[q;]u cdh;;q’lk

Texte intégral

(1)c[[IPo. 7toyflkf -Ao,afmtpq,lbdlk xumu @. IP[IP’3fpG. IDl clj; ,6o]tfqd lyo me,tgm3; [q;]u cdh;;q’lk vjvocdh; lu;yw-. g0Qk\hk3fpG. [q;]u cdh;;q’lk. d;fcdh3fpG. cv39 vkfzkl5d ly’7e dyj’ltfk. dt-;’lbdlkmydko c]t du]k lt4k[ao7Qo7;hk ;ymtpklkf dkolbdlk ;P’9ao @W!!.

(2)

(3) 7eoe xB,c[[IPo7toyflkf ,D@ gsÂooU wfhIP[IP’0Bogrnjvlvo8k,sâdl6f0v’ 7toyflkf-Ao,afmtpq,lbdlk8vo8Qolt[a[xa[x5’xu @WW_D .o0Ao,afmtpq,lbdlk xumulv' X,D@? cdo0v’sâdl6fxtdv[,u ( rkd 7nG 9eo;oxqddt8y c]t 9eo;o 9y’F dkog7yj’7nF g]dde]a’ c]t leo;oF lq,zqo c]t vtlq,zqo0Ao|bj’muj,u|bj’ 8q; ]a[F I66[lujc9{I6[lk,c9 c]t ;q’,qoF vaf8klj;orq;rao{g]dlj;oIhvp c]t ,kf 8klj;oF sâd gdo8kc]afF lt4y8yD xB,c[[IPosq;oU wfhIa[dkoIP[IP’0Bo grnjv-j;p.sh76lvo8k,sâddko dkoIPo{dkolvoc[[gvqkoadIPogxao.9dk’ 3fp.shz6hIPo7Qorq[7;k,I6hfh;p 8qogv’ rkp.8hdko-Uoe0v’76D faj’oAoF c8j]t[qf0v’xB,c[[IPosq;oU 9bj’xtdv[ fh;p lk,0Ao8vo7nG dyf9tdeF .97;k, c]t [qfg/ydsafD grnjv-j;p.shoadIPowfh IPoI6hfh;p8qogv’s^kp0BoF .odkoIP[IP’xN,sq;oU 9bj’wfh[ao95gvqk7ecotoe s^n 7e8v[0v’[qfg/ydsafw;hmhkpgsÂ,D dkola[-hvogoNv.o c]t [qfc,jovu’.ljltrk ;t7;k,gxao9y’0v’dkoIPo{dkolvo.oxtgmfgIqkF sâddko0v’dkoIPo 7toyflkfle]a[-Ao ,D@F sâd9yf8t;ymtpkdjP;da[7;k,Ia[I6hF sâd;y-k768k,95f \kp c]t sâd;y-k76muj9ecod7;k,c8d8jk’ ]ts;jk’[5d7qo 3fplq,mq[da[;ymu dko96’.9oadIPoD 1jk’.fd=8k, dkoIP[IP’xB,sq;oUc,jo.-hg;]klAo c]t d+7q’[+xklt9kd 0=h0kf8qd[qdrjv’wfhF ltoAo 9bj’0=7;k,Ij;,,noemjkoz6hmujoe.-hxB,sq;oU 9qj’-j;pg[yj’ 95f[qdrjv’ s^n 95fzyfrkf c]h;lqj’7e7yf7egsao0v’mjkowxpa’lt4k[ao7Qo7;hk ;ymtpklkfdkolbdlkD r;dgIqk9t4n;jkm5d7e7yfgsao0v’mjkogxao0=h,6omuj,u75o7jk c]t gxaodko-j;pxa[x5’75ootrk[dkolbdlkr;dgIqk.shl6’0BoD lt4k[ao7Qo7;hk;ymtpklkfdkolbdlk. d.

(4) ! ! & ( !! !& !_ @# @( #! #& #_ $# $# $( &! &_ &_ *& *( (! (& )! )! )& _#. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. rkdmu ! 9eo;oxqddt8y c]t 9eo;o9y’DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu ! g]dlj;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @ mt;u76o c]t v5xt76o0v’g]dlj;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu # dkosp=h c]t dko0Bor6fIj;,g]dlj;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu $ dkoxP[mP[g]dlj;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu & dko[;d c]t dko]q[g]dlj;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu * dko[;d c]t dko]q[g]dlj;o X8+? DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu ( dko76og]dlj;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu ) dkoskog]dlj;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu _ 9eo;oxqddt8y DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !W 9eo;oxqddt8y X8+?DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !! 9eo;ovtxqddt8y DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD rkdmu @ dkog7yj’7n DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !@ dkog7yj’7nmP[.ljg,af|bj’ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !# dkog7yj’7nmP[.ljg,af|bj’ X8+? DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !$ dkog7yj’7n0v’,5,mP[.ljcdo c]t mP[.ljg,af DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD rkdmu # g]dde]a’ c]t leo;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !& g]dde]a’ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !* g]dde]a’ X8+? DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !( leo;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !) dko7eo;odjP;da[leo;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu !_ dkocpdlj;o76o.oleo;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD rkdmu $ lq,zqo c]t vtlq,zqo0Ao|bj’mju,u|bj’8q;]a[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @W ltg\uzqo DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @! lq,zqo0aho|bj’muj,u|bj’8q;]a[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @@ dkocdh39fdjP;da[lq,zqo0Ao|bj’muj,u|bj’8q;]a[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD. ສດິ. lk]t[ko. 0.

