ມ.2_ຄະນິດສາດ_Math

d=]toumu @ • g,af M [+g8a’da[g,af OD M O M' { 0ufg7yj’glAo-nj [MO) { 0ufmvj odq’j mu,j ug,af.9dk’16j O c]t ]aflt\\u OM. { mvj odqj’8af [MO) 16jg,af M’. M’ c,oj g,afg7y’j 7n0v’ M mP[.lj O. 8q;1jk’G lhk’I6[g7y’j 7n A’B’C’D’ A C 0v’I6[ ABCD mP[.ljg,af O. B ;ymulhk’G { lhk’g7y’j glAo-nj [AO), [BO), [CO) c]t [DO). D O D' { 0ufmjvodqj’mu,j ug,af.9dk’ O c]t ]aflt\\u OA, OB, OC, OD 8k,]efa[D ສະຫງສ.ວວນ.ລິສຂະສິດ { [aofkmjvodq’j 8af [OA), [OB), B' [OC) c]t [OD) 16gj ,af A' , B', C' C' A' c]t D' 8k,]efa[D { 0uf8+g,af A' sk B'F B' sk C', C' sk D' c]t D' sk A' . gIqkwfhI6[ A'B'C'D' c,joI6[g7y’j 7nda[ ABCD mP[.ljg,af O muj8hv’dkolhk’D #D [qfg/ydsaf C !D 9q’j lhk’I6[ B' c]t B''g7yj’7n d1 B B da[I6[ B XI6[ !? mP[.ljglAo (I6[1) A D d2 -nj d1c]t d2 8k,]efa[D (I6[2) @D 9qj’lhk’g,afg7y’j 7nda[g,af A, B c]t C mP[.ljg,af D muj .shdvj o XI6[ @?D I 45

#D .shI6[lk,c9 ABC c]t g,af I D $D 9qj’lhk’I6[g7yj’7nda[I6[ A mP[da[g,af O muj.shdjvo 9qj’lhk’I[6lk,c9 A'B'C'g7y’j 7nda[ I6[lk,c9 ABC mP[.lgj ,af I D A AOສະຫງສ.ວວນ.ລິສຂະສິດ B I C &D .shI6[lk,c9 ABC 8k,.9D 9qj’lhk’g,af I c]t J g7y’j 7nda[g,af A mP[.ljg,af B c]t C 8k,]efa[D 9qj’xP[mP[];’pk;0v’mjvo-nj [ IJ ] c]t [ BC ] fh;pdko;afcmdD *D .shI6[lk,c9 ABC 8k,.9D M c,oj g,af|bj’mu1j 6j.oI6[lk,c9 ABC. dD 9q’j lhk’G { g,af I g7y’j 7nda[ M mP[.lgj ,af A. { g,af J g7y’j 7nda[ M mP[.lgj ,af B { g,af K g7y’j 7nda[ M mP[.ljg,af C. 0D 9qj’xP[mP[];’Iv[0v’I6[lk,c9 IJK c]t ];’Iv[0v’I6[lk,c9 ABC fh;pdko;afcmdD (D lv’glAo-nj p c]t q 8A’lkddao16j O. A c,jog,af|b’j 8k,.9[+1j6 p c]t q D dD 9qj’lhk’G { g,af A' g7y’j 7nda[g,af A mP[.lgj lAo-nj q. { g,af A\" g7yj’7nda[ A' mP[.ljglAo-nj p. 0D 9q’j -Uc9’;jk g,af A'' c]t A g7yj’7ndaomP[.lgj ,af O. 46

[qfmu !# dkog7yj’7nmP[.ljg,af|b’j X8+? !D dyf9tde dyf9tdemu ! .shglAo-nj d F g,af O, A c]t B Bd O g-yj’;jkg,af A c]t B 16glAo-nj d. { 9q’j lhk’g,afg7yj’7n A' c]t B' A 0v’ A c]t B mP[.lgj ,af O. { 9q’j 0ufglAo-nj d' zkj og,af A' c]t B'. { la’gdfgsaoco;.fdPj ;da[lv’glAo-nj d c]t d 'D { I6[lujc9 ABA'B' gxaoI6[lucj 9-toyf.fL dyf9tdemu @ i .shmvj o-nj [AB] c]t g,af O. B ສະຫງສ.ວວນ.ລິສຂະສິດ { 9q’j lhk’g,af A' c]t B' g7yj’7nda[ M g,af A c]t B mP[.lgj ,af O. A O { 9q’j -Uc9’;kj mvj o-nj [AB] c]t [A'B'] 0t|kodao c]t gmqjkdaoD i .sh M c,oj g,afg7y’j dk’0v’mjvo-nj [AB]. { 9qj’lhk’g,afg7y’j 7n M' 0v’ M mP[.ljg,af O c]h;d;f7nog[y’j ;kj g,af M' 16jmvj o-nj A'B' s^n [+L { 9qj’xP[mP[];’pk;0v’ MA da[ M'A'F MB da[ M'B' c]t M'A' da[ M'B' D dyf9tdemu # gryjog;Qk;kj I6[|bj’,ul6odk’g7yj’7n s^n cdog7y’j 7n g,njvI6[oAog8a’da[I6[g7yj’7n0v’ ,aogv’mP[.llj 6odk’g7y’j 7n s^n cdog7yj’7n0v’,aoD { .o[aofkI6[]5j,oU 9q’j [vd9eo;ocdog7y’j 7n c]t l6odk’g7y’j 7n0v’,ao X4hk,u?D 47

#D .97;k, !D dkog7y’j 7n0v’glAo-njmP[.lgj ,af|b’j D d 4hklv’glAo-ngj 7y’j 7ndaomP[.lgj ,af|b’j B c,oj lv’glAo-njoAo 0t|kodaoD d' { glAo-nj d c]t d' g7y’j 7ndaomP[.lgj ,af A O A' O D gIqkwfh d // d'D 8q;1jk’G lhk’glAo-nj k ' g7yj’7nda[ k B' mP[.lgj ,af O muj.shdjvoD ;ymulhk’G A { ;k’lv’g,af A c]t B 8k,.916jglAo-nj k. Bk { lhk’g7yj’glAo-nj [AO) c]t [BO) D O { 0uflv’mjvodq’j muj,ug,af.9dk’ O c]t ]aflt\\u AO c]t BO 8k,]efa[D B' A' k ' { [aofkmjvodq’j 8af [AO) c]t [BO) 16gj ,af A' c]t B'D ສະຫງສ.ວວນ.ລິສຂະສິດ { 0ufglAo-njzjko A' c]t B' 9twfhglAo-nj k ' mugj 7yj’7nda[glAo-nj k mP[.lgj ,af OD @D dkog7yj’7n0v’mvj o-nj c]t g,afg7yj’dk’ M B A' 0v’,aomP[.ljg,af|bj’D A O lv’mjvo-ngj 7yj’7ndaomP[.ljg,af|bj’ r;d,ao 0t|kodao c]t ,u];’pk;gmqjkdaoD M' { lv’mvj o-nj [AB] c]t [A'B'] g7yj’7ndaomP[.lj B' g,af O D gIqkwfh [AB]//[A'B'] c]t [AB] = [A'B']D { M c,oj g,afg7yj’dk’0v’mvj o-nj [AB] gIqkwfh B A' M' c,jog,afg7yj’dk’0v’mvj o-nj [A'B'] D M { I6[lujc9 ABA'B' c,oj I6[lucj 90hk’0t|koD O 8q;1jk’G lhk’mvj o-nj [M'N'] g7y’j 7nda[mjvo-nj A B' [MN] mP[.ljg,af P muj.shdvj oD ;ymulhk’G 48

{ 0ufg7yj’glAo-nj [MP) c]t [NP) D N M' { 0uflv’mjvodqj’muj,ug,af.9dk’16j P P fh;p]aflt\\u MP c]t NP 8k,]efa[D M N' { lv’mvj odqj’8af [MP) c]t [NP) 16jg,af M' c]t N' { 0uf8+ M' sk N' gIqkwfhmjvo-nj [M'N'] g7yj’7nda[mjvo-nj [MN] mP[.ljg,af P D #D l6odk’g7yj’7n0v’I6[|bj’D gryoj g;Qk;jkG <I6[\\bj’,ul6odk’g7yj’7n< d=8+g,njvc8j]tg,af0v’I6[,ug,afg7yj’7n16j.oI6[ oAogv’D 8q;1kj ’G I6[ 9eo;ocdog7yj’7n 0 3 4 ສະຫງສ.ວວນ.ລິສຂະສິດ0 9eo;ol6odk’g7y’j 7n 0 1 1 1 #D [qfg/ydsaf !D 9qj’lhk’glAo-nj d' g7yj’7nda[ d mP[.lgj ,af OD d O @D 9qj’lhk’glAo-nj g7yj’7nda[glAo-nj d1 c]t d2 D d1 dD mP[.lgj ,af A D B A d2 0D mP[.lgj ,af B D #D 9qj’lhk’g7yj’glAo-nj [A'X') g7yj’7nda[g7yj’glAo-nj [AX) mP[.ljg,af O .oc8]j td=]tou8+wxoUD 49

dD O X 0D XO A A X 7D O ’D ’D A X OA $D 9q’j -vdskl6odk’g7y’j 7n0v’I6[]5j,oUD la’gdfgsaoco;.fL dD 0D 7D &D 9qj’c8h,I6[g7y’j 7n0v’I6[]5j,oUmP[.lgj ,af OD 9D ສະຫງສ.ວວນ.ລິສຂະສິດlD dD 0D 7D ’D •O O• O• •O O• O• 7D *D 9q’j [vd9eo;ocdog7y’j 7n c]t l6odk’g7yj’7n0v’[aofkI6[]5j,oUD dD 0D 50

