QKNO= e~äÑ=^åÖäÉ=cçêãìä~ë==94= QKNP= e~äÑ=^åÖäÉ=q~åÖÉåí=fÇÉåíáíáÉë==94= QKNQ= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=mêçÇìÅí==95= QKNR= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=pìã==97=== QKNS= mçïÉêë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==98= QKNT= dê~éÜë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==99= QKNU= mêáåÅáé~ä=s~äìÉë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==102= QKNV= oÉä~íáçåë=ÄÉíïÉÉå=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==103= QKOM= qêáÖçåçãÉíêáÅ=bèì~íáçåë==106= QKON= oÉä~íáçåë=íç=eóéÉêÄçäáÅ=cìåÅíáçåë==106= = = 5 j^qof`bp=^ka=abqbojfk^kqp= RKN= aÉíÉêãáå~åíë==107= RKO= mêçéÉêíáÉë=çÑ=aÉíÉêãáå~åíë==109= RKP= j~íêáÅÉë==110= RKQ= léÉê~íáçåë=ïáíÜ=j~íêáÅÉë==111= RKR= póëíÉãë=çÑ=iáåÉ~ê=bèì~íáçåë==114= = = 6 sb`qlop= SKN= sÉÅíçê=`ççêÇáå~íÉë==118= SKO= sÉÅíçê=^ÇÇáíáçå==120= SKP= sÉÅíçê=pìÄíê~Åíáçå==122= SKQ= pÅ~äáåÖ=sÉÅíçêë==122= SKR= pÅ~ä~ê=mêçÇìÅí==123= SKS= sÉÅíçê=mêçÇìÅí==125= SKT= qêáéäÉ=mêçÇìÅí=127= = = 7 ^k^ivqf`=dbljbqov= TKN= låÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==130=

vi

TKO= qïç=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==131= TKP= píê~áÖÜí=iáåÉ=áå=mä~åÉ==139= TKQ= `áêÅäÉ==149= TKR= bääáéëÉ==152= TKS= eóéÉêÄçä~==154= TKT= m~ê~Äçä~==158= TKU= qÜêÉÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==161= TKV= mä~åÉ==165= TKNM= píê~áÖÜí=iáåÉ=áå=pé~ÅÉ==175= TKNN= nì~ÇêáÅ=pìêÑ~ÅÉë==180= TKNO= péÜÉêÉ==189= = = 8 afccbobkqf^i=`^i`rirp= UKN= cìåÅíáçåë=~åÇ=qÜÉáê=dê~éÜë==191= UKO= iáãáíë=çÑ=cìåÅíáçåë==208= UKP= aÉÑáåáíáçå=~åÇ=mêçéÉêíáÉë=çÑ=íÜÉ=aÉêáî~íáîÉ==209= UKQ= q~ÄäÉ=çÑ=aÉêáî~íáîÉë==211= UKR= eáÖÜÉê=lêÇÉê=aÉêáî~íáîÉë==215= UKS= ^ééäáÅ~íáçåë=çÑ=aÉêáî~íáîÉ==217= UKT= aáÑÑÉêÉåíá~ä==221= UKU= jìäíáî~êá~ÄäÉ=cìåÅíáçåë==222= UKV= aáÑÑÉêÉåíá~ä=léÉê~íçêë==225= = = 9 fkqbdo^i=`^i`rirp= VKN= fåÇÉÑáåáíÉ=fåíÉÖê~ä==227= VKO= fåíÉÖê~äë=çÑ=o~íáçå~ä=cìåÅíáçåë==228= VKP= fåíÉÖê~äë=çÑ=fêê~íáçå~ä=cìåÅíáçåë==231= VKQ= fåíÉÖê~äë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==237= VKR= fåíÉÖê~äë=çÑ=eóéÉêÄçäáÅ=cìåÅíáçåë==241= VKS= fåíÉÖê~äë=çÑ=bñéçåÉåíá~ä=~åÇ=içÖ~êáíÜãáÅ=cìåÅíáçåë==242= VKT= oÉÇìÅíáçå=cçêãìä~ë==243= VKU= aÉÑáåáíÉ=fåíÉÖê~ä==247= VKV= fãéêçéÉê=fåíÉÖê~ä==253= VKNM= açìÄäÉ=fåíÉÖê~ä==257= VKNN= qêáéäÉ=fåíÉÖê~ä==269=