(5) ສດິ. __ !W& !!! !!& !!_ !@# !@# !@( !#! !#& !#_ !$# !&!. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. [qfmu @# vtltg|uzqo DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @$ vtlq,zqo0Ao|bj’muj,u|bj’8q;]a[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @& dkocdh39fdjP;da[vtlq,zqo0aho|bj’muj,u|bj’8q;]a[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @* lq,zqomuj,u7jkle[6o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @( vtlq,zqomuj,u7jkle[6o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD rkdmu & I6[lujc9F I6[lk,c9 c]t ;q’,qo DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @) I6[lujc90hk’0t|ko DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu @_ I6[lujc9lkd DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu #W I6[fvd9ao c]t I6[9t85]af DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu #! I6[7k’\6 DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu #@ I6[lk,c9 DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu ## glAomujle7ao.oI6[lk,c9 DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu #$ ];’Iv[ c]t goNvmuj0v’I6[lujc9lkdF I6[9t85]af c]t I6[fvd9aoDDD [qfmu #& ];’Iv[ c]t goNvmuj0v’I6[lujc90hk’0t|koF I6[lk,c9 c]t I6[ 7k’\6 DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu #* ,5,.o c]t ,5,ovd0v’I6[lk,c9 DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu #( ,5,.o0v’I6[lujuc9l;f DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu #) ;q’,qo c]t czjo,qoDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD rkdmu * vaf8klj;orq;raoF g]dlj;oIhvp c]t ,kf8klj;o DDDDDDDDDDDDDDD [qfmu #_ |hkrP’8q;xtlko DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu $W dkogxaovaf8klj;orq;raodq’ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu $! vaf8klj;orq;raoxUo DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu $@ g]dlj;oIhvp DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu $# ,kf8klj;o DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD rkdmu ( sâdgdo8kc]af DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu $$ dkolkp8k,];’muj.sh,k DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 7. !&& !&_ !*# !*( !(! !(! !(_ !)& !_! !_( @W# @W#.

(6) @W( @!! @!_ @@# @@( @@( @#& @$#. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. [qfmu $& glAo-njzjkog,afg7yj’dk’0v’lv’0hk’0v’I6[lk,c9 DDDDDDDDDDDDDDDDDDDDDDDD [qfmu $* sâdgdo8kc]af (Thalès) DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu $( I6[lk,c97hkp7ndao DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu $) d=]tou7hkp7n0v’I6[lk,c9 DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD rkdmu ) lt4y8y DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu $_ dkogda[de0=h,6o c]t dkoltg|u0=h,6ofh;p8k8t]k’ DDDDDDDDDDDDDDDDDDDDDDDD [qfmu &W dkoltg|u0=h,6ofh;pI6[c8h, DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD [qfmu &! dkoltg|u0=h,6ofh;pI6[c8h, X8+? DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD. ’.

(7) rkdmu ! 9eo;oxqddt8y c]t 9eo;o9y’ [qfmu ! g]dlj;o dyf9tde dyf9tdemu ! 9qj’-vdg]dlj;omujgmqjkda[9eo;omqfltoypq,8+wxoUG 8,5 = 0, 2 =. 10 10. = =. 100. =. 1000. 0,75 =. 2 =. 1 …. 1,25 =. 100. 100. = =. 750. 500. = =. 4. 4. ສດິ. dyf9tdemu @ gryjo,ug0Qk|q,lk,dhvofaj’I6[]j5,oUG. A. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !D. C. B. dD 9qj’c8h,g0Qk|q, A 7no.\jc]h;mklurkdlj;oltcf’. 1 4. c]t rkdlj;oltcf’. 1 8 1 6. S. 1 0v’g0Qk|q, 2. 0D 9qj’c8h,g0Qk|q, B 7no.\jc]h;mklurkdlj;oltcf’ 7D 9qj’c8h,g0Qk|q, C 7no.\jc]h;mklurkdlj;oltcf’. S. 3 8 2 6. c]t c]t. dyf9tdemu # dD g0Qk|q, |ad &*W gF rkdlj;o 0D. 2 0v’g0Qk|q,9t|adgmqjk.fF 5. 3 8. |adgmqjk.fL. 4hkg0Qk|q,dhvooAoskd|ad $&W gL 1. 1 4 5 6.

(8) dyf9tdemu $ g0Qk|q,c8j]tdhvo]j5,oU|ad *#Wg D 9qj’-vdskoE|ad0v’rkdlj;omuj.ljlu0v’ g0Qk|q,D dD 0D. .97;k, !D g]dlj;o. 9eo;or6f0v’g]dlj;o 4 9. 0v’g0Qk|q,. 4 9. ສດິ. r6f0v’g]dlj;o 0v’g0Qk|q,. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. @D. @D dko76o9eo;o|bj’da[g]dlj;o 76o9eo;o|bj’da[g]dlj;o. a b. c,joxt8y[affaj’oUG. ¾ 76o9eo;ooAoda[ a c]h;sko.sh b s^n ¾ gvqk9eo;ooAosko.sh b c]h;76oda[ 8q;1jk’ !D. @D. 4 200 × 4 800 = = = 160 5 5 5 s^n 200 × 4 = 200 × 4 = 40 × 4 = 160 5 5 5 1,2 × 5 6 1,2 × = = 7 7 7 200 ×. 2. a.

(9) #D. [qfg/ydsaf. !D. 9qj’0Po9eo;o8+wxoUrkp.8hI6[c[[g]dlj;oD 3) 10,5. 4) 2,14. 5) 10,811. 6) 2,201. 9qj’0Po9eo;o8+wxoU rkp.8hI6[c[[9eo;omqfltoypq, 1). 5 8. 2). 4). 2 125. 5). 9qj’76og]d8+wxoUG 1). 18 25 6 75. 14 ×3 15. 5 4 5 4) 21 × 7 2) 2,3 ×. 10 1000 17 5) 3,12 × 12. 3) 0,1 ×. $D. &D. 6) 4,03 ×. 3 1000. .oI6[]j5,oUF 9qj’ltcf’g]dlj;omujdq’da[rkdlj;omujmkluD dD 0D 7D ’D. 9D. 1j6mjvo-nj [AB] muj,u];’pk; 75 mm . 9qj’;k’G dD g,af C 3fp.sh. *D. 831 15 10 6) 125. 3). ສດິ. #D. 2) 3,4. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. @D. 1) 2,1. AC =. 2 AB 3. 0D g,af D 3fp.sh. 9qj’8nj,xt3spd7eg;Qk]j5,oUG. dD 1 0v’,5,rP’,uDDDDDDDDDDDDDvq’lk. 0D. 7D. ’D. 4 1 0v’,5,rP’,uDDDDDDDDDDDDDvq’lk 5. 3. 1 3 1 6. BD =. 3 AB 5. 0v’,5,lkd,uDDDDDDDDDDDDDvq’lk 0v’,5,lkd,uDDDDDDDDDDDDDvq’lk.