[qfmu !$ dkog7y’j 7n0v’,5,mP[.ljcdo c]t mP[.lgj ,af !D dyf9tde dyf9tdemu ! .sh,5, xAy F glAo-nj Δ c]t g,af O D y O dD x y' { 9q’j lhk’g,af A' g7y’j 7nda[ A mP[.lgj lAo-nj Δ D y { 9q’j lhk’g7yj’glAo-nj [A'x') c]t [A'y') g7yj’7nda[ [Ax) c]t [Ay) mP[.lgj lAo-nj Δ D A { 9q’j xP[mP[,5, xAy da[ x'A'y' fh;pdko;afcmdD Δ 0D { 9qj’lhk’g,af A'' g7y’j 7nda[ A mP[.ljg,af O D { 9q’j lhk’g7yj’glAo-nj [A\"x\") c]t [A''y\") g7y’j 7nda[[Ax) c]t [Ay) mP[.lgj ,af O D ສະຫງສ.ວວນ.ລິສຂະສິດ { 9q’j xP[mP[7kj ;afcmd0v’,5, xAy da[ x''A''y'' fh;pdko;afcmdD dyf9tdemu @ x .shlv’glAo-nj (xy) c]t (x'y') 8afdao1j6g,af O. O { 3fpmP[.llj 6odk’g7yj’7n O F 9qj’deoqfg7yj’glAo -njmujg7y’j 7nda[ [Ox) F [Ox') c]t ,5,mugj 7yj’7nda[ xOx'D x' { 9q’j xP[mP[7jk;afcmd0v’,5, x'Ox da[ y'Oy c]t x'Oy da[ xOy' fh;pdko;afcmdD { 4hkgryjogvUo,5, xOx' c]t y'Oy ;kj lv’,5,0hk,9v,D .o[aofk,5,]5j,oU ,u,5,.fc,oj ,5, 0hk,9v,D (1) (2) (3) (4) 51

dyf9tdemu # v .shglAo-nj (uv) 8aflv’glAo-n0j t|kodao Ay (xy) c]t (x'y') 16lj v’g,af A c]t B D x { 9qj’;k’g,af I 16jmvj o-nj AB 3fp;jk glAo-nj (x'y') g7yj’7nda[ (xy) mP[.lj I. x' B y' { 3fpmP[.llj 6odk’g7y’j 7n I F 9q’j deoqf u g,afg7yj’7n0v’ A F g7yj’glAo-njmujg7yj’7nda[[Ax) c]t ,5,muj g7yj’7nda[ xAB. { 9q’j -Uc9’;kj xAB = ABy' { 9qj’[vd[aofk,5,mugj 7yj’7ndaomP[.ljg,af I. { 9qj’-Uc9’;jk ABy' = x'Bu; xAB = x'Bu; c]t vAy = ABy' D i 4hklv’glAo-nj0t|kodao4nd8affh;pglAo-n|j b’j 9txtdv[gxaos^kp,5,gmqjkdao8k, c8]j txtgrffa’j oUG ,5,lts^a[.o ,5,-hvo0t\\koສະຫງສ.ວວນ.ລິສຂະສິດ d d'' d'' 12 41 d 43 32 12 12 d' d' 4 3 43 d d d'' d'' 12 12 43 4 3 d' 1 2 12 d' 3 3 4 4 { 9q’j oe.-h75o]adltot0hk’gmy’oUgrnjv0Po[vd[aofk,5,lts^a[.o c]t ,5,-hvo0t \\komugj mqkj dao ma’\\qf.odyf9tdemu #D 52

@D .97;k, !D dkog7y’j 7n0v’,5,mP[.ljglAo-n|j b’j c]t g,af|bj’D x A x A y A' y Δ O y' A' y' x' x' dkog7yj’7n0v’,5,mP[.lgj lAo-nj dkog7y’j 7n0v’,5,mP[.lgj ,af i lv’,5,g7yj’7ndaomP[.lgj lAo-nj|b’j sn^ g,af|bj’gmqjkdaoD 8q;1jk’G ,5, xAy c]t x'A'y' g7y’j 7ndaomP[.ljglAo-nj Δ s^n mP[.ljg,af O. gIqkwfh xAy = x'A'y' ສະຫງສ.ວວນ.ລິສຂະສິດ @D ;ymulhk’,5,g7yj’7nD dD mP[.ljglAo-nj|b’j D 8q;1jk’G lhk’,5, A'O'B' g7y’j 7nda[ AOB mP[.lgj lAo-nj d mu.j shdjvoD ;ymulhk’G d { 0ufg7yj’glAo-nj [Ox) F [Ay) c]t [Bz) AB 8A’lkdda[ d 16gj ,af I F L c]t M 8k,]efa[D { 0ufmvj odqj’mu,j ug,af.9dk’16j I, L c]t M muj M B' ,u]aflt\\u OI, AL c]t BM F mvj odq’j 8af OL z [Ox), [Ay) c]t [Bz) 16gj ,af O', A' c]t B'D I { 0uf8+ B' sk O' c]t O' sk A' 9twfh,5, O' A' y A'O'B' g7y’j 7nda[,5, AOB mP[.lgj lAo-nj d D x 53

0D mP[.lgj ,af|bj’ 8q;1jk’G lhk’,5, x'O'y' g7y’j 7nda[ xOy mP[.lgj ,af R. ;ymulhk’G { lhk’g7y’j glAo-nj [OR), [xR) c]t [yR). xy { 0uflk,mjvodq’j muj,ug,af.9dk’16j R muj,u]aflt\\u OR O' OR, xR c]t yR. [aofkmvj odqj’8af [OR), [xR) c]t [yR) 16g,af O', x' c]t y' D { 0uf8+ O' sk x' c]t O' sk y' 9twfh,5, y' x' x'O'y' g7y’j 7nda[,5, xOy mP[.lgj ,af R. #D ,5,0v’lv’glAo-nj8afdaoD dD oypk,G g,nvj lv’glAo-nj (xy) c]t (x'y') x y ' 8afdao16gj ,af O F ,5, xOx' c]t yOy' O gvUo;jk <,5,0hk,9v,<D x ' y 0D 75o]adltotD 7jk;afcmd0v’lv’,5,0hk,9v,gmqkj daoD ສະຫງສ.ວວນ.ລິສຂະສິດ 8q;1kj ’G xOx' c]t yOy' c,jo,5,0hk,9v, gIqkwfh xOx' = yOy' D xOy' c]t x'Oy c,jo,5,0hk,9v, gIqkwfh xOy' = x'Oy $D ,5,mujwfh9kddko8afdao0v’glAo-n|j b’j da[lv’glAo-nj0t|kodao lv’glAo-nj0t|kodao8afda[glAo-n|j n’j xtdv[gxao,5,faj’oUG dD ,5,lts^a[.oD z i oypk,G lv’,5,mujwfh9kddko8afdao0v’ glAo-n|j n’j 8afda[lv’glAo-nj0t|kodaogvuho;kj G x A y ,5,lts^a[.o g,njvG { r;d,ao[+8yfcxtdaoD x' B y ' { r;d,ao[+16jrkdfP;dao0v’glAo8afD { r;d,ao16jrkp.og0fmudj eoqffh;plv’ z ' glaho-noj AoD 54

8q;1jk’G xAz' c]t zBy' c,oj ,5,lts^a[.oD z i 75o]adltotD 4hklv’glAo-n0j t|kodao,5,lts^a[.og-y’j xt x A x ' dv[fh;pglAo8afoAo8hv’gmqjkdaoD 8q;1jk’G xAz' = zBy ' c]t x'Az' = zBy i 75o]adltotxUo7noD y y ' 4hklv’glAo-nxj tdv[da[glAo8afgxao,5,lts^a[ B .ogmqjkdao lv’glAo-njoAo8hv’0t|kodaoD z' 8q;1jk’G 4hk;kj x'Az' = zBy gIqkwfh (x'x) // (y'y) 0D ,5,-hvo0t|koD x i oypk,G lv’,5,muwj fh9kddko8afdao0v’ z glAo-n|j bj’da[lv’glAo-nj0t|kodaogvuho;jkG A ,5,-hvo0t|ko g,njvG ສະຫງສ.ວວນ.ລິສຂະສິດ y { r;d,ao[+8yfcxtdaoD { r;d,ao16jrkdfP;dao0v’glAo8afD x' { ,5,|b’j 16rj kp.o c]t ,5,|bj’16jrkpovd B 0v’g0fmujdeoqffh;plv’glAo-nj0t|kooAoD y' z' 8q;1jk’G zAx c]t zBy c,jo,5,-hvo0t|koD i 75o]adltotD z 4hklv’glAo-n0j t|kodao ,5,-hvo0t|kog-y’j xtdv[fh;pglAo8afoAo8hv’gmqkj daoD x A x ' 8q;1jk’G zAx = zBy c]t xAz' = yBz' i 75o]adltotxUo7noD y y ' 4hklv’glAo-njxtdv[da[glAo8afgxao,5,-hvo0t B |kogmqjkdao lv’glAo-noj Ao8hv’0t|kodaoD z ' 8q;1jk’G 4hk;kj xAz' = yBz' gIqkwfh (x'x) // (y'y) 55

#D [qdg/ydsaf Δ A !D 9q’j lhk’,5,g7yj’7n0v’,5, xAy ]5,j oUmP[.lcj do Δ D Δ A x xy y X0? Xd? Δ Δ x x A ສະຫງສ.ວວນ.ລິສຂະສິດAy y X7? X’? @D 9qj’lhk’,5,g7yj’7n0v’,5, AOB ]5,j oUmP[.lgj ,af P. A A PO P P B B B A X0? Xd? X7? 56

#D 9qj’lhk’glAo-njg7yj’7nda[ d1 c]t d2 mP[.lgj ,af I .oc8j]td=]tou]5,j oU D d2 d1 d2 d1 O I O I (1) (2) $D 9q’j 0Po,5,0hk,9v,c8j]t76mj ujltcf’16jI6[]5j,oUG yA z x' z x B x A x' (1) C yB y' z' y' z' (2) &D .shglAo-nj d c]t d' 8afdao16gj ,af O Xfa’j I6[?D ສະຫງສ.ວວນ.ລິສຂະສິດ 9qj’-vdsk7kj ;afcmd0v’,5, 1; 2 c]t 3D 2 1 55 3 d d' *D 9q’j [vd7kj 0v’,5, a, b, c, d, e, f c]t g 16Ij 6[]5,j oUD 60 a 120 a cb cb gd gd fe fe (1) (2) 57

(D .shlk,glAo-nj d1; d2 c]t d3 4nd8affh;plv’glAo-nj d4 c]t d5 Xfa’j I6[?D 9qj’-vdsk 7jk;afcmd0v’,5, 1; 2; 3; 4; 5 c]t 6 D d1 56 d2 110 60 4 d3 1 32 d4 d5 ສະຫງສ.ວວນ.ລິສຂະສິດ )D gryoj .shI6[lk,c9mjP’ ABC(AB = AC). dD 9q’j -vdsk7jk0v’[aofk,5,.o0v’I6[lk,c9fa’j dkj ;I6h;kj ,5, B gmqjklv’ gmnjv,5, A D 0D 9q’j 0ufglAoc[’j g7yj’,5,0v’,5,8jk’M8+wxoUG [BD] le]a[,5, ABC, [DE] le]a[,5, BDC, [EF] le]a[,5, DEC c]t [FG] le]a[,5, EFC D 7D 9q’j -vdsk7kj 0v’,5, BDA c]t BDC D ’D .oI6[lk,c9mPj ’ ABC ,uI6[lk,c9mjP’vnoj 9adI6[L glAo-nj0t|koda[ AB ,u9adglAo-nLj da[ BD ,u9adglAo-njL 9qj’.shgsfzqorhv,D 58

rkdmu # g]dde]a’ c]t leo;o [qfmu !& g]dde]a’ !D dyf9tde dyf9tdemu ! dD 9q’j 7yfw]j x 7 + 7 + 7 = .........; 3× 7 = ........... ltcf’;kj G 7 + 7 + 7.....3× 7 i 6, 2 + 6, 2 + 6, 2 + 6, 2 = 4×..... = ..... i 2 + 2 + 2 + 2 + 2 = 5×..... = ..... 3 3 3 3 3 0D 9q’j la’gdfg[yj’8q;1kj ’c]h;8n,j 9eo;omujg\\ktlq,.lj[jvo9Eg,afD 8q;1kj ’G 4× 4× 4 = 64 = 43 i 3× 3× 3× 3 = ...... = 3.... i(−5)×(−5)×(−5) = ...... = (.....)3 ສະຫງສ.ວວນ.ລິສຂະສິດ i(0, 2)× (0, 2) = ...... = ...... i 4 × 4 × 4 = ...... = ...... 5 5 5 dyf9tdemu 2 gryjo.sh9eo;o a,b 8jk’l6o c]t 9eo;o4h;o[;d m, n dD 9q’j 0Po8nj,.l8k8t]k’]5,j oU8k,8q;1jk’.oc4;muj|bj’D m n am an am × an m + n am+n 2 1 a2 a1 a2 × a1 3 a3 32 22 Š 9kd8k8t]k’F gIqkwfhG am × an = a.....+..... 59