vii

VKNO= iáåÉ=fåíÉÖê~ä==275= VKNP= pìêÑ~ÅÉ=fåíÉÖê~ä==285= = = 10 afccbobkqf^i=bnr^qflkp= NMKN= cáêëí=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==295= NMKO= pÉÅçåÇ=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==298= NMKP= pçãÉ=m~êíá~ä=aáÑÑÉêÉåíá~ä=bèì~íáçåë==302= = = 11 pbofbp= NNKN= ^êáíÜãÉíáÅ=pÉêáÉë==304= NNKO= dÉçãÉíêáÅ=pÉêáÉë==305= NNKP= pçãÉ=cáåáíÉ=pÉêáÉë==305= NNKQ= fåÑáåáíÉ=pÉêáÉë==307= NNKR= mêçéÉêíáÉë=çÑ=`çåîÉêÖÉåí=pÉêáÉë==307= NNKS= `çåîÉêÖÉåÅÉ=qÉëíë==308= NNKT= ^äíÉêå~íáåÖ=pÉêáÉë==310= NNKU= mçïÉê=pÉêáÉë==311= NNKV= aáÑÑÉêÉåíá~íáçå=~åÇ=fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë==312= NNKNM= q~óäçê=~åÇ=j~Åä~ìêáå=pÉêáÉë==313= NNKNN= mçïÉê=pÉêáÉë=bñé~åëáçåë=Ñçê=pçãÉ=cìåÅíáçåë==314= NNKNO= _áåçãá~ä=pÉêáÉë==316= NNKNP= cçìêáÉê=pÉêáÉë==316= = = 12 mol_^_fifqv= NOKN= mÉêãìí~íáçåë=~åÇ=`çãÄáå~íáçåë==318= NOKO= mêçÄ~Äáäáíó=cçêãìä~ë==319= = = =

= =

viii

= qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK=

=

1

Chapter 1

Number Sets = = = =

1.1 Set Identities = pÉíëW=^I=_I=`= råáîÉêë~ä=ëÉíW=f= `çãéäÉãÉåí=W= ^′ = mêçéÉê=ëìÄëÉíW= _^⊂ == bãéíó=ëÉíW=∅ = råáçå=çÑ=ëÉíëW= _^∪ = fåíÉêëÉÅíáçå=çÑ=ëÉíëW= _^∩ = aáÑÑÉêÉåÅÉ=çÑ=ëÉíëW= _y^ = =