(10) 9qj’9afg]dlj;o8+wxoU.lj8k8t]k’ g]dlj;o|hvpdj;k !. g]dlj;ogmqjkda[ !. mkh; le]u,u\kd[u @!W |j;pS ]k;gIafglp 3 D 4k,;kj\kd[u]k;glp9ad|j;pL 7. c]t pa’gs^nv9ad|j;pL _D. g]dlj;os^kpdj;k !. -k;l;ozh6|bj’gda[\kdoafwfh. 240 kg S. ]k;0kpc]h; 3 D 5. 4k,;jk\kdoaf]k;pa’gs^nvgmqjk.fL !WD g0Qk|q,vq,4q’|bj’,u _* dhvoS dD 4hkdyowxc]h; 0D 4hkdyowxc]h;. 5 F 4k,;jkg0Qk|q,vq,pa’gs^nv9addhvoL 12 3 F 4k,;jkg0Qk|q,vq,pa’gs^nv9addhvoL 8. ສດິ. )D. 21 17 19 21 19 15 17 ; ; ; ; ; ; 23 17 17 19 19 19 19. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. (D. !!D ws|j;p|bj’[ao95oEwfh !@ ]ufD dD 4hk[ao95c8j 0D 4hk[ao95c8j. 2 3 3 4. 9t,uoE9ad]ufL 9t,uoE9ad]ufL. !@D .o’ko;k’ltcf’]qfgdqjkcsj’|bj’ ,u]qfma’\qf (@W 7aoD ]qflu2hk,u 1 S ]qflucf’,u 3. 1 4. c]t ]qflu0k;,u. 5 D 12. 9qj’-vdsk9eo;o]qfc8j]txtgrf Xlu?D. !#D mkh; d=]tdaoF mkh;la’mv’ c]tok’ verk Ij;,dao]q’mbogIafm5]tdyf,u,6o7jk !! W!* WWWdu[D c8j]t7qo,umbo 4k,;jkmboc8j]t7qo,ugmqjk.fL !$D 9vdgxqjk|j;p|bj’,u[=]y,kf[ao95. 5 S 1 c]t 2 8k,]efa[D 24 8 3. 20 dl F. gryjo4vdoE\kdw,hlqf8jk’M.lj9vdfaj’. djk;faj’oUG oE\kddhP’ 1 S oE \kdoaf 4. 3 S 10. oE\kd,j;’ 2 S oEvhvp. 4k,;jk [=]y,kf0v’oE\kdw,hc8j]t-toyf,ugmqjk.fL 4. 5. 1 D 20.

(11) [qfmu @ mt;u76o c]t v5xt76o0v’g]dlj;o !D. dyf9tde dyf9tdemu ! .shlv’g]dlj;o. 2 5. c]t. 4 D 15. dD 9qj’[vdmt;u76oIj;,|hvpl5f0v’ @ c]t $D 0D 9qj’[vdv5xt76oIj;,s^kpl5f0v’ & c]t !&D mt;u76oIj;,|hvpl5f0v’ @ c]t $ … 7D 9qj’8nj,.lj[jvo,9Eg,afG = v5xt76oIj;,s^kpl5f0v’ & c]t !& … 4 gvUo;jkmt;u76oIj;,|hvpl5f0v’ 2 c]t 4 D 5. 5. 15. dyf9tdemu @ 2 3. c]t 4 D 7. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. dD 9qj’[vdv5xt76oIj;,s^kpl5f0v’ @ c]t $D 0D 9qj’[vdmt;u76oIj;,|hvpl5f0v’ # c]t (D v5xt76oIj;,s^kpl5f0v’ @ c]t $ … 7D 9qj’8nj,.lj[jvo,9Eg,afG = mt;u76oIj;,|hvpl5f0v’ # c]t ( … 2 gvUo;jkv5xt76oIj;,s^kpl5f0v’ 2 c]t 4 D. ສດິ. .shlv’g]dlj;o. 21. @D. 3. 7. .97;k, !D mt;u76oIj;,|hvpl5f0v’g]dlj;o mt;u76oIj;,|hvpl5f0v’lv’ s^n s^kpg]dlj;o gmqjkda[zqosko]ts;jk’mt;u76oIj;, |hvpl5f0v’9eo;or6f c]t v5xt76oIj;,s^kpl5f0v’r6fD mt;u76oIj;,|hvpl5f0v’g]dlj;o 8q;1jk’ !D. =. mt;u76oIj;,|hvpl5f0v’9eo;or6f v5xt76oIj;,s^kpl5f0v’r6f. -vdmt;u76oIj;,|hvpl5f0v’ 5. 2 9. c]t. 5 D 12.

(12) gIqk,u mt;u76oIj;,|hvpl5f0v’ 2 c]t 5 c,jo 10 v5xt76oIj;,s^kpl5f0v’ 9 c]t 12 c,jo 3 faj’oAo. 10 3. c,jomt;u76oIj;,|hvpl5f0v’. 2 9. c]t. 5 D 12. @D v5xt76oIj;,s^kpl5f0v’g]dlj;o v5xt76oIj;,s^kpl5f0v’lv’ s^n s^kpg]dlj;o gmqjkda[gmqjkda[zqosko]ts;jk’v5xt 76oIj;,s^kpl5f0v’9eo;or6f c]t mt;u76oIj;,|hvpl5f0v’r6fD v5xt76oIj;,s^kpl5f0v’g]dlj;o. v5x76oIj;,s^kpl5f0v’9eo;or6f mt;u76oIj;,|hvpl5f0v’r6f. 12 c]t 2 5 25 v5xt76oIj;,s^kpl5f0v’ 12 c]t 2. -vdv5xt76oIj;,s^kpl5f0v’. 9bj’wfh. #D. [qfg/ydsaf. !D. .shlk,g]dlj;o. 2 25. 5. c]t. 25. c,jo. c,jov5xt76oIj;,s^kpl5f0v’. 12 5. 2 3 4 , c]t D 5 10 15. dD 9qj’[vdmt;u76oIj;,|hvpl5f0v’ @F # c]t $D 0D 9qj’[vdv5xt76oIj;,s^kpl5f0v’ &F !W c]t !&D 7D 9qj’4vogvqkmt;u76oIj;,|hvpl5f0v’. 2 7. 9qj’-vdv5xt76oIj;,s^kpl5f c]t mt;u76oIj;,|hvpl5f0v’. #D. 2 7 32 D , c]t 3 24 45 9qj’-vdv5xt76oIj;,s^kpl5f0v’ 25 , 15 c]t 10 D 3 7 9 5 21 11 9qj’-vdr6fIj;,0v’g]dlj;o , c]t D 48 120 90 9qj’7af9hvog]dlj;o 126 3fp.-hv5xt76oIj;,s^kpl5f0v’ 126 189. &D *D. 25. c]t. 2 25. 2 3 , c]t 4 D 5 10 15. @D. $D. c,jo 2 D. c]t. 8 D 21. 9qj’-vdmt;u76oIj;,|hvpl5f0v’. 6. ສດິ. gIqk,u mt;u76oIj;,|hvpl5f0v’. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 8q;1jk’ @D. =. c]t 189 D.