0D 9qj’0Po8n,j .l8j k8t]k’]5,j oU8k,8q;1jk’.oc4;mu|j n’j D m n am an am m − n am−n an 5 2 a5 a2 a5 3 a3 a2 41 33 Š 9kd8k8t]k’ gIqkwfhG am = a.......−....... X.og’nvj ow0 m > n c]t a ≠ 0 ? an Š 4hk m = n c]t a ≠ 0 F gIqkwfhG am am an = am = am−m = a...... vudfhko|bj’ am =1 faj’oAo gIqkwfhG a0 = ........ ສະຫງສ.ວວນ.ລິສຂະສິດ am 7D 3fpvu’.lzj qowfhIa[9kd0=h Xd? c]t X0?D 9qj’0Po8n,j .l[j vj o9Eg,afD ( )i a4 2 = a4 × a4 = a......+...... = a4 ×...... ( )i a3 4 = a3 ×.....×.....×..... = a....+....+....+.... = a3 ×..... Š faj’oAo gIqkwfhG (am )n = a....×.... i ⎛ a ⎞4 = a × ..... × ..... × ..... ⎜⎝ b ⎟⎠ b = a × .... × ....× a b × .... × ....× b a.... = b.... 60

faj’oAo gIqk9twfhG ⎛ a ⎞n = an ⎜⎝ b ⎟⎠ bn i (ab)3 = ab ×.....×..... = a×a×a×b×b×b = a...b... faj’oAogIqk9twfhG (ab)n = a...b... @D .97;k, !D oypk,G .sh9eo;o a 8kj ’l6o c]t 9eo;o4h;o[;d n D an = a × a × a ×.....× a vjko;jk a 0Node]a’ n a gvUo;kj rNo c]t n gvUo;kj 8q;de]a’D n gmnvj 8q;1kj ’ !D 43 = 4× 4× 4 = 64 Xvjko;jk $ 0Node]a’ # gmqkj *$?ສະຫງສ.ວວນ.ລິສຂະສິດ @D (−5)3 = (−5)× (−5)× (−5) = −125 Xvjko;jk {& 0Node]a’ # gmqjkda[ {!@&? @D 75o]adltotD le]a[m5dM9eo;o a c]t b 8kj ’l6oF m c]t n c,jo9eo;o4h;oleraf gIqk9twfhG Š zqo76o0v’g]dde]a’ am × an = am+n 8q;1jk’ !D 52 ×53 = 52+3 @D 23 × 24 × 2−5 = 23+4+(−5) = 55 = 22 =4 = 3125 #D (xy)9 (xy)10 (xy)11 = (xy)9+10+11 #D x5 × x6 × x7 = x5+6+7 = (xy)30 = x18 4hk m > n, Š zqosko0v’g]dde]a’ g,njv a ≠ 0 gIqkwfhG am = am−n an 61

am 1 4hk m < n, an = an−m am = am = a0 =1 4hk m = n. an am 8q;1kj ’ !D 45 = 45−3 @D 73 × 85 × 97 97 85 73 43 96 × 85 × 74 = 96 × 85 × 74 = 42 = 97−6 ×1× 1 74−3 = 16 = 9 × 1 7 = 9 7 X dko76og]dl;j o a × c = a×c ? b d b×d Š de]a’0v’de]a’ ສະຫງສ.ວວນ.ລິສຂະສິດ (am )n = am×n 8q;1kj ’ !D (22)2 = 22×2 @D ((−3)2 )3 = (−3)2×3 = 24 = (−3)6 = 16 = 729 #D (a4 )5 = a4×5 $D (xa+1)2 × (x2 )1−a = x(a+1)×2 × x2×(1−a) = a20 = x2a+2 × x2−2x Š de]a’0v’zqo76o = x(2a+2)+(2−2a) (a × b)n = an ×bn = x2a+2+2−2a = x4 62

8q;1jk’ !D (4× 5)2 = 42 × 52 @D [(−2)4 ×53]2 = (−2)4×2 × 53×2 = 16× 25 = (−2)8 ×56 = 400 = 256×3125 #D (−4b)3 = (−4)3b3 = 800000 $D (2ab2c3)4 = 24 a4b2×4c3×4 = −64b3 = 16a4b8c12 Š de]a’0v’zqosko ⎛ a ⎞n = an g-yj’;jk b ≠ 0 ⎝⎜ b ⎠⎟ bn 8q;1jk’ !D ⎛ 2 ⎞2 = 22 @D ⎛ −3n3 ⎞3 = (−3)3 (n3 )3 ⎝⎜ 5 ⎟⎠ 52 ⎜⎝⎜ 4 ⎠⎟⎟ 43 = 4 = −27n3×3 25 64 Š ly’j mu7j ;ogvqk.9.ljD 27n9 ສະຫງສ.ວວນ.ລິສຂະສິດ = − 64 i a1 = a i a−1 = 1 . a−1 vkj o;kj 9eo;oxUo0v’ a a i a−n = 1 . a−n vjko;jk9eo;oxUo0v’ an an i 0n = 0, n ≠ 0 vkj o;kj l6o0Node]a’ n 8q;1kj ’G i (−5)0 = 1; (−0,38)0 = 1; 270 = 1; (π )0 = 1 i 81 = 8; (−0, 25)1 = −0, 25; x1 = x; (−13)1 = −13 i 5−1 = 1 ; ⎛ a ⎞−1 = b 5 ⎝⎜ b ⎟⎠ a i 7−2 = 1 = 1; 3−4 = 1 = 1 72 49 34 81 i 9eo;oxUo0v’ 5 c,jo 1 s^n 5−1S 9eo;oxUo0v’ a c,jo b s^n ⎛ a ⎞−1 D 5 b a ⎝⎜ b ⎟⎠ 63

#D [qfg/ydsafD !D 9qj’0Pozqo76o]5j,oUgxaog]de]a’D dD 2× 2× 2× 2× 2 0D (−3) × (−3) × a × a × b × b 7D (x + y)(x + y)(x + y)(x + y) ’D (− 3 )(− 3 )(− m )(− m )(−t )(−t ) @D 9qj’7eo;og]d]5j,oUD 4 4 n n dD 22 × 33 × 44 0D (−3)3 × (−32 )2 × (−52 ) 7D 2×(22 )2 × 3× (32 )3 ’D 22 × 32 × 62 9D ⎛ 2 ⎞3 × ⎛ 3 ⎞3 × ⎛ 5 ⎞3 lD ⎛ 1 ⎞2 − ⎛ 1 ⎞2 − ⎛ 1 ⎞2 ⎜⎝ 5 ⎠⎟ ⎜⎝ 2 ⎠⎟ ⎝⎜ 3 ⎟⎠ ⎜⎝ 2 ⎟⎠ ⎝⎜ 4 ⎠⎟ ⎜⎝ 6 ⎟⎠ #D 9q’j 0af9hvog]d]5,j oUD dD x4 × x6 0D y2 × y3 × y4 7D (ab)10 (ab)11(ab)12 ’D −4x(5x3)2 9D t15 lD x2n ສະຫງສ.ວວນ.ລິສຂະສິດ t14 xn -D (n2 )a (na )2 pD [(x2 )−3]−4 fD (axb ycz )2 8D ⎛ x ⎞100 ⎛ y ⎞101 ⎛ z ⎞102 $D 7e4k,7nda[0=h #D ⎜ y ⎟ ⎝⎜ z ⎟⎠ ⎝⎜ x ⎟⎠ ⎝ ⎠ ( )dD8×102 2 0D −13x8 y 16×105 26x4 y5 7D 14a20b21 'D (an−1)2 (a2 )1−n 14a19b20 lD [(a2n+1)0 ]10 9D 22m × 4m &D .sh N = 1× 22 ×33 × 44 ×55 × 66 ×88 ×99 ×1010 D 9qj’-vdsk7kj 0v’ n, m c]t p D N = (2× 2× 2×.........× 2) × (3× 3× 3×..........× 3)× (5× 5× 5..........×5) n l;j o76o m lj;o76o p l;j o76o 64

[qfmu !* g]dde]a’ X8+? !D dyf9tde dyf9tdemu ! dD 9qj’0Pog]dde]a’8+wxoUgxao9eo;o4h;oD 102; 105; 106 c]t 100 0D 9qj’0Po9eo;o]5,j oUgxaoI6[c[[g]dde]a’D 1000; 10 000; 100 000; 10; 1 7D9q’j 8nj,9eo;omujg\\ktlq,.l[j jvo9Eg,afG • 45000 = 4,5×..... = 4,5×10.... • 565, 01 = ............×103 • 0, 75 = 75×10...... • 88, 008 = 8800,8×10....... @D .97;k, ສະຫງສ.ວວນ.ລິສຂະສິດ !D g]dde]a’rNo 10 dD le]a[ n gxao9eo;o4h;o[;d 10n = 10×10 ×.....×10 = 100............0 n gmnjv n 8q; Š 9eo;o8q;g]d 0 ,u n 8q;D ∗ 10−n = 1 = 0, 00..............01 10n n 8q;s^a’\\kp95f Š .o9eo;o n 8q;g]ds^a’\\kp95f 9t,u 8q;g]d0 n −1 8q; c]h;]q’mhkpfh;pg]d1D 8q;1kj ’G i 101 = 10; 103 = 1000; 108 = 100 000 000 i 10−3 = 1 = 1 = 0, 001 103 1000 i 10−5 = 1 = 1 = 0, 00 001 105 100 000 i 10−6 = 1 =1 1 000 = 0, 000 001 106 000 65

0D dko76o9eo;o|nj’da[ 10n 1kd76o9eo;o|nj’da[ 10n .shphkp\\kp95f XF? wxg[Nv’0;k n 8q;g]d 8q;1kj ’ !D 7, 258×102 = 725,8 phkp\\kp95fwxg[Nv’0;klv’8q;g]dD @D 0,0759×103 = 75,9 phkp\\kp95fwxg[Nv’0;klk,8q;g]dD 7D dko76o9eo;o|n’j da[ 10−n 1kd76o9eo;o|nj’da[ 10−n .shphkp\\kp95f XF? wxg[Nv’-hkp n 8q;g]d 8q;1jk’ !D 520, 45×10−2 = 5, 2045 phkp\\kp95fwxg[Nv’-hkplv’8q;g]dD @D 0,58×10−3 = 0,00058 phkp\\kp95fwxg[Nv’-hkplk,8q;g]dD @D mk’fhko;ymtpklkf gryoj oypq,0Po9eo;os^kp.oI6[c[[G A×10n g-y’j ;kj 1 ≤ A <10 8q;1kj ’G |;j p3]d,uI6[Ijk’.dh7P’da[|j;p,qog-yj’,u]aflt\\ugmqkj da[ 6380000 m. gryjo0Po 6,38×1000000 = 6,38×106 . fa’j oAo ]aflt\\u0v’|j;p3]dpk; 6,38×106 m. ສະຫງສ.ວວນ.ລິສຂະສິດ #D [qfg/ydsafD !D 9q’j 0Po9eo;o]j5,oU.oI6[c[[ A×10n g-yj’ 1≤ A <10 D dD 6950 000; 46 020 000 0D 0, 0052; 25,125; 1250, 02 D 7D 0, 43×103; 0, 0045×10−4; 155, 002×102; 75, 25×10−3 D @D 9qj’0Poleo;o.oxt3spd]5j,oUgxao9eo;o4h;oD dD Iu3fic-o|bj’dik,,u9eo;o3,g]d5oxt,ko 6,0238×1023 3,g]d5oD 0D glAol6ol6f0v’3]d pk;xt,ko 3,93×107 m D 7D mtg]-kp-kIkik gxaomtg]-kpmuj.spmj ulj 5f.o3]d ,ugoNvmujxt,ko 8,96×107 km2 D #D .oxu !_@_ mjko g;uokfl%du X Vernadsky) g-y’j c,oj oad-u;t;ymtpk wfhlaooy4ko;kj ,;olko0v’ dkl vqd-uc-o 16jgmy’|hk3]d,uxt,ko 150×1019 dik,D 9qj’0Po ,;olko0v’dkl vqd-uc-o 16gj my’|hk3]drkp.8hI6[c[[ A×10n g-yj’ 1≤ A <10 D 66