= 1. f^⊂ =

= 2. ^^⊂ =

= 3. _^ = =áÑ= _^⊂ =~åÇ= ^_⊂ .=

= 4. bãéíó=pÉí=

^⊂∅ = =

5. råáçå=çÑ=pÉíë== { }_ñçê^ñöñ_^` ∈∈=∪= =

=

CHAPTER 1. NUMBER SETS

2

===== = =

Figure 1. =

6. `çããìí~íáîáíó= ^__^ ∪=∪ =

= 7. ^ëëçÅá~íáîáíó=

( ) ( ) `_^`_^ ∪∪=∪∪ = = 8. fåíÉêëÉÅíáçå=çÑ=pÉíë=

{ }_ñ~åÇ^ñöñ_^` ∈∈=∪= = = =

===== = =

Figure 2. =

9. `çããìí~íáîáíó= ^__^ ∩=∩ =

= 10. ^ëëçÅá~íáîáíó=

( ) ( ) `_^`_^ ∩∩=∩∩ = =

CHAPTER 1. NUMBER SETS

3

11. aáëíêáÄìíáîáíó= ( ) ( ) ( )`^_^`_^ ∪∩∪=∩∪ I= ( ) ( ) ( )`^_^`_^ ∩∪∩=∪∩ K=

= 12. fÇÉãéçíÉåÅó=

^^^ =∩ I== ^^^ =∪ =

= 13. açãáå~íáçå=

∅=∅∩^ I= ff^ =∪ =

= 14. fÇÉåíáíó=

^^ =∅∪ I== ^f^ =∩

= 15. `çãéäÉãÉåí=

{ }^ñöfñ^ ∉∈=′ =

16. `çãéäÉãÉåí=çÑ=fåíÉêëÉÅíáçå=~åÇ=råáçå f^^ =′∪ I== ∅=′∩^^ =

= 17. aÉ=jçêÖ~å∞ë=i~ïë

( ) _^_^ ′∩′=′∪ I== ( ) _^_^ ′∪′=′∩ = =

18. aáÑÑÉêÉåÅÉ=çÑ=pÉíë { }^ñ~åÇ_ñöñ^y_` ∉∈== =

=

CHAPTER 1. NUMBER SETS

4

===== = =

Figure 3. =

19. ( )_^y_^y_ ∩= =

20. ^_^y_ ′∩= =

21. ∅=^y^ =

22. ^_y^ = =áÑ= ∅=∩_^ . =

===== = =

Figure 4. =

23. ( ) ( ) ( )`_y`^`_y^ ∩∩=∩ 24. ^yf^ =′ 25. `~êíÉëá~å=mêçÇìÅí

( ){ }_ó~åÇ^ñöóIñ_^` ∈∈=×= = =

CHAPTER 1. NUMBER SETS

5

1.2 Sets of Numbers = k~íìê~ä=åìãÄÉêëW=k= tÜçäÉ=åìãÄÉêëW= Mk =

fåíÉÖÉêëW=w= mçëáíáîÉ=áåíÉÖÉêëW= +w = kÉÖ~íáîÉ=áåíÉÖÉêëW= −w = o~íáçå~ä=åìãÄÉêëW=n= oÉ~ä=åìãÄÉêëW=o== `çãéäÉñ=åìãÄÉêëW=`== = =

26. k~íìê~ä=kìãÄÉêë `çìåíáåÖ=åìãÄÉêëW { }KIPIOINk = K=

27. tÜçäÉ=kìãÄÉêë

`çìåíáåÖ=åìãÄÉêë=~åÇ=òÉêçW= { }KIPIOINIMkM = K= = 28. fåíÉÖÉêë

tÜçäÉ=åìãÄÉêë=~åÇ=íÜÉáê=çééçëáíÉë=~åÇ=òÉêçW= { }KIPIOINkw ==+ I=

{ }NIOIPIw −−−=− K I= { } { }KK IPIOINIMINIOIPIwMww −−−=∪∪= +− K=

= 29. o~íáçå~ä=kìãÄÉêë

oÉéÉ~íáåÖ=çê=íÉêãáå~íáåÖ=ÇÉÅáã~äëW==

  