(13) [qfmu # dkosp=h c]t dko0Bor6fIj;,g]dlj;o !D. dyf9tde dyf9tdemu ! dD 9qj’8nj,9eo;omujg\ktlq,.lj[jvo9Eg,afD 210 42 × = 165 33 ×. =. 42 14 × = 33 11 ×. =. 0D 9qj’7af9hvog]dlj;o]5j,oUG 30 24 42 ; ; ; 36 32 18. 27 ; 81. 14 11 75 45. dD 0D. 2 … … = = 3 6 9 2 2 ×… 2 ×… … × 4 2 ×… = = = = 3 3 × 2 3 ×… … × 4 … × 5. =. … …. =. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. dyf9tdemu @ 8k,I6[c8h,]5j,oUF 9qj’8nj,9eo;omujg\ktlq,.lj[jvo9Eg,afD. … …. dyf9tdemu # dD 9qj’8nj,9eo;omujg\ktlq,.lj[jvo9Eg,afD 5 5 ×… … ; = = 6 6 × … 12. 7 7 ×… … = = 4 4 × … 12. 0D 9qj’0Bor6fIj;,g]dlj;o]5j,oUG. @D. 3 ; 4. .97;k,. 7 ; 8. 3 ; 24. 5 2. !D dko7af9hvog]dlj;oD dko7af9hvog]dlj;o c,jodko7aflj;o76oIj;,0v’9eo;or6f c]t 0v’r6fvvdD 7.

(14) 8q;1jk’ !D @D. 30 6/ × 5 5 = = 36 6/ × 6 6 138 2/ × 3/ × 23 23 = = 24 2/ × 3/ × 4 4. @D dko0Bor6fIj;,0v’g]dlj;oD dko0Bor6fIj;,0v’g]dlj;o c,jodkogIaf.shg]dlj;o,ur6fgxao9eo;ofP;daoD. #D g]dlj;ogmqjkdaoD 4hk. a b. ສດິ. @D. 7 c]t 5 18 12 7 7 × 2 14 = gIqk,u = 18 18 × 2 36 c]t 5 = 5 × 3 = 15 12 12 × 3 36 9bj’wfh 7 c]t 5 ,ur6fIj;,fP;daoc,jo #*D 18 12 0Bor6fIj;,g]dlj;o 8 c]t 5 3 24 8 8 × 8 64 = gIqk,u = 3 3 × 8 24 9bj’wfh 8 c]t 5 ,ur6fIj;,fP;daoc,jo @$D 3 24. 0Bor6fIj;,g]dlj;o. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 8q;1jk’ !D. gxaog]dlj;o|bj’S k gxao9eo;ome,t-kf8k,.98jk’ WS ka a = kb b 48 c]t 16 15 5 48 3/ × 16 16 gIqk,u = = 15 3/ × 5 5 ltcf’;jk 48 = 16 15 5 5 c]t 15 4 12 5 5 × 3 15 gIqk,u = = 4 4 × 3 12. gIqkwfh. 8q;1jk’ !D. @D. 8.

(15) ltcf’;jk. 5 15 = 4 12. #D. [qfg/ydsaf. !D. 9qj’8nj,9eo;omujg\ktlq,.lj[jvo9Eg,afD. @D. 9qj’8nj,g7njv’\kp < = s^n dD 0D. #D. 7D. 0D 7D. 100 0 …… 1000 10 48 8 …… 18 3. 8 18 32 ; ; 12 33 96 33 15 27 ; ; 44 20 24. 0D ’D. 5 c]t 4 7 c]t 8 9 c]t 21. 10 ; 3 11 ; 16 1 ; 3. − 2 − 120 81 ; ; 4 − 360 − 9 24 14 36 ; ; 30 16 24. 7 c]t 1 ; 5 4 5 c]t 2 ; 18 27 −3 5 c]t 4 6. .og]dlj;o]5j,oU ,ug]dlj;o0=h.fgmqjkdaoL dD 7D 9D. *D. .lj[jvo9Eg,afD. 72 8 …… ; 63 9 400 1 …… ; 800 2. 9qj’0Bor6fIj;,g]dlj;o]j5,oUG dD. &D. ≠<. 9qj’7af9hvo]dlj;o]5j,oUG dD. $D. 6 42 …… ; 5 35 12 1 …… ; 32 3. 7 28 … = = 5 … 100 16 … 1,6 = = 100 25 …. dD. 28 c]t 35 12 c]t 16 55 c]t 33. 32 40 24 18 10 6. 8 c]t 10 12 15 ’D 33 c]t 11 21 3 7 14 lD c]t 16 32 2 9qj’0Pom5dMg]dlj;omujgmqjkda[ c]t ,ur6f|hvpd;jk 3. 0D. 9. @!D. ສດິ. 0D. 76 … 19 = = ; 12 6 … 1,5 … 0,3 ; = = 2,5 25 …. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. dD.