[qfmu !( leo;o !D dyf9tde !D zqo[;dxtdv[fh;prqfF zqo76oxtdv[fh;pl;j o76o c]t zqoskoxtdv[ fh;p8q;8A’sko c]t 8q;sko Xs^n 9eo;or6f c]t r6f?D .o[aofkleo;o]5,j oUF 9q’j [vd;kj leo;o.f c,oj zqo[;dF leo;o.fc,oj zqo76o c]t leo;o.fc,oj zqoskoL dD 5x −3 0D 2x − 3( x + 2) x +1 7D (3x −1)( x + 3) ’D 4x ( x − 3) 3 9D ( x − 3)2 lD 3( x +1) − ( x +1)2 -D x ( x +1) + 3 pD ⎛ x − 3 ⎞ x +3 ⎝⎜ x ⎟⎠ ສະຫງສ.ວວນ.ລິສຂະສິດ fD 1 3 x− x +1 @D 9qj’la’gdf8q;1jk’F 9kdoAoltcf’xt3spd7eg;Qk.o0=h8+wxgxaoleo;orbf-t7toyfD 8q;1kj ’G ok’ ohvp-NxN, @ sq;F sq;]t x du[ c]t [yd # dhkoF dhko]t y du[D gIqklk,kfltcf’9eo;og’yoma’\\qfmuoj k’ohvp-NxN, c]t [yd faj’oUG i 9eo;og’yomuj-NxN,c,jo 2x i 9eo;og’yomuj[ydc,oj 3y i 2x + 3y 9eo;og’yoma’\\qfmuj ok’ ohvp-NxN, c]t [ydD dD ]k7k\\kddhP’ m \\j;pF \\;j p]t &WW du[ gvqk.l.j odt8jk ]k7k @WWW du[D 0D ]qf3fplko7ao|b’j ,umujoa’j le]a[lv’7qo a c4; c]t mujoa’j le]a[lk,7qo c4;D 9qj’0Po9eo;omulj tcf’g4y’7qo3fplkoma’\\qf.o]qf7aooAoD 7D goNvmu0j v’I6[lucj 9lkdgmqjkda[];’pk;76o];’dhk’D #D 9q’j la’gdf8q;1kj ’dkoxjPoleo;orbf-t7toyf gxaoxt3spd7eg;Qk]5j,oUD 8q;1jk’G 7kj [afzkj oxt86g0Qk-q,7voglufc,jo x du[ le]a[zh6.spj c]t y du[ le]a[gfadohvpD 67

i leo;o 3x + 4y Xdu[? ltcf’g4y’zqo[;d0v’7jk[afzjkoxt86g0Qk-q, 7vogluf 0v’z6h.splj k,7qo c]t gfadohvplu7j qoD i vy’.lj0=h,6,.o8q;1jk’ 9q’j ltcf’leo;orbf-t7toyf]5j,oUgxaoxt3spd 7eg;QkD dD 2x; 0D x + 2y 7D 5x − y @D .97;k, !D oypk,G leo;oc,joc4;9eo;omuj,us^kprqf8+daofh;pg7nvj ’\\kp [;d (+) F ]q[(−) F 76o (×) s^n sko(÷)D 8q;1kj ’G dD (18 + 2) ÷ (9 − 4); 3x −1 leo;omugj xaozqosko 1− x leo;omujgxaozqo[;d 0D a + b − c; 2x2 + 5x − 6 7D (5 − 2x)(12 − x); (8 + 4)(13 − 5) leo;omujgxaozqo76o @D dkoltcf’xt3spd7eg;Qk.shgxaoleo;orbf-t7toyfD ສະຫງສ.ວວນ.ລິສຂະສິດ 8q;1jk’ !D ,u\\kd[ko a \\j;pF c8]j t|j;p|ad !&W g c]t w,h8udav[vao |b’j |ad (WW g. 4k,;jk,;olko0v’v5xtdvody]kma’lv’-toyf,ugmqjk.fL ;ymu8v[G .sh M c,jo,;olko0v’v5xtdvody]kma’\\qf Xsq;|j;pgxao g? |kd[ko ! |j;p ,u,;olko !&W |kd[ko a |;j p ,u,;olko 150× a = 150a w,h8udav[ ! vao ,u,;olko (WW ,;olko0v’v5xtdvody]kma’\\qf gmqjk ,;olko0v’|kd[ko a |j;p [;dda[ ,;olko0v’w,h8udav[ ! vaoD fa’j oAoF gIqkwfhG M = 150a + 700 8q;1kj ’ @D xN,7toyflkf x sq;F sq;]t ) WWW du[ c]t w,h[aomaf y vaoF c8j ]tvao]k7k @ WWW du[D 4k,;kj ]k7kxN, c]t w,h[aomafma’\\qfc,oj gmqjk.fL ;ymu8v[G .sh P c,jo]k7kxN, c]t w,h[aomafma’\\qf Xsq;|;j pgxao du[? xN, 1sq; ]k7k ) WWW du[ 68

xN, x sq; ]k7k 8000x du[ [aomaf 1 vao ]k7k 2000 du[ w,h[aomaf y vao ]k7k 2000y du[ ]k7kxN, c]t w,h[aomafma’\\qf gmqjkda[ ]k7k0v’xN, x sq;[;dda[]k7k0v’ w,h[aomaf y vaoD fa’j oAoF gIqkwfhG 8000x + 2000y 8q;1kj ’ #D lv’gvNvpohv’-Nfvdd5s^k[grnjvv;prvo76.o;aomu ( 85]kD gvNvp-N 5 fvd c]t ohv’-N 2 fvd .o]k7kfvd]t du[D gIqklk,kfltcf’9eo;og’yomujlv’gvNvpohv’-Nfvdd5s^k[wfhfa’j oUG ⋅ .sh P c,jo9eo;og’yomujlv’gvNvpohv’-Nfvdd5s^k[ma’\\qf Xdu[?. ;ymumuj 1G ⋅ 9eo;og’yomujgvNvp-Nfvdd5s^k[5 fvd c,oj 5x du[ ⋅ 9eo;og’yomuoj hv’-Nfvdd5s^k[ 2 fvd c,jo 2x du[ ສະຫງສ.ວວນ.ລິສຂະສິດ faj’oAoF gIqkwfhG P = 5x + 2x = 7x ;ymumuj 2 G ma’lv’gvNvpohv’-Nfvdd5s^k[ma’\\qf 7 fvd .o]k 7x du[D faj’oAoF gIqkwfhG P = 7x #D dkoltcf’leo;orbf-t7toyf.shgxaoxt3spd7eg;QkD 8q;1jk’ !D g,njv n ltcf’g4y’9eo;ome,t-kfD i 2n ltcf’g4y’mt;u76o0v’lv’ s^n 9eo;o76Dj i 2n −1 ltcf’g4y’9eo;o7udD 8q;1jk’ @D .sh a, b c]t h c,jo];’pk;0v’0hk’F rNo c]t ];’l6’0v’I6[lucj 9 0hk’0t|ko 8k,]efa[D leo;o8+wxoU ltcf’g4y’G i 2(a + b) ];’Iv[D i bh goNvmujD 8q;1jk’ #D 4hk m Xdu[? c,jo]k7kdt8=hs;kp|bj’|;j p c]t n Xdu[? c,jo]k7k \\kd[ko|b’j |j;p D leo;o8+wxou ltcf’g4y’G 69

i 5m + 2n zqo[;d]k7k\\kddt8=hs;kp & \\;j p c]t \\kd[ko @ |;j pD i 10m + 8n − 200 -N\\kddt8=h !W |;j p c]t \\kd[ko )F mk’Ihkomvo g’yo.sh @WW du[D #D [qfg/ydsaf !D 9q’j ltcf’xt3spd7eg;Qk]5,j oU.shgxaoleo;orbf-t7toyfD dD 9eo;ooadIPo.oshv’|b’j ,uz6h-kp a 7qo c]t z6hpy’ b 7qoD 0D ]k7kl=fe x dhkoF dhko]t 1000 du[ c]t [yd y dhkoF dhko]t 1500 du[D 7D .odt8jk ,u\\kdc8’lk,dy3]F dy3]]t n du[F l;j o]k7k0v’dt8jkc,jo 5000 du[D ’D g’yomvomuj mhk; 7e -NxN,c[[IPo7toyflkflk,sq;F sq;]t x du[D I6h;kj ]k; gvqkg’yo.[ 50000 du[ .shc,7j hkD 9D 0hk’0v’I6[9t85]afmuj,u];’Iv[gmqkj y (cm). lD goNvmuj0v’I6[lk,c9 g-y’j 0hk’rNoc,jo b c]t ];’l6’ h. ສະຫງສ.ວວນ.ລິສຂະສິດ -D -Ng0Qklko x dy3]dik,F dy3]dik, ]t 5000 du[D c,j7hk.9fu s^5f.sh 2000 du[D @D leo;o8+wxoUltcf’g4y’spa’ g,nvj n c,jo9eo;ome,t-kfD dD 3n; 0D 2n +1 #D I6[da[lkd|bj’ ,u0t|kf0v’rNoc,oj a c]t b ];’l6’ c D leo;o8+wxoUltcf’g4y’spa’L c b dD 4a + 4b + 4c 0D ac + bc + ac + bc 7D ab ’D abc a $D ]k7klQ’|b’j znoc,jo m c]t n c,jo]k7kglNv|bj’znoD leo;o8+wxoUltcf’g4y’spa’L dD m + n 0D 2m + 3n 7D 8m + 6n − 4000 &D 9q’j ltcf’leo;omu[j vdgonvmu0j v’I6[]5,j oUD B b2 h hA C X!? a X@? b1 X#? D 70