   ≠∈∈== MÄ~åÇwÄ~åÇw~~åÇ

Ä

~ ñöñn K=

= 30. fêê~íáçå~ä=kìãÄÉêë

kçåêÉéÉ~íáåÖ=~åÇ=åçåíÉêãáå~íáåÖ=ÇÉÅáã~äëK =

CHAPTER 1. NUMBER SETS

6

31. oÉ~ä=kìãÄÉêë== råáçå=çÑ=ê~íáçå~ä=~åÇ=áêê~íáçå~ä=åìãÄÉêëW=oK=

= 32. `çãéäÉñ=kìãÄÉêë

{ }oó~åÇoñöáóñ` ∈∈+= I== ïÜÉêÉ=á=áë=íÜÉ=áã~Öáå~êó=ìåáíK

= 33. `onwk ⊂⊂⊂⊂ =

=

=== = =

Figure 5. = = = = = =

CHAPTER 1. NUMBER SETS

7

1.3 Basic Identities = oÉ~ä=åìãÄÉêëW=~I=ÄI=Å=

= =

34. ^ÇÇáíáîÉ=fÇÉåíáíó= ~M~ =+ =

= 35. ^ÇÇáíáîÉ=fåîÉêëÉ=

( ) M~~ =−+ = =

36. `çããìí~íáîÉ=çÑ=^ÇÇáíáçå= ~ÄÄ~ +=+ =

= 37. ^ëëçÅá~íáîÉ=çÑ=^ÇÇáíáçå=

( ) ( )ÅÄ~ÅÄ~ ++=++ = = 38. aÉÑáåáíáçå=çÑ=pìÄíê~Åíáçå=

( )Ä~Ä~ −+=− = =

39. jìäíáéäáÅ~íáîÉ=fÇÉåíáíó= ~N~ =⋅ =

= 40. jìäíáéäáÅ~íáîÉ=fåîÉêëÉ=

N ~

N ~ =⋅ I= M~ ≠

= 41. jìäíáéäáÅ~íáçå=qáãÉë=M

MM~ =⋅ =

42. `çããìí~íáîÉ=çÑ=jìäíáéäáÅ~íáçå= ~ÄÄ~ ⋅=⋅

= =

CHAPTER 1. NUMBER SETS

8

43. ^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå= ( ) ( )ÅÄ~ÅÄ~ ⋅⋅=⋅⋅

= 44. aáëíêáÄìíáîÉ=i~ï=

( ) ~Å~ÄÅÄ~ +=+ = = 45. aÉÑáåáíáçå=çÑ=aáîáëáçå=

Ä

N ~

Ä

~ ⋅= =

= = =

1.4 Complex Numbers = k~íìê~ä=åìãÄÉêW=å= fã~Öáå~êó=ìåáíW=á= `çãéäÉñ=åìãÄÉêW=ò= oÉ~ä=é~êíW=~I=Å= fã~Öáå~êó=é~êíW=ÄáI=Çá= jçÇìäìë=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=êI= Nê I= Oê = ^êÖìãÉåí=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=ϕ I= Nϕ I= Oϕ =

= =

ááN = = ááR = = áá NåQ =+ = NáO −= = NáS −= = Ná OåQ −=+ = ááP −= = ááT −= = áá PåQ −=+ =

46.

NáQ = = NáU = = Ná åQ = = = 47. Äá~ò += = = 48. `çãéäÉñ=mä~åÉ=

=

CHAPTER 1. NUMBER SETS

9

===== = =

Figure 6. =

49. ( ) ( ) ( ) ( )áÇÄÅ~ÇáÅÄá~ +++=+++ = = 50. ( ) ( ) ( ) ( )áÇÄÅ~ÇáÅÄá~ −+−=+−+ = = 51. ( )( ) ( ) ( )áÄÅ~ÇÄÇ~ÅÇáÅÄá~ ++−=++ = =

52. á ÇÅ

~ÇÄÅ

ÇÅ

ÄÇ~Å

ÇáÅ

Äá~ OOOO ⋅

+ −

+ + +

= + +

=

= 53. `çåàìÖ~íÉ=`çãéäÉñ=kìãÄÉêë=

Äá~Äá~ |||||||

−=+ = = 54. ϕ= Åçëê~ I= ϕ= ëáåêÄ ==

=

CHAPTER 1. NUMBER SETS

10

= =

Figure 7. =

55. mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë= ( )ϕ+ϕ=+ ëáåáÅçëêÄá~ =

= 56. jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê=

fÑ= Äá~ + =áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå= OO Ä~ê += =EãçÇìäìëFI==

~

Ä ~êÅí~å=ϕ =E~êÖìãÉåíFK=

= 57. mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

( ) ( )OOONNNON ëáåáÅçëêëáåáÅçëêòò ϕ+ϕ⋅ϕ+ϕ=⋅ = ( ) ( )[ ]ONONON ëáåáÅçëêê ϕ+ϕ+ϕ+ϕ= =

= 58. `çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

( ) ( ) ( )[ ]ϕ−+ϕ−=ϕ+ϕ ëáåáÅçëêëáåáÅçëê |||||||||||||||||||||

= = 59. fåîÉêëÉ=çÑ=~=`çãéäÉñ=kìãÄÉê=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

( ) ( ) ( )[ ]ϕ−+ϕ−=ϕ+ϕ ëáåáÅçëê N

ëáåáÅçëê

N =

CHAPTER 1. NUMBER SETS

11

60. nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ( ) ( ) ( ) ( )[ ]ONONO

N

OOO

NNN

O

N ëáåáÅçë ê

ê

ëáåáÅçëê

ëáåáÅçëê

ò

ò ϕ−ϕ+ϕ−ϕ=

ϕ+ϕ ϕ+ϕ

= =

= 61. mçïÉê=çÑ=~=`çãéäÉñ=kìãÄÉê=

( )[ ] ( ) ( )[ ]ϕ+ϕ=ϕ+ϕ= åëáåáåÅçëêëáåáÅçëêò ååå = = 62. cçêãìä~=±aÉ=jçáîêÉ≤=

( ) ( ) ( )ϕ+ϕ=ϕ+ϕ åëáåáåÅçëëáåáÅçë å = = 63. kíÜ=oççí=çÑ=~=`çãéäÉñ=kìãÄÉê=

( )   