(16) 0D 9qj’0Pom5dMg]dlj;omujgmqjkda[. 5 4. c]t ,ur6f|hvpd;jk $WD. 16 c]t ,u9eo;or6fgmqjk @D 24 ’D 9qj’0Pog]dlj;omujgmqjkda[ 18 c]t ,ur6fgmqjk &D 30 gryjo.shg]dlj;o 7 ; 6 ; 70 ; 9 ; 63 ; 21 3 4 30 5 27 9 .o[aofkg]dlj;ogmy’oAo ,ug]dlj;o.fmujgmqjkda[ 35 ? 15 6 0,1 4 15 6 6 ; ; ; .o[aofkg]dlj;o8+wxoUG ; ; ,ug]dlj;o.fL 24 8 12 0,2 16 20 0D gmqjkda[ 1 S 7D gmqjkda[ 3 D dD gmqjkda[ 1 S 4 2 4 1 9qj’-vdg]dlj;omujgmqjkda[ c]t ,ur6fc,joG 2. 7D 9qj’0Pog]dlj;omujgmqjkda[. _D. dD $S. 0D !WS. 7D !)S. !WD 9qj’-vdg]dlj;omujgmqjkda[ dD )S. 0D !@S. 3 4. 7D @)S. ’D #WD. c]t ,ur6fc,joG ’D #*D. 10. ສດິ. )D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. (D.

(17) [qfmu $ dkoxP[mP[g]dlj;o !D. dyf9tde dyf9tde ! dD 9qj’8nj,9eo;omujg\ktlq,.lj[jvo9Eg,af grnjv.shwfhg]dlj;ogmqjkdaoD 3 = 5. 75. ;. 7 91 = 3. 0D 9qj’7af9hvog]dlj;o]5j,oUG. dD 0D. 1 5 ; 3 3 6 6 ; 10 5. = < .lj[jvo9Eg,afD 6 8 13 5 5 5 3 3 8 2. 7 5 ; 12 12 12 1; 12. dyf9tdemu # dD 9qj’0Bor6fIj;,g]dlj;o]5j,oUG 9 ; 4. 2 ; 5. 7 ; 20. 5 2. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. dyf9tdemu @ 9qj’8nj,g7njv’\kp < > S < sn. ສດິ. 32 35 42 30 90 ; ; ; ; 36 60 108 20 75. 0D 9qj’9af]Pog]dlj;o0hk’gmy’oAoc8j|hvpsks^kpD. @D. .97;k, dkoxP[mP[g]dlj;o dD g]dlj;omuj,ur6f7ndao { 4hkg]dlj;o.f,u9eo;or6fs^kpd;jk c,jog]dlj;ooAos^kpd;jkD 8q;1jk’ !D. xP[mP[ gIqk,u. 7 3. c]t. 7 > 5F. 5 3. faj’oAo. 7 5 > 3 3. 11.

(18) @D. xP[mP[ gIqk,u. 4 15. c]t. 4 < 11 F. 11 15 4 11 < 15 15. faj’oAo. 0D g]dlj;omuj,u9eo;or6f7ndao { 4hkg]dlj;o.f,ur6fs^kpd;jk c,jog]dlj;ooAo|hvpd;jkD xP[mP[ gIqk,u @D. c]t. 8 > 5F. xP[mP[ gIqk,u. 9 8. 3 17. 9 5 9 9 < 8 5. faj’oAo c]t. 17 > 14 F. 3 14. faj’oAo. 3 3 < 17 14. 8q;1jk’ !D. xP[mP[. 4 5. c]t. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 7D g]dlj;omuj,ur6f8jk’dao 1kdxP[mP[g]dlj;omuj,ur6f8jk’dao gIqk8hv’0Bor6fIj;,djvoc]h;9bj’xt8y[af dkoxP[mP[7nda[g]dlj;omuj,ur6f7ndaoD 3 7. 4 4 × 7 28 c]t 3 = 3 × 5 = 15 = = 5 5 × 7 35 7 7 × 5 35 gsao;jk 15 < 28 faj’oAo 3 < 4 35 35 7 5 xP[mP[ 7 c]t 5 18 12 .og,njv 7 = 7 × 2 = 14 c]t 5 = 5 × 3 = 15 18 18 × 2 36 12 12 × 3 36 14 15 gsao;jk < F faj’oAoF gIqkwfh 7 < 5 36 36 18 12. .og,njv. @D. \kpgsfG gIqklk,kfxP[mP[g]dlj;o fh;pdko76ow0;jD 4 c]t 3 7 5 4 3 > 5 7. 8q;1jk’G xP[mP[ faj’oAo. ສດິ. 8q;1jk’ !D. gIqk,u. 4 × 7 = 28. 12. c]t. 5 × 3 = 15 gsao;jk 28 > 15.

(19) #D. [qfg/ydsaf. !D. 9qj’9af]Pog]dlj;o]5j,oUc8j|vhpsks^kpG dD 7D ’D. 9qj’xP[mP[g]dlj;o]j5,oUG 0D 7D. 1 c]t 7 ; 6 36 1 c]t 1,9 ; 1,5 3 1 c]t 0,1 ; 10. dD gryjo[;d !W .lj9eo;or6f c]t 0D 9qj’xP[mP[. $D. 7 10. 1 8. c]t. 0D 7D. 3 ; 4 2 ; 5 11 ; 12. 5 ; 8 7 ; 10 2 ; 3. 7 48. 3 c]t 7 5,1 10,2 1 c]t 2 0,5 r6f0v’g]dlj;o 7 D 9qj’0Pog]dlj;o.\jmujwfhIa[D 10. c]t g]dlj;o.\joAo. 9qj’0Bor6fIj;,g]dlj;o]j5,oU c]h;9af]Poc8j|hvpsks^kpG dD. &D. 1 c]t 5 ; 3 18 1 c]t 5 ; 0,8 3,2 1 c]t 10 ; 0,1. ສດິ. dD. #D. 3 2. 4 9 5 ; ; 16 2 3 9 3 1 ; ; 20 4 2 5 3 7 ; ; 6 4 12. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. @D. 5 7 1 3 12 0D 3 ; 3 ; 3 ; 3 ; ; ; ; ; 8 8 8 8 8 8 12 9 6 2,5 1 20 1,5 4 1,7 ; ; ; ; ; 7 7 70 7 7 7 12 1 3 23 1,5 2 ; ; ; ; ; 4,3 43 4,3 4,3 43 4,3. 9qj’8nj,9eo;ofP;dao.lj[jvo,9Eg,afD dD. … 6 … < < 8 … 5. 4 … 7 < < … 5 … 2 3 ; ,ad[ko[h;’ c]t 3 4. 0D. *D. oadIPoshv’ ,@ shv’|bj’,adg8t[ko. (D. 4k,;jkoadIPoshv’oU,addy]kxtgrf.fs^kpd;jkL lv’I6[lujc9lkd R1 c]t R2 ,u];’d;hk’gmqjkdao c]t gmqjkda[. 13. ,ad[kolqj’. 7 12. 3,5 cm c]t. ,u.