[qfmu !) dko7eo;odjP;da[leo;o !D dyf9tde !D 9q’j 0Po8n,j .l8j k8t]k’]5,j oUD dD a b c b+c a −(b + c) a−b−c −2 3 4 42 −3 3 −4 2 0D a b c b−c a −(b −c) a−b+c −2 3 4 4 2 −3 3 −4 2 9kdkola’gdfzqowfhIa[.o8k8t]k’ 9q’j 0Po8n,j .llj tg\\uzqo]5,j oU le]a[m5dM7jk0v’ a, b, c gIqkwfhG ສະຫງສ.ວວນ.ລິສຂະສິດ a − (b + c) = .......................... a − (b − c) = .......................... @D 9qj’7yfw]jleo;o8+wxoUD A = (a +b)−(c −b)−(a −c) B = (a − 2b) − (c − 3b) + 2c @D .97;k, !D dko7eo;odPj ;da[leo;o dD leo;o,u;q’g]a[D dko7eo;osk7kj 0v’leo;o,u;q’g]a[F ,udko[;dF dko]q[F dko76oF dkosko c]t g]dde]a’16oj edao dvj ovnjo\\qf 8hv’7eo;og]dmu1j 6j.o;q’g]a[dvj o 9kdoAo 71

9bj’7eo;og]dde]a’F g]d76o c]t g]dsko l5fmhkp 9b7j eo;og]d[;d c]t g]d]q[ 8k,]efa[D 8q;1kj ’G !D E = 36 ÷ (3 + 9) − 7 + 3(8 ÷ 2)3 = 36 ÷ 12 − 7 + 3× 43 = 36 ÷  12 − 7 + 3N× 64 = 3 − 7 + 192 = − 4 +192 = 188 @D M = 3(2x +1) − 4(x − 7) = 3× 2x + 3×1− 4× x − 4× (−7) = 6x + 3 − 4x + 28 = 6x − 4x + 3N+ 31 ສະຫງສ.ວວນ.ລິສຂະສິດ = 2x − 31 0D dkogvqk;q’g]a[vvd9kdleo;oD ⋅ g,njv,ug7nvj ’\\kp[;d (+) 168j +|hk;q’g]a[gryoj lk,kfgvqk;q’g]a[vvd9kd leo;o wfh3fpmujg7nvj ’\\kp0v’g]dmu1j 6j.o;q’g]a[pa’7ngdqjkD ⋅ g,njv,ug7nvj ’\\kp]q[ (−) 168j +|hk;q’g]a[gryjolk,kfgvqk;q’g]a[vvd9kdleo;owfh 3fpdkoxjPo g7njv’\\kp0v’m5dM8q;g]dmuj16j.o;q’g]a[D 8q;1jk’G A = 3 + (a − 2 + b) B = 3 − (a − 2 + b) = 3+ a − 2+b = 3− a + 2−b =1+ a +b =5−a−b E = 4x2 + (2x − 3) − 7(x +1) = 4x2 + 2x −3− 7x − 7 = 4x2 − 5x −10 7D dko7eo;oleo;omuj[+,u;q’g]a[D dko7eo;osk7kj 0v’leo;omuj[+,u;q’g]a[g-yj’,udko[;dF dko]q[F dko76oF dko sko c]t g]dde]a’16joedao djvovnjo\\qfgIqk8hv’7eo;og]dde]a’lkdjvo 72

8+,k 9bj’7eo;og]d76o c]t g]dsko l5fmhkp9bj’7eo;og]d[;d c]t g]d]q[ 8k,]efa[D 8q;1jk’G M = 3 × 52 × 2 − 22 +6 N = 5x + 7 − 3x 4 = 5x − 3x + 7 = 3× 25× 2 − 4 + 6 4             = 150 −1 + 6 = 2x + 7 = 155 &D 7kj 0v’leo;o 8q;1kj ’ !D ];’Iv[0v’I6[lujc90hk’0t|ko ltcf’fh;pleo;o E = 2(a + b) D 9qj’-vdsk];’Iv[0v’,ao g,njv dD a = 4; b = 6 0D a = 12cm; b =18cm grnvj 7yfw]7j kj 0v’leo;o gIqkcmo7jk0v’ a c]t b c]h;7eo;oD cdhG dD E = 2(a + b) 0D E = 2(a + b) = 2(4 + 6) = 2(12 cm +18cm) ສະຫງສ.ວວນ.ລິສຂະສິດ = 20 Xsq;|;j p];’pk;? = 60 cm 8q;1kj ’ @D mtok7ko7yfw]jfvdg[hp Xfvdg[hpme,tfk? fh;pl6f I= PRT g-yj’;kj 100 I c,jofvdg[hpF P c,oj 9eo;og’yomu/j kd Xmbo?F R c,oj vaf8kfvdg[hp c]t T c,jow]ptg;]kmuj/kdg’yoD 4hk mhk; mv’;yw] /kdg’yo 5000000 du[ 16mj tok7kodko7hk8kj ’xtgmf fh;p vaf 8kfvdg[hp 8% 8+xuD g,njv7q[Iv[|bj’xuF ]k; 9twfhfvdg[hpgmqkj .fL [qfcdhG 9kdl6f I = PRT g,njv P = 5000000 du[F R = 8% = 0, 08 c]t T =1 xu 100 gIqkwfhG I = 5 000 000× 0, 08 ×1 100 = 400000 du[ 8q1jk’ #D l6f7yfw]jgoNvmuj|hk|j;p,qoc,jo A = 4πr2 g-yj’;kj π = 3,14 c]t r c,oj ]aflt\\uD 9qj’-vdsk7jk A g,nvj r = 6cm S r =18dm [qfcdhG cmo7kj r = 6cm .llj 6f 9twfhG g,nvj r =18dm gIqkwfhG A = 4 × 3,14× 62 = 452,16 cm2 A = 4× 3,14×182 = 4069, 44 dm2 73

#D [qfg/ydsaf !D 9q’j 7yfw]jleo;o]5,j oUD A = 2,1− (3,5 + 2,1) + (5 + 3,5) C = −(251× 3 + 281) + 3× 251− (1− 28) B = (5,3 − 2,8) − (4 + 5,3) D = − ⎛ 3 + 3 ⎞ − ⎛ − 3 + 2 ⎞ ⎝⎜ 5 4 ⎟⎠ ⎜⎝ 4 5 ⎠⎟ @D 9qj’gvqk;q’g]a[vvd9kdleo;o c]h;7yfw]jD A = 12 − (a − b) + (8 − b); E = 3(2x +1) − 4(x − 7) B = 2a − (a − (b + a)); F = 8(x − 2) + 4(2x + 6) C = (6 − a) + (9 + b) − (12 − a); G = 5(x − 3) − 2(x +1) D = −(b − 2b) + 2(b − a) − 5(5 − b); H = 2 y(1− y) + y(1+ 2 y) #D 9qj’7eo;oleo;o]j5,oUD A = 3 × 5 − 52 ; C = −2,1 + 2,1× 3 − 3 4 4 4 4 B = 3 + 9 × 3 −1; D = 1−1÷ 3 +1÷ (3 +1) 4 5 2 $D .oleo;o8+wxoU leo;o.fmu,j u7kj gmqjkda[ !@S *S W s^n − * L ສະຫງສ.ວວນ.ລິສຂະສິດ A = 3 + 3 + 3 + 3; D =3−3−3−3 B = 3 + 3 + 3 − 3; E = 3 + 3 − (3 + 3) C = 3 + 3 − 3 − 3; F = 3 − (3 + 3 + 3) &D 9qj’7yfw]7j kj 0v’leo;o]5j,oUD A = 4b + 7 + 3b − 2 + 2b g,nvj b = −5; B = 36b − 4b2 + 3b3 g,nvj b = −2 C = 3x + 2x2 − 6(x − 4) + 8x3 g,nvj x = 2; D = 4 y −12 + 128 −16 y g,njv y = 4 y2 y3 *D 7e4k,7ndaoda[0=h &D dD F = am g,nvj a = 8 c]t m = 150 0D F = 9C + 32 g,njv C = 45 5 7D P = RT g,njv R = 48;T = 20 c]t V = 6 ’D I = E g,njv E = 220 S R = 80 V R 9D V = 1 π r 2h g,njvπ = 3,14; r = 6 c]t h = 5 lD h = v2 g,nvj v = 15 S g = 9,8 3 2g 74

[qfmu !_ dkocpdlj;o76o.oleo;o !D dyf9tde !D gIqk9t,u;ymu.fcfgj rnvj -vdskgoNvmuj0v’I6[lucj 9lkd MNPQ ]5,j oUD I6h;jk goNvmu0j v’I6[lujc9lkd gmjqk];’pk;76oda[];’d;hk’D dD a Ib P 9qj’-vdskgoNvmuj0v’I6[lucj 9lkd N MNPQ fh;plv’;ymu 3fp.sh k a = 5; b = 2 c]t k = 4 i AMNPQ = AMJIN + DDDDDDDDDDDDDD M JQ = DDDDDDDDDDD + DDDDDDDDDDDDDD = DDDDDDDDDDDD i AMNPQ = (5 + 2)× DDDDDDDD = DDDDDDDDDDDDDDDD ສະຫງສ.ວວນ.ລິສຂະສິດ 9q’j xP[mP[zqowfh0v’lv’d=]tou0hk’gmy’ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 9kdoAoF 9qj’-vdskgoNvmu0j v’I6[lujc9lkd MNPQ fh;plv’;ymu 8k, a, b c]t k D c]h;4vf4vogvqkzqo76o k(a + b) = DDDDDDDDDDDD + DDDDDDDDDDDDDDDD 0D a I b Pc N g-aoj fP;daoda[0=h Xd?F 9qj’oe.-h7jk L EK 0v’ a = 5; b = 3; c = 2 c]t d d = 4 grnvj -vdskgoNvmu0j v’I6[luj c9lkd MNPQ fh;plv’;ymuD M JQ i 7yfw]gj oNvmu0j v’ MNPQ fh;plv’;ymu8k, a, b, c c]t d 9kdoAo 4vogvqk zqo76o (a + b)(c + d) = DDDDDDDD + DDDDDDDD + DDDDDDDD + DDDDDDDDD @D 9qj’oe.-hdqfgdo0v’dkoc9dl;j o0v’dko76o k(a + b + c) = ka + kb + kc grnvj 7yfw]jg]d]5j,oUD 75

i 70×11+ 70× 22 + 70×17 = 70(...... + ...... + ......) = ........... i 2a × 3 + 2a × 5 − 2a × 6 = ...................................................... = ...................................................... i 2(a +1) − 3(a +1) + 4(a +1) = ............................................. = ............................................ i m2 − m2 (m − 2) + m2 (m − 3) = ............................................ = ........................................... #D 9qj’la’gdfdko7eo;o0v’leo;o]5j,oUD A = (2x −1)(x −1) − (2x −1)(x + 2) + 2x(2x −1) = (2x −1)[(x −1) − (x + 2) + 2x] = (2x −1)[x −1− x − 2 + 2x] = (2x −1)(2x − 3) 9kdoAo9a[76zj qo[;dmujgmqjkda[zqo76o.o8k8t]k’]5j,oUD leo;omugj xaozqo[;d leo;omujgxaozqo76o ສະຫງສ.ວວນ.ລິສຂະສິດ A = 2x(x −1) − (2 + x)(x −1) + (2x +1)(x −1) ...... 2x(2x + 2) B = 2x(x + 2) + 2x(2x −1) − 2x(x −1) ...... (x + 2)(−3x) C = (x + 2)(x −1) − 2x(x + 2) − (x + 2)(2x −1) ...... (2x −1)(2x − 3) D = (x −1)(2x −1) − (2x −1)(x + 2) + 2x(2x −1) ...... (x −1)(3x −1) $D 9kddyf9tde0=h X@? c]t X#?D i ka + kb + kc ,u k c,jolj;o76oIj;,D i (2x −1)(x −1) − (2x −1)(x + 2) + 2x(2x −1) ,u 2x −1 c,oj l;j o76oI;j ,D 9q’j -vdskl;j o76oI;j ,0v’leo;o]5j,oUD i 3×5 + 3×7 + 3×11+ 3×13 i 4 + 8 +12 +16 + 20 + 24 i 2ab + 2ac + 2ad + 2ae + 2af i (1− n)(m − n) − (m − n)(2 − n) i b + b2 + b3 + b4 i ab − (ab)2 76