   π+ϕ+

π+ϕ =ϕ+ϕ=

å

âO ëáåá

å

âO ÅçëêëáåáÅçëêò ååå I==

ïÜÉêÉ== NåIIOINIMâ −= K K==

= 64. bìäÉê∞ë=cçêãìä~=

ñëáåáñÅçëÉáñ += = = =

12

Chapter 2

Algebra = = = =

2.1 Factoring Formulas = oÉ~ä=åìãÄÉêëW=~I=ÄI=Å== k~íìê~ä=åìãÄÉêW=å=

= =

65. ( )( )Ä~Ä~Ä~ OO −+=− = =

66. ( )( )OOPP Ä~Ä~Ä~Ä~ ++−=− = = 67. ( )( )OOPP Ä~Ä~Ä~Ä~ +−+=+ = = 68. ( )( ) ( )( )( )OOOOOOQQ Ä~Ä~Ä~Ä~Ä~Ä~ ++−=+−=− = =

69. ( )( )QPOOPQRR Ä~ÄÄ~Ä~~Ä~Ä~ ++++−=− = = 70. ( )( )QPOOPQRR Ä~ÄÄ~Ä~~Ä~Ä~ +−+−+=+ = = 71. fÑ=å=áë=çÇÇI=íÜÉå=

( )( )NåOåOPåOåNååå Ä~ÄÄ~Ä~~Ä~Ä~ −−−−− +−−+−+=+ K K== = 72. fÑ=å=áë=ÉîÉåI=íÜÉå==

( )( )NåOåOPåOåNååå Ä~ÄÄ~Ä~~Ä~Ä~ −−−−− +++++−=− K I==

CHAPTER 2. ALGEBRA

13

( )( )NåOåOPåOåNååå Ä~ÄÄ~Ä~~Ä~Ä~ −−−−− −+−+−+=+ K K= = = =

2.2 Product Formulas oÉ~ä=åìãÄÉêëW=~I=ÄI=Å== tÜçäÉ=åìãÄÉêëW=åI=â= = =

73. ( ) OOO Ä~ÄO~Ä~ +−=− = =

74. ( ) OOO Ä~ÄO~Ä~ ++=+ = =

75. ( ) POOPP Ä~ÄPÄ~P~Ä~ −+−=− = =

76. ( ) POOPP Ä~ÄPÄ~P~Ä~ +++=+ = =

77. ( ) QPOOPQQ Ä~ÄQÄ~SÄ~Q~Ä~ +−+−=− = =

78. ( ) QPOOPQQ Ä~ÄQÄ~SÄ~Q~Ä~ ++++=+ = = 79. _áåçãá~ä=cçêãìä~=

( ) IÄ`~Ä`Ä~`Ä~`~`Ä~ åååNåNååOOåOåNåNååMåå +++++=+ −−−− K

ïÜÉêÉ= ( )>âå>â >å

`â å

− = =~êÉ=íÜÉ=Äáåçãá~ä=ÅçÉÑÑáÅáÉåíëK=

=

80. ( ) ÄÅO~ÅO~ÄOÅÄ~ÅÄ~ OOOO +++++=++ = =

81. ( ) ++++++=+++++ OOOOOO îìÅÄ~îìÅÄ~ KK = ( )ìîÄîÄìÄÅ~î~ì~Å~ÄO +++++++++++ KKK =

CHAPTER 2. ALGEBRA

14

2.3 Powers = _~ëÉë=EéçëáíáîÉ=êÉ~ä=åìãÄÉêëFW=~I=Ä== mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=

= =

82. åãåã ~~~ += = =

83. åã å

ã

~ ~

~ −= =

=

84. ( ) ããã Ä~~Ä = = =

85. ã

ãã

Ä

~

Ä

~ =

 

  

=

=

86. ( ) ãååã ~~ = = = 87. N~M = I= M~ ≠ = = 88. N~N = = =

89. ã

ã

~

N ~ =− =

=

90. å ãå ã

~~ = = = = = = =

CHAPTER 2. ALGEBRA

15

2.4 Roots = _~ëÉëW=~I=Ä== mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=

MÄI~ ≥ =Ñçê=ÉîÉå=êççíë=E âOå = I= kâ∈ F= = =

91. ååå Ä~~Ä = = =

92. åã åããå Ä~Ä~ = = =

93. å

å

å

Ä

~

Ä

~ = I= MÄ ≠ =

=

94. åã å

ã

åã å

åã ã

ã

å

Ä

~

Ä

~

Ä

~ == I= MÄ ≠ K=

=

95. ( ) å ãééå ã ~~ = = =

96. ( ) ~~ åå = = =

97. åé ãéå ã ~~ = = =

98. å ã

å ã ~~ = = =

99. ãåã å ~~ = = =

100. ( ) å ããå ~~ = = =

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