(20) )D. goNvmujgmqjkda[ 19,6 cm 2 c]t 19,25 cm 2 8k,]efa[D 4k,;jkI6[lujc9lkd.f,u];’pk; pk;d;jkL oadIPoshv’ ,@}! .ljc;jo8k !$ 7qo.o9eo;o #W 7qoF lj;oshv’ ,@}@ .lj !) 7qo.o9eo;o #) 7qoD 4k,;jkoadIPoshv’.f.ljc;jo8ks^kpd;jkdaoL. _D. .odko3po\kd[u|j;p|bj’]q’05, • mhk; cl’3po]q’ & gmnjv.odko3po _ gmnjv • mhk; 7e,u3po]q’ & gmnjv.odko3po ) gmnjv • mhk; ]5o3po]q’ $ gmnjv.odko3po _ gmnjv • mhk; gof3po]q’ * gmnjv.odko3po ) gmnjv 4k,;jk.o9eo;oluj7qooU c,jo.z3pogdaj’d;jkdaoL !WD .oI6[lk,c9lkdF .sh A c,jogoNvmujF b c,jorNo c]t h c,jo];’l6’mujrq; 2A b. b (cm). #. (. !!. h (cm). h. h’. h”. ສດິ. h=. h. 9kd8k8t]k’0hk’gmy’F gryjo.shlk,I6[lk,c9muj ,ugoNvmujgmqjkdao c]t gmqjkda[ 13 cm 2 dD 9qj’7yfw]j];’l6’0v’c8j]tI6[lk,c9.oI6[Ijk’g]dlj;oD 0D 9qj’9af]Polk,g]dlj;ooAoc8j|hvpsks^kpD 7D 4k,;jkI6[lk,c9.f,u];’l6’ l6’d;jkdaoL. 14. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. raodaofh;pl6f. A. b.

(21) [qfmu & dko[;d c]t dko]q[g]dlj;o dyf9tde dyf9tdemu ! .oI6[]5j,oUF 9qj’8nj,g]dlj;omujg\ktlq,.lj[jvo9Eg,afD. dD rkdlj;omuj0uf]kpdq’da[g]dlj;o DDDDDDDDDD 0D rkdlj;omujmkludq’da[g]dlj;o DDDDDDDDDD 7D rkdlj;omuj0uf]kp c]t rkdlj;omujmkludq’da[g]dlj;o DDDDDDDDD s^n DDDDDDDDDD \kp7;k,;jk. 1 5 " " + = = 10 10 10 ". \kp7;k,;jk. 10 6 " " − = = 10 10 10 ". dyf9tdemu @ ok’ 8kfegIafoExajogrnjv.shohv’MdyoS mevyf]k;xajo\kdoaf xajo\kd]t,5f8nj,. 6 7. ]ufD. dD 4k,;jk ok’ 8kfegIafoExajowfhma’\qf9ad]ufL 0D ohv’lk;0v’]k;dyowxc]h;. 2 7. ]ufF. 4k,;jkoExajomujok’8kfegIafpa’gs^nv9ad]ufL dyf9tdemu # 9qj’7eo;og]dlj;o]j5,oUG dD 0D. 15 18 + ; 7 7 13 8 + ; b b. 11 5 23 18 10 7 − ; + ; − 12 12 2 2 23 23 a b 8 6 a b + ; − ; − 9 9 b b 3 3. 15. ສດິ. ’D rkdlj;omuj[+mklu c]t [+0uf]kpdq’da[g]dlj;o DDDDDDDDD s^n DDDDDDDDDD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !D. 4 7. ]ufF 8+,k]k;.

(22) @D. .97;k,G !D dko[;dg]dlj;omuj,ur6f7ndao dko[;dg]dlj;omuj,ur6f7ndao c,jogryjo[;d9eo;or6foedaoc]h;Iadlkr6foAow;hD 4hk;jk. 8q;1jk’. a c]t c c,jog]dlj;omuj,ur6fc,jo b b b a c a+c + = b b b 12 8 12 + 8 20 !D + = = 11 11 11 11 2 13 2 + 13 15 5 @D + = = = 9 9 9 9 3. g-yj’. b≠0. gIqkwfhG. @D. 8 1 8 −1 7 − = = 3 3 3 3 12 17 12 − 17 − 5 1 − = = =− 25 25 25 25 5. #D. [qfg/ydsaf. !D. 9qj’7eo;og]dlj;o]j5,oUG 4 3 + ; 5 5 4 3 d= − ; 5 5 a=. @D. 5 3 + 16 16 5 3 f = − 16 16 c=. 9qj’[;dc]h;7af9hvog]dlj;o]5j,oUG 6, 4 1,5 ; + 24 24 8,5 0,5 d= − ; 12 12. a=. #D. 7 5 + ; 4 4 7 5 e= − ; 4 4 b=. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 8q;1jk’ !D. ສດິ. @D dko]q[g]dlj;omuj,ur6f7ndao dko]q[g]dlj;omuj,ur6f7ndaoc,jogryjo]q[9eo;or6foedaoc]h;Iadlkr6foAow;hD. 6, 2 2,8 ; + 21 21 7,8 1,8 e= − ; 15 15. b=. 9qj’8nj,9eo;omujg\ktlq,.lj[jvo9Eg,afD 16. 7 5 + 1,2 1,2 28 3 f = − 0,5 0,5. c =.