@D .97;k, ສະຫງສ.ວວນ.ລິສຂະສິດ !D dko0tspkpzqo76oD 1kd76o9eo;o|n’j da[zqo[;d gIqk76o9eo;ooAoda[c8]j trqf0v’zqo[;dc]h;[;d zqowfhIa[g0qhkdaoD dqfgdoc9dl;j o i k (a + b) = ka + kb i k (a − b) = ka − kb 8q;1kj ’ !D 7yfw]lj eo;o E = 3( x −1) − 5( x +1) + x = 3× x − 3×1− 5× x − 5×1+ x = 3x − 3− 5x − 5 + x = 3x + x − 5x − 3− 5 = −x −8 @D A = 3(2x + 3y) − (x + 5y) = 3×2x + 3×3y − x − 5y = 6x +9y − x −5y = 6x − x +9y −5y = 5x − 4y Š 1kd76ozqo[;dda[zqo[;d gIqk76ozqo[;d0v’c8j]trqfmevyf da[c8j]trqf0v’ zqo[;dmulj v’ c]h;[;dzqowfhIa[g0QkdaoD (a +b)(c + d) = a(c + d)+b(c + d) = ac + ad + bc + bd (a +b)(c + d +e) = a(c + d +e)+b(c + d +e) = ac + ad + ae + bc + bd + be 8q;1jk’ !D A = (5x −1)(4 − x) = 5x (4 − x) −1(4 − x) = 20x − 5x2 − 4 + x = −5x2 + 21x − 4 @D B = (3 − 2x)(x2 + 5x + 6) = 3(x2 + 5x + 6) − 2x(x2 + 5x + 6) = 3x2 +15x +18 − 2x3 −10x2 −12x = −2x3 − 7x2 + 3x +18 77

#D I = m + 2m + n − m − 2n 12 4 3 = m + 3(2m + n) − 4(m − 2n) 12 = m + 6m + 3n − 4m + 8n = 3m −11n 12 12 @D dkocpdleo;omujgxaozqo[;d vvdgxaolj;o76oD dkocpdzqo[;dvvdgxaolj;o76o c,jodkoxPj ozqo[;d.shgxaozqo76o c8j7kj 0v’ leo;opa’7ngdqkj 3fpdkocpdgvqkl;jo76oIj;,vvdgxaolj;o76oD zqo[;d lj;o76oIj;, zqo76o ka + kb k k (a +b) ka − kb k k (a −b) ka − kb + kc k k (a − b + c) Š 0=h7;ola’gdfG .shgIqk4n;jk k =1× k g-ajo;kj G ສະຫງສ.ວວນ.ລິສຂະສິດ ka + k = k (a +1) ka − k = k (a −1) 8q;1kj ’ !D x + 2x + 3x + 4x ,u x gxaolj;o76oI;j , faj’oAo gIqkwfhG x + 2x + 3x + 4x = x(1+ 2 + 3 + 4) = 10x @D 2n(n +1) − 2n(2 − n) ,u 2n gxaolj;o76oI;j , gIqkwfhG 2n(n +1) − 2n(2 − n) = 2n[(n +1) − (2 − n)] = 2n[n +1− 2 + n] = 2n(2n −1) #D 11b2 +12b2 +13b2 ,u b2 gxaolj;o76oI;j , gIqkwfhG 11b2 +12b2 +13b2 = b2 (11+12 +13) = 36b2 $D (x − 5)(2x −1) + 2(x − 5)(3x +1) + 4(x − 5) ,u x − 5 gxaol;j o76oIj;, gIqkwfhG (x − 5)(2x −1) + 2(x − 5)(3x +1) + 4(x − 5) 78

= (x − 5)[(2x −1) − 2(3x +1) + 4] = (x − 5)[2x −1− 6x − 2 + 4] = (x − 5)[−4x +1] = (x − 5)(1− 4x) Š [k’leo;omujgxaozqo[;d gIqkvkf9t[+gsaol;j o76oIj;,muj9tc9h’wfh 9nj’9egxao 8hv’wfh-vdskl;j o76oIj;,oAodjvoD 8q;1jk’G 1. 7a2 −14a3 + 21a5 = 7 × a2 − 7 × 2× a2 × a + 7 × 3× a2 × a3 ( )= 7a2 1− 2a + 3a3 lj;o76oI;j ,c,jo 7a2 2. a (2x +1) − (4x + 2) = a (2x +1) − 2(2x +1) = (2x +1)(a − 2) lj;o76oI;j ,c,oj 2x +1 #D (1− x)(4x −1) + (x −1)(3x − 2) = (1− x)(4x −1) + (−1)(1− x)(3x − 2) = (1− x)[(4x −1) + (−1)(3x − 2)] = (1− x)[4x −1− 3x + 2] = (1− x)(x +1) Xl;j o76oI;j ,c,jo 1− x ? ສະຫງສ.ວວນ.ລິສຂະສິດ $D x2 + 3x + 2x + 6 = (x× x + 3x) + (2x + 2×3) = x(x + 3) + 2(x + 3) lj;o76oI;j ,c,jo x + 3 = (x + 3)[x + 2] = (x + 3)(x + 2) &D m2 − m3 + m4 m2 m2 m m2 m2 pn pn2 p2n2 = pn − pn × n + pn × pn = m2 (1 − m + m2 ) l;j o76oI;j ,c,oj m2 pn n pn pn #D [qfg/ydsaf !D 9qj’7yfw]zj qo76o]5,j oUD dD 5(x +1) 0D 2a(a +1) 7D 7(a2 −1) ’D ⎛ 1+ y ⎞⎛1− y + y2 ⎞ 9D ⎛ 5+n ⎞⎛ n−5⎞ lD ⎛ 3x −12 ⎞ ⎛ 2x2 − 4 ⎞ ⎜⎜⎝ 1+ y + y2 ⎟⎟⎠⎜⎜⎝ 1− y ⎟⎟⎠ ⎜⎝ 5−n ⎟⎠ ⎜⎝ n + 5 ⎟⎠ ⎝⎜ 4x2 −8 ⎟⎠ ⎝⎜⎜ 9x − 36 ⎟⎠⎟ -D (a + b)(a − b) pD (1− x)(1+ x + x2 ) fD (m + n)(m2 − mn + n2 ) 79

@D 9q’j 7eo;og]d]5,j oUD dD 3(2x + 3y) − (x + 5y) 0D x(a + b) − x(a + 3b) 7D 2(5a + 3b) + 3(a − 2b) ’D 2m(m − 5) − m(m − 2) − 3m(m − 5) 9D 3(a − b) − 2(2a − 3b) + 4(a − 3b) lD 3n(n2 + 7n −1) − 2n(2n2 + 3) − 3(n2 + 5) -D (a + b)(a − b + 2) − 2(a + b) pD (m + n)(m − n −1) − (m − n)(m + n) fD ( a − 2 )( a + 2 ) − ( a − 3)( a + 3) 8D (1 + b )(1 − b+ b2 ) − (1 − b )(1 + b+ b2 ) 2 2 3 3 5 5 5 5 #D 9qj’-vdskl;j o76oI;j ,0v’leo;o]5j,oUD dD10 +15 + 25 + 35 0D 3 + 6 + 9 +15 7D 22 + 33 + 44 − 55 − 66 ’D 4x + 4 y 9D 2a + 4ab + 6abc lD 4mn − 6mnp + 8mnpt -D 2ab − 3a2b pD 15abcd + 21a2c2 fD 12x2 y3 − 8x3 y2 8D x(x −1) − 2x(x − 2) 4D m(x + y) + n(x + y) mD n(m − n) − np(m − n) ສະຫງສ.ວວນ.ລິສຂະສິດ $D 9qj’cpdleo;o]5j,ohuvvdgxaolj;o75oD dD 7x + 7 y 0D 3a + 3b + 3c 7D 2ab − 2ac − 2ad ’D 4n − 6mn 9D 22ab + 33ac − 44ad lD 10xy + 5x2 y2 -D 36b2 − 9b pD 5a2 −10a3 +15a4 fD 15m4 − 25m2 &D 7e4k,7nda[0=h $D dD 3m5 − 3m4 +12m3 − 9m2 0D 4a2b2c2 + 8ab2c −12a2b3c3 7D 9x3 y − 6x2 y2 + 3xy3 ’D 5n3 −10n2m +15nm2 *D 7e4k,7n0=h $D dD ax + ay + bx + by 0D pm + pn − mq − nq 7D x2 + 2x + 5x +10 ’D 4ax + 6ay − 4bx − 6by 9D x2 − 3x − 2x + 6 lD a2c2 + acd + acd + d 2 -D 3n(n + 2) − 3n(3 − 2n) pD (x − 5)(n + m) − (x − 5)(n − 2m) fD (1− 3x)(2 + 3x) − (3x −1)(1+ x) 8D (2x + 4)(3x − 5) − 2(3x + 6)(x −1) 4D mn m2n mn2 mD a3 − a2b − a2c 5 + 10 − 15 2x3 4x4 6x3 80

ສະຫງສ.ວວນ.ລິສຂະສິດrkdmu $ lq,zqo c]t vtlq,zqo0Ao|b’j muj,u|bj’8q;]a[ [qfmu @W ltg\\uzqo !D dyf9tde dyf9tde ! gryoj 76o & da[9eo;o.f|b’j c]h; ]q[vvd @ gmqkj da[ !#D gryoj 0Poxt3spdlaopt]adfa’j oU (5×...) − 2 = 13 { 9qj’0Poxt3spdlaopt]ad0hk’gmy’oU7no 3fpcmo a .l[j vj o9Eg,af (5×...) − 2=13 xt3spdlaopt]adoUgvUo;kj G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD { 9qj’la’gdfltg\\uzqo 2a = 10 c]h;8v[7e4k,8+wxoUD • 4hkcmo a fh;p9eo;o * 9t.fhltg\\uzqo4nd s^n zyfL • 4hkcmo a fh;p9eo;o & 9t.fhltg\\uzqo4nd s^n zyfL dyf9tde @ d?D   |adgmqjkdao S 4hk.lj gmy’zk’-y’0hk’]t @ vaozk’-y’9t’jP’ wx0hk’.f0hk’|bj’ s^n [+L { 9qj’8v[7e4k,8+wxoU Xle]a[m5dM 9eo;o a , b c]t c ? • 3+1 = 4 gxaoltg\\uzqo4nd s^n zyfL • ( 3+1 )+ 2 = 4 + 2 gxaoltg\\uzqo4nd s^n zyfL • ( 3+1 )+ a = 4 + a gxaoltg\\uzqo4nd s^n zyfL • .sh a = b faj’oaho ( a + 7 ) = ( b + 7 ) gxaoltg\\uzqo4nd s^n zyfL 81 