(23) 15 11 " 18 " = + = − 7 7 7 7 7 8 " 20 " c) 1 = + = − 11 11 11 11. 19,5 11 " 30 " = + = − 13 13 13 13 13 17 " 35 " d) 2 = + = − 8 8 8 8. a). 9qj’7eo;og]dlj;o]5j,oUG. &D. 9qj’7eo;og]dlj;o]5j,oUG ⎛ 2 4 ⎞ 13 a =⎜ + ⎟+ ; ⎝3 3⎠ 3 ⎛5 1⎞ c = 2+⎜ + ⎟ ; ⎝4 4⎠ ⎛7 ⎞ 5 e = ⎜ + 1⎟ + ; ⎝ 12 ⎠ 12. *D. 15 ⎛ 13 1 ⎞ +⎜ − ⎟ ; 2 ⎝ 2 2⎠ ⎛ 29 1 ⎞ c = ⎜ + ⎟ − 2; ⎝ 12 12 ⎠. 16 ⎛ 4 1 ⎞ −⎜ − ⎟ 3 ⎝ 3 3⎠ ⎛ 25 9 ⎞ 1 d =⎜ − ⎟+ ⎝ 4 4⎠ 4 b=. 9qj’7eo;og]dlj;o]5j,oUG 17 2 + 2× ; 15 15 ⎛ 9 5 ⎞ c = 7×⎜ − ⎟ ; ⎝ 17 17 ⎠ a=. )D. 5 ⎛4 9⎞ +⎜ + ⎟ 8 ⎝8 8⎠ ⎛ 8 15 ⎞ d = ⎜ + ⎟ +1 ⎝3 3 ⎠ 7⎞ 4 ⎛ f = ⎜5 + ⎟ + 9⎠ 9 ⎝ b=. 9qj’7eo;og]dlj;o]5j,oUG a=. (D. 2 8 7 + + 13 13 13 7 2 4 d= + − 15 15 15 2 1 11 f = − + 9 9 9 b=. ສດິ. 13 5 1 + + ; 4 4 4 20 2 17 c= − − ; 3 3 3 12 8 2 e= − + ; 7 7 7 a=. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. $D. b). ⎛ 11 1 ⎞ b = 5×⎜ + ⎟ ⎝ 5 5⎠ ⎛7 9⎞ d = 15 × ⎜ + ⎟ ⎝ 15 15 ⎠. 9qj’7eo;og]dlj;o]5j,oUG ⎛1 4⎞ ⎛7 5⎞ a = ⎜ + ⎟+⎜ + ⎟ ; ⎝3 3⎠ ⎝ 3 3⎠ 1 ⎞ ⎛ 13 3 ⎞ ⎛ 9 c = ⎜ − ⎟+⎜ + ⎟ ; ⎝ 20 20 ⎠ ⎝ 20 20 ⎠. ⎛5 7⎞ ⎛8 1⎞ b =⎜ + ⎟+⎜ − ⎟ ⎝6 6⎠ ⎝6 6⎠ ⎛ 7 9⎞ ⎛ 1 9⎞ d = ⎜ + ⎟−⎜ + ⎟ ⎝ 19 19 ⎠ ⎝ 19 19 ⎠. 17.

(24) ⎛ 19 7 ⎞ ⎛ 9 6 ⎞ e =⎜ − ⎟+⎜ − ⎟ ; ⎝ 2 2⎠ ⎝2 2⎠. 9qj’0uf8+.lj7e8v[muj4nd8hv’D 7 13 + 12 12 19 5 − 28 28 7 23 + 20 20 9,1 0,1 − 15 15 7,3 4,7 + 18 18. •. •. •. •. •. •. •. •. •. •. 3 5 1 2 5 3 2 3 3 2. 7 30 + 11 30 =. +. − +. 13 30 − 1 30 =. = +. = =. 13 24 − 1 24 =. =. − +. = =. = −. 18. 5 24 +. =. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !WD 9qj’7yfw]j8k,c4;8A’ c]t c4;ovog]dlj;o.o8k8t]k’]5j,oUG dD 0D. ສດິ. _D. ⎛ 15 1 ⎞ ⎛ 8 2 ⎞ f = ⎜ − ⎟−⎜ − ⎟ ⎝ 9 9⎠ ⎝9 9⎠. − 7 24 =.

(25) [qfmu * dko[;d c]t dko]q[g]dlj;o X8+? dyf9tde dyf9tdemu ! .oI6[]5j,oUF 9qj’8nj,g]dlj;omujg\ktlq,.lj[jvo9Eg,afD. dD rkdlj;omuj0uf]kpdq’da[g]dlj;o DDDDDDDDDD 0D rkdlj;omujmkludq’da[g]dlj;o DDDDDDDDDD s^n DDDDDDDDDDD 7D rkdlj;omuj0uf]kp c]t rkdlj;omujmkludq’da[g]dlj;o DDDDDDDDDD \kp7;k,;jk. 1 3 1 " " + = + = 20 10 20 20 20. ’D rkdlj;omuj[+mklu c]t [+0uf]kpdq’da[g]dlj;o DDDDDDDDDD 10 7 " 7 " − = − = 10 20 20 20 ". ສດິ. \kp7;k,;jk dyf9tdemu @. c,j8h6 ve,k]k ,ug0Qk|q,vq,|bj’4q’grnjvc[j’.shs^kolk; 0v’g0Qk|q,ma’\qfD dD 4k,;jkc,j8h6c[j’g0Qk|q,.shs^koma’lv’gmqjk.fL 0D g0Qk|q,c,j86hpa’gs^nvgmqjk.fL dyf9tde # 9qj’7yfw]jG dD 0D. 4 7 + ; 15 30 3 14 + ; 25 75. 13 7 − ; 6 18 7 7 − ; 2 8. 5 1 5 + ; 2− 12 3 6 5 1 7 2 + ; − 4 3 5 3. 19. 2 5. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !D. c]t s^ko-kp. 3 25.