• .sh a = b faj’oaho ( a + c ) = ( b + c ) gxaoltg\\uzqo4nd s^n zyfL 0?D 9qj’0Po8nj,.lj[jvo9Eg,af grnjv.shltg\\uzqo]5,j oUgxaoltg\\uzqo4ndc]h;d;f7no 3fp cmo7jk0v’ a , b c]t c fh;p9eo;o8k,.9D • .sh a = b fa’j oaho a − c = ........ • .sh a = b faj’oaho a × c = ........ • .sh a=b fa’j oaho a × 1 = ........ (c ≠ 0) c 7?D gryoj .shltg\\uzqo a +1 = 4 c]t a + 7 = 10 9q’j [;drkd8+rkd0v’ltg\\uzqofh;pdko0Po.lj[vj o9Eg,afD a +1 = 4 + a + 7 = 10 .... + .... = ..... { 9qj’d;f7no;jkltg\\uzqo.\\mj ujwfh,kgxaoltg\\uzqo4nd sn^ zyf3fpdkocmo7jk.sh a fh;p9eo;o8k,.9D @D .97;k, ສະຫງສ.ວວນ.ລິສຂະສິດ !D oypk,G ltg\\uzqoc,jolv’leo;omu8j +daofh;pg7jnv’\\kp \" = \" c]t ,aogxao9y’le]a[m5dM 7jk0v’8q;vadlvo X pqdg;ahoc8j7kj 8hv’shk, ?D 8q;1kj ’G (a + b) 2 a 2 2 • = + 2ab + b (a − b) 2 a 2 2 ltg\\uzqo7;o9nj • = − 2ab + b • a 2 − b 2 = (a − b)(a + b) • 7 + 5 = 15 − 3 • 1 + 2 + 3+ 4 = (1+ 4)× 2 @D 75o]adltotG d?D g,njv[;dma’lv’rkd0v’ltg\\uzqoda[9eo;ofP;dao gIqk9twfhltg\\uzqo.\\j mP[ gmqjkda[ltg\\uzqogfu,D { le]a[m5dM9eo;o a , b, c 4hk a = b gIqk9twfh a + c = b + c 82 

0?D g,jnv76oma’lv’rkd0v’ltg\\uzqo da[ 9eo;ofP;dao gIqk9twfhltg\\uzqo.\\jmP[ gmjqkda[ltg\\uzqogfu,D { le]a[m5dM 9eo;o a , b c]t c 4hk a = b gIqk9twfh a × c = b× c { le]a[m5dM 9eo;o a , b c]t c 8jk’l6o, 4hk a = b gIqk9twfh a × 1 = b× 1 c c 7?D g,njv[;drkd8+rkd0v’lv’ltg\\uzqooedaogIqk9twfhltg\\uzqo.\\mj P[gmqjkda[lt g\\uzqogfu,D { le]a[m5dM9eo;o a , b , c 4hk a = b c]t c = d gIqkwfhG a + c = b + d #D [qfg/ydsafG ສະຫງສ.ວວນ.ລິສຂະສິດ !D 9qj’0Poxt3spd]5j,oUgxaoxt3spdlaopt]adD • g9af[;dlk,gmjqkda[ly[ • cxf]q[vvdlv’gmqjkda[sqd • lv’gmjnv0v’lk,95fshkgmqkj da[g9af • g7yj’|bj’0v’-k;lujgmqjkda[ly[lv’ @D 9q’j 0Po9eo;omugj \\ktlq,.lj[vj o9Eg,afgrjnvxjPoltg\\uzqo Xd? gxao X0? gxao X7? c]h; [vd75o]afltotmujoe.-h.odkoxPj oltg\\uzqofaj’dkj ;D dD 5a − 20 = 40 5a − 20 + ......... = 40 + ......... 0D 5a = 60 5a × ............. = 60 × ................ 7D a = 12 #D 9qj’-Uc9’ltg\\uzqo 5a − 3 = 3b − 4 9tgxaoltg\\uzqo4ndg,njv a = 7 c]t b = 14 { 4hk a = 7 c]t b = 14 ltg\\uzqo 5a − 3 = 3b − 4 9tgxaoltg\\uzqo4nd s^n lt g\\uzqozyfL { ovd9kd a = 7 c]t b = 14 c]h; pa’,u7kj .f0v’ a c]t b vudmugj Iaf.sh ltg\\uzqo 5a − 3 = 3b − 4 gxaoltg\\uzqo4ndL $D 9q’j 0Poxt3spdlaopt]adgxaoxt3spd7eg;qhk3fpdkocmo a fh;p7e;kj G “9eo;o.f|b’j ” 83 

8 + a = 11 S 2a + 4 = 10 S a − 3 = 0 2 &D 9q’j -vdsk7jk0v’ a mugj Iaf.shltg\\uzqo 9a = 91 + 2 gxaoltg\\uzqo4ndD *D oadIPolv’7qooe.-h75o]adltot0v’ltg\\uzqog0qhk.odko-vdsk7jk0v’ a F9q’j [vd ;kj oadIPo7qo.fcdh4nd c]t oadIPo7qo.fcdhzyfL zyf16j[vj o.fL oadIPo7qomu ! oadIPo7qomu @ − 2a + 3 = 15 − 2a + 3 = 15 − 2a = 18 − 2a = 12 a = 20 a = −6 (D oadIPo7qooU[+g0qhk.9[qfIPo9b’j cdhg]dzyfrkfD 9q’j -;j p]k;cdhg]d.sh4nd8hv’rhv, ma’vtmy[kp;ymudkooe.-h75o]adltot0v’ltg\\uzqo.odkoxjPoltg\\uzqo Xd? gxao X0? gxao X7? dD − 2a − 3 = 15 dD − 3a + 12 = 120 0D − 2a = 18 0D 3a = 108 7D a = 20 7D a = 36 ສະຫງສ.ວວນ.ລິສຂະສິດ 84 

[qfmu @! lq,zqo0aho|bj’muj,u|bj’8q;]a[ !D dyf9tde dyf9tdemu ! .shoadIPo0Poxt3spd]5,j oUgxaoxt3spdlaopt]ad0v’7toyflkfD  sqdgmqjkda[ x  x [;dda[lk,gmqjkda[ly[shk  lv’gmnjv0v’9eo;o.f|bj’[;dda[shkgmjqkda[ly[gvaf  9eo;o.f|bj’[;dda[shkgmqkj da[lv’gmnjv0v’9eo;ooAo]q[vvd|bj’ dyf9tdemu @ dD 9q’j -vdsk7kj 0v’8q;vadlvo grnjvgIaf.shltg\\uzqo]5j,oUgxao9y’D mk’g0Qk 15  3x 49  1 6 p 1 k  15 mk’vvd 4 y  3  15 3d  24 3 ສະຫງສ.ວວນ.ລິສຂະສິດ 24  3z  3 1c38 1m3 2 9 n  2n  15 14  4t  42 3b  b  18 2a  6a  88 0D grnvj g0Qkg[y’j ]t7voF c8]j t7qo8hv’wx8k,glAomk’ 3fp;kj G 7kj 0v’8q;vad lvo8hv’gxao9eo;o7udD 9q’j c8h,glAomk’fa’j djk;D dyf9tdemu #D 9qj’0Po9eo;omujg\\ktlq,8nj,.lj8k8t]k’]5j,oU 8k,c[[1jk’0v’c4;mu !D lq,zqo 0ufvvh,gvqk.9zqo m5d9eo;omujgxao.9zqo xx 1  0 W s^n ! 3x 1x  2  0 1; 0; 2; 3; 4  2; 1 ; 0; 1 ; 3 DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD x  52x  7  0 33 DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD  5; 7 ; 0; 7 ; 5 22 85 

@D .97;k,D !D oypk, ltg\\uzqomuxj tdv[fh;p9eo;o|bj’muj[+maoIh6-bj’ltcf’fh;p8q;vadlvogvUo;jkG lq,zqoF 9eo;omu[j +maoIh6gvUo;kj G 8q;]a[0v’lq,zqoD ax  b  c .ooAo a , b c]t c gxao9eo;omujgIqkIh6c]h; c]t a 8hv’8jk’l6o X a  0 F x c,oj 9eo;omu[j +maoIh6gvUo;jkG “ 8q;]a[ ”; 7jk0v’ x mu8j v[lt|v’da[ lq,zqo gvUo;jkG “ .9zqo0v’lq,zqo ”. X ax  b  c ; a  0 gvuho;jk lq,zqo0aho |bj’mu,j u|b’j 8q;]a[ ?D 8q;1jk’G  3x  4  2  2x  10  0  5x  9  x 7 @D dkocdhlq,zqo ສະຫງສ.ວວນ.ລິສຂະສິດ d?D dkocdhlq,zqo.oI6[c[[ ax  b  c / a  0 3fpmqj;wxc]h;dkocdhlq,0aho|b’j mu,j u|bj’8q;]a[gruoj .-h75o]adltot0v’ltg\\uzqogrjnvcdhD 8q;1jk’G 9qj’cdhlq,zqo x  5  12 X!? ;ymucdhG x  5 5  12  5 X@? x  5  5  12  5 X#? x 7 X$? d;f7nofh;p;ymucmo x  7 .ljlq,zqo X!? gIqk9twfhG x  5  12 7  5  12 ltcf’;kj x  7 c,oj .9zqo0v’lq,zqo x  5  12 8q;1jk’G 3x  2  x  8 X!? ;ymucdhG 3x  2  x  8 86 

3x   x  2  x   x 8 X@? X#? 2x  2  8 X$? X&? 2x  2   2  8   2 2x  6 76oma’lv’rkd0v’ X&? da[ 1 gIqk9twfhG 2 1 2x  1 6 X*? 22 X(? x 3 ;ymud;f7noG gIqkcmo7jk0v’ x  3 .lj X!? gIqk9twfhG 3x  2  x  8 X!? 33  2  3  8 11  11 ສະຫງສ.ວວນ.ລິສຂະສິດ ltcf’;jk x  3 c,jo.9zqo0v’lq,zqo 3x  2  x  8  4hklq,zqogxaog]dl;j omu,j ur6f8jk’dao grujo0Bor6fIj;,c]h;xtr6fD 8q;1kj ’G x  5  11  3x 76o.sh # ma’lv’rkdgIqk9twfhG 3 x  5  3  3  11  3x 3 x  5  33  9x phkprqfmu,j u8q;]a[wxg[Nv’-hkp c]t phkprqfmuIj h6 c]h;wxg[Nv’0;kgIqk9twfhD x  9x  33  5 10x  28 76o.sh 1 ma’lv’rkdgIqkwfhD 10 1 10x  1  28 10 10 x  28  2,8 10 \\kpgsfG [k’d=]toulq,zqovkf9t[+,u.9zqo s^n ,u.9zqo[+lyhol5fD 87 

8q;1jk’G d=]tou,u.9zqo[+lYol5f 2x  3  3 x  2  x  9 d=]tou[+,u.9zqo 4x  5  6  41  x 2x  3  3x  6  x  9 4x  5  6  4  4x 2x  3  2x  3 4x  5  2  4x 2x  2x  3  3 0x   3 0x  0 lq,zqo[+,u.9zqo grkt;kj lq,zqo,us^kp.9zqo[+lYol5f [+,u7kj .f0v’ x 76oda[ W grktm5dM7kj 0v’ x 76oda[ W c]h;gmjqkda[ (3) D c]h;gmqjkda[ WD  dkocdhlq,zqo ax  b  0 gIqklk,kfcdhfh;p;ymufaj’]5,j oUD lq,zqo g’njvow0 .9zqo ສະຫງສ.ວວນ.ລິສຂະສິດ ax  b  0 a 0 ,u.9zqofP; x   b a  0; b  0 a a  0; b  0 [+,u.9zqo ,u.9zqo[+lYol5f 0?D lq,zqo.9I6[c[[ A  B  0 zqo76ogmqjkl6o 4hk;kj |b’j .o[aofkl;j o76o0v’,aogmqjkl6o A  B  0 \\kp7;k,;kj G A  0 s^n B  0 8q;1kj ’G 9qj’cdhlq,zqo]5,j oUG !D  x 1 x  2  0 gIqkwfhG x  1  0 s^n x  2  0 x 1 x  2 ltoAo lq,zqo  x 1 x  2  0 ,u.9zqoc,joG x  1 c]t x  2 @D  3x5x  3  0 gIqkwfhG 88 

3x  0 s^n 5x  3  0 x 0 0 5x  3 3 x3 5 ltoAo lq,zqo  3x5x  3  0 ,u.9zqoc,oj G x  0 c]t x  3 5 #D  x  2 x  3  x  2 [qfcdhG djvovnjo8hv’0Po x  2 x  3  x  2 .sh16j.oI6[c[[ A  B  0 3fpdkocpdgxaolj;o76oD 9kd  x  2 x  3  x  2 x  2x  4x  2  0        x  2  x  4 1  0                      x  2 x  3  0 3fpdkocdh7ndaoda[0=h ! c]t @ 0hk’gmy’ gIqkwfhG x  2 s^n x  3 7?D dko0Po.9zqo s^n dkoltcf’.9zqo0v’lq,zqo.o8q;1kj ’gmy’lk,kfltcf’ fh;pd5j,.9zqo g-ajoG d5j,.9zqo0v’8q;1jk’ ! c,jo S  1;2 ສະຫງສ.ວວນ.ລິສຂະສິດ d5j,.9zqo0v’8q;1kj ’ @ c,oj S  0; 3  d5j,.9zqo0v’8q;1jk’ #  5   c,oj S  3;2 #D [qfg/ydsaf !D 9q’j 0Poxt3spd]5j,oU gxaoxt3spdlaopt]adG  zqo[;d0v’9eo;o.f|bj’da[ $ gmqjkda[lv’gmnjv0v’9eo;ooAoD  zqo]q[0v’9eo;o.f|b’j da[ @ gmqjkda[ @)D  vud & xu vkp50v’0hvp9tgmqjkda[ !* xuD @D 9qj’xPj oxt3spdlaopt]ad]5,j oU gxaoxt3spd7eg;qhk3fp.sh x cmo9eo;o.f|b’j D  x20  2x  3  13  x 1 8  2x  x  6 89 

#D xt3spd7eg;qhk]5j,oUdq’da[lq,zqo.f L dD lugj mjnv0v’9eo;o.f|bj’gmqjkda[9eo;ooAo[;dda[lv’  4x  2  x  4x  x  2  x4x2  2x  x  4 0D vud @&W du[ g’yo0v’0hvpgmqkj da[ @&WW du[  250x  2500  2500  x  250  250  x  2500  250x  2500  x 7D zqo[;d0v’9eo;o.f|bj’da[ !& gmqkj da[lk,gmjnv0v’9eo;ooAo ]q[vvd #D  3x 15  x  3  x 15  3x  3  x 15  3x  3  3x 15  3x  3 $D .o9eo;o8kj ’M 8+wxoUG  1 ; 0; 1 c]t ! 9eo;o.fgxao.9zqo0v’lq,zqo 33 2x 3  5x  4L &D 9q’j g]nvdgvqk7jk0v’ x mujgxao.9zqo0v’lq,zqo dD 2x  3  5 aD ! bD $ cD @ ສະຫງສ.ວວນ.ລິສຂະສິດ 0D 3x 1  6x  3 aD ! bD @ cD 4 a D 4 bD $ 7D 4x  3  2x  5 3 *D 9qj’cdhlq,zqo8+wxoUD c D 8  1  x  3 1S   7 x 3 1 10 20 5 2  17 x  4  2  1 x  x  5  11S 52 2  2x  1  11S  102 x  103 6 (D mhk; r5flk7vo cdh[qfg]dfa’j ]5j,oUD 6x  36  90 x  36  90  15 6 x  15  36 x  21 7e8v[0hk’gmy’oU4nd s^n zyf L fh;pgsfzqo.f L 90 

)D 9q’j cdhlq,zqo8+wxoUG a. 3x 1  23 b. 8x  7  63 c. y 1  6 d. y  4  88 5 3 e. 8x 12  5x  21 f . 3x  4  8x  3  32 g. 2y  3  9  4y h. 7z  294  3z  14 i. 6a  31 a  24 j. 4b  38  2b 1 k. 2 x  3  2 x  4  24 l. 50  5 x  2  4 x 1 m. 3n  2  5  8  2n  4 n. 7 a 1  3a  4a 1  5 o. 23d  2 11  3d  4 p. 6x  36  5x  3x  10  42  x _D 7e4k,7n0=h ) le]a[G a. 5y  1  2 y  3 32 b. 4 y  1  3y  4 ສະຫງສ.ວວນ.ລິສຂະສິດ 10 5 c. n  n  7 36 2 d. s  s  s  13 234 e. 3x 1  2x 1  4x 1 23 6 f . 1 4x  x 1  3x  5 5 4 20 4 g. x 1  x  x  2  5   x  33x 1 3 22 6 h. 1 4x  6  1 8x 12  1 2x  4 24 2 i. 1  x  3  1  x  7  0 56 j. x  2  2x 1  3  x  1 345 k. 3  5 5  x  6 3  x  1091 56 7 210 91 

!WD 7e4k,7n0=h ) le]a[G a. 5x  7  2x 1  3 x  5  4 x  3  b. 3x 1 2 1 2x2  51 x  4x2  4 c.  x  2 x  2  x  1  x 1 x 1  d. 1 1 2x2  3  x  x2 22 !!D 7e4k,7n0=h ) le]a[G dD  x 1 x  5  0 0D x  4  x  1   0  2  7D  x  2 x  3 x  4  0 ’D 2x 2x 12x  2  0 9D  x  3 x  5  x  5 lD 1 x2x 1  1 x x  6  0 ສະຫງສ.ວວນ.ລິສຂະສິດ -D 2x  4 x  4   x  23x 1  0 !@D 9q’j d;fg[y’j ;jk[aofklq,zqo]5j,oU lq,zqo.f,us^kp.9zqoL [+,u.9zqoL dD x  5   x  7  2 0D  x 1 x 1   x  2 x  2 7D 2 y  3  14  2 y 12 ’D 9n  47  n  5n  28 !#D 9qj’9a[76lj q,zqomu,j u.9zqofP;daoD dD 4x  10 a. x  3  13 0D 30x  156 b. 0,5  x 1 7D 15x  51 c. 4  1, 2  x ’D 0, 2x  0,86 d. 10  x  6, 6 92 

[qfmu @@ dkocdh39fdjP;da[lq,zqo0Ao|bj’mu,j u|bj’8q;]a[ ສະຫງສ.ວວນ.ລິສຂະສິດ !D dyf9tde dyf9tdemu ! mhk; mv’fe,uvkp5gmqkj da[|b’j l;j olk,0v’vkp5c,Fj lj;oc,j0v’]k;,uvkp5gmqkj da[g7y’j |bj’vkp50v’c,jg4qhkF vkp50v’ma’lk,7qoI;,g0qhkdaogmqkj 110xuD 9q’j -vdskvkp50v’c8]j t7qoD dD grujocmovkp50v’mhk; mv’fe fh;p x F g,nvj x gxqo8q;]a[D • x 9tgxao9eo;o]q[wfh[+L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD • x 9ts^kpd;kj !!W wfh[=L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 0D g,jnvI6h;jkvkp50v’mhk; mv’fe c,oj x • vkp50v’c,j mhk; mv’fe c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD • vkp50v’c,jg4Qk mhk; mv’fe c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD • vkp50v’ma’lk,7qoI;,g0hqkdaoma’\\qf c,oj DDDDDDDD + DDDDDDD + DDDDDDDD = 110 7D fh;pdkocdhlq,zqo0hk’gmy’gIqkwfh • vkp50v’mhk; mv’fe c,jo x =........ xu • vkp50v’c,j c,oj DDDDDDDDDDDDDDD xu • vkp50v’c,gj 4Qk c,jo DDDDDDDDDDDDDDDxu ’D 9q’j d;f7nog[yj’;kj vkp5ma’lk,7qoI;,g0qhkdaogmqkj da[ !!W xu s^n [+L • ltoAo vkp5mu4j nd8hv’0v’ mhk; mv’feF c,j c]t c,gj 4Qk c,joG DDDDDDDDDDDDDxu F DDDDDDDDDDDDDDDDxu F DDDDDDDDDDDDDDDDxu dyf9tdemu @ gryoj .shlu9j eo;oF c8j]t9eo;os^5f]noj dao9eo;o]t|b’j sq;|;j pD zqo[;d0v’lu9j e o;ooAogmqjkda[ &$*D 9qj’-vdskc8j]t9eo;ofaj’dkj ; 3fpxt8y[af8k,0Ao8vo.odyf 9tdemu !D dyf9tdemu # ];’Iv[0v’I6[lucj 9lkd|bj’gmqkj $@ cm, I6h;jk ];’pk;gmqkj lv’gmnjv0v’];’ d;hk’D 9q’j -vdsk0t|kf0v’I6[lucj 9lkdfaj’dkj ;D   93

@D .97;k, ສະຫງສ.ວວນ.ລິສຂະສິດ 0aho8vo.odkocdh39fD dkocdh39f[aosk7;oxt8y[af8k,luj0Ao8voG !D g]nvd8q;]a[ rhv,ma’;k’g’njvow00v’,aoD @D lhk’lq,zqoD #D cdhlq,zqoD $D d;f7no.9zqomujwfhIa[ c]h; .sh7e8v[D 8q;1kj ’ !D ok’ l5rkrvo ,ug’yo 10800 du[F mhk; graflt\\vo ,ug’yo9300 du[F ma’lv’7qowx -NxN,7toyflkfzh6]tsq;D Ih6;kj g’yomugj s^nv0v’ok’l5rkrvogmqjkda[lv’gmnvj 0v’ g'yomujgs^nv0v’ mhk; graflt\\voD 4k,;kj ]k7kxN,7toyflkfc,jogmqjk.fL [qfcdhG • g]nvd8q;]a[ { ;k’.sh x Xdu[? c,jo]k7kxN, g-y’j x gxao9eo;o[;dD { 9eo;og’yo0v’ ok’l5rkrvo pa’gs^nv c,jo 10800 − x { 9eo;og’yo0v’ mhk; graflt\\vo pa’gs^nv c,oj 9300 − x • lhk’lq,zqoG Ih6;jk 9eo;og’yomugj s^nv0v’ok’l5rkrvo gmqkj lv’gmnvj 9eo;og’yomugj s^nv0v’ mhk; graflt\\vo gIqk9twfhlq,zqo 10800 − x = 2(9300 − x) • cdhlq,zqo 10800 − x = 2(9300 − x) 10800 − x = 18600 − 2x    x = 18600 − 10800 x = 7800 du[D   94