(26) @D. .97;k,G @D dko[;dg]dlj;omuj,ur6f8jk’dao dko[;dg]dlj;omuj,ur6f8jk’dao c,jogryjo0Bor6fIj;,djvoc]h;9bj’xt8y[afdko [;d7nda[dko[;dg]dlj;omuj,ur6f7ndaoD 8q;1jk’ !D @D. 5 3 5 × 2 3 × 3 10 9 10 + 9 19 + = + = + = = 6 4 6 × 2 4 × 3 12 12 12 12 4 5 8 35 43 + = + = 7 2 14 14 14. @D. 11 7 44 21 23 − = − = 6 8 24 24 24 7 2 35 6 29 − = − = 3 5 15 15 15. #D. [qfg/ydsaf. !D. 9qj’7yfw]j c]h;7af9hvog]dlj;o]5j,oUG a). c). @D. 4 7 + ; 5 10 2 13 + ; 7 28. 5 5 + 4 12 1 7 d) + 6 30. b). 9qj’7af9hvo]dlj;o]j5,oUF c]h;7yfw]jG 15 2 − ; 20 8 26 11 d= + ; 48 24. a=. #D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 8q;1jk’ !D. 49 9 + ; 35 15 14 25 e= − ; 12 30. b=. ltg|uzqo]5j,oU 4nd s^n zyfL 0 = 0; 9 1 9 c) − =0; 3 27. a). ສດິ. $D dko]q[g]dlj;omuj,ur6f8jk’dao dko]q[g]dlj;omuj,ur6f8jk’dao c,jogryjo0Bor6fIj;,djvoc]h;9bj’xt8y[afdko ]q[g]dlj;omuj,ur6f7ndaoD. 24 36 11 − + 8 16 44 12 15 9 f = − + 12 30 6. c=. 13 = 13 1 8 1 9 d) + = 5 5 10. b). 20.

(27) e). $D. 7 1 6 − = ; 12 6 6. f ) 3+ 3 4 + ; 5 15 1 e = 5− ; 5 2 h = 3− ; 3. 4 20 + ; 9 27 1 d = 1+ ; 4 3 g = 6− ; 20. b=. 9qj’[;dg]dlj;o]5j,oUG 13 4 1 + + ; 6 5 2 13 4 1 d= − − ; 6 5 2. 9 =0 3. 9qj’-vdsk7jk0v’. 2 3 2 f = 3− 5 15 k = −2 4. c =7−. 8 2 3 + + ; 5 3 4 8 2 3 e= − − ; 5 3 4. a=. *D. g) 3 −. 9qj’7yfw]jleo;o]j5,oUG a=. &D. 3 =4; 3. 20 2 12 + + 3 7 21 20 2 12 f = − − 3 7 21. b=. x. c=. grnjv.shltg\uzqo]5j,oUgxao9y’D. 10 2 7 b) x + 2 = = ; 5 5 3 2 1 1 3 d) −x= c) x − = ; 8 4 7 14 ltg\uzqo 3 y − 4 x = 2 4nd s^n zyfL 4hk;jkG dD x = 1 c]t y = 1 4 0D x = 3 c]t y = 5 2 2 7D x = 5 c]t y = 3 8 2 1 ’D x = c]t y = 67 100 100 9D x = 0 c]t y = 2 3. )D. ok’ ,k]y,ug0Qk|q,g7addhvo|bj’F ]k;c[j’dyo8vog-Qk. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. (D. ສດິ. a) x +. 1 , 4. dyo8vol;kp. 4k,;jkg0Qk|q,muj ok’ ,k]ygs^nvw;hdyo8voc]’,ugmqjk.fL _D. 3I’IPo,afmtpq,csj’|bj’ wfhc[j’g’yo[6]tot3I’IPofaj’oUG -Ao ,! wfh 21. 7 . 12. 1 ; 6.

(28) -Ao ,@. 5 ; -Ao 24. ,#. 1 4. S -Ao ,$. 1 12. c]t lj;omujpa’gs^nv,v[.sh-Ao,=xkpD. dD -Ao,=8Qowfhg’yoma’\qfgmqjk.fL 0D -Ao,=xkpwfhg’yogmqjk.fL !WD gryjolk,kfg;Qkwfhco;.fdjP;da[g]dlj;o]j5,oUG dD 0D. 7 ; 5 2 ; 25. 28 ; 20 10 ; 125. 49 35 6 75. c]t c]t. 21 15 8 100. !!D 9qj’7eo;og]dlj;o.o8k8t]k’]5j,oU a+b. 3× a + 2 × b. 3× a − 2 × b. ສດິ. b 1 8 1 12 3 20. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. a 1 2 1 3 3 5. 22.

(29) [qfmu ( dko76og]dlj;o dyf9tde dyf9tdemu ! 9qj’7yfw]j c]t 8nj,9eo;omujg\ktlq,.lj[jvo9Eg,afD 3 × 40 ; 10. c = 15 = 18 ×. b = 28 × " ; 6. 5 7. d = 21 = 35 ×. " 5. dyf9tdemu @ .shI6[9t85]af|bj’muj,u0hk’gmqjk 1mD 1j6.oI6[ 9t85]af gryjolhk’I6[lujc9lkd ABCD 3fp.sh0hk’. 2 AB = m 3. c]t 0hk’. 1m. 4 AD = m . 5 2. 9qj’8nj,9eo;omujg\ktlq,.lj[jvo9Eg,afD dD goNvmuj0v’I6[lujc9lkd ABCD gmqjkda[ AB × AD =. 3. 1m. m A. " " 2 × m " ". 0D goNvmuj0v’I6[lujc9lkdohvpmujmklufegmqjkda[. C. B. 1 2 m ". ສດິ. a=. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !D. 4 m 5. D. 7D goNvmuj0v’I6[lujc9lkd ABCD gmqjkda[ DDDDDDD gmnjv0v’goNvmuj0v’I6[lujc9 lkdohvpmujmklufe g-yj’gmqjkda[. " 2 m , ". ltcf’;jk. dyf9tdemu # 9qj’7yfw]jzqo76o]5j,oU dD. 0D. 3 2 × ; 5 7 7 3 d= × ; 3 7. 4 3 4 3 × ; c= × ; 9 5 9 5 7 5 e= × 8 3 9qj’8nj,leo;o8+wxoU a × c = " b d " a=. b=. 23. 2 4 " × = 3 5 ".

(30) .97;k, !D dko76og]dlj;o dko76og]dlj;o c,jogryjo76o9eo;or6fda[9eo;or6f c]t 76or6fda[r6f c]h;7af9hvo X4hk7af9hvowfh?D .shlv’g]dlj;o. a b. c]t. c d. g-yj’ b ≠ 0 c]t. d ≠0. a c a×c × = b d b×d 4 5 4 × 5 20 = 8q;1jk’ !D × = 3 7 3 × 7 21 7 2 7 ×2 2 × = = @D 3 7 3× 7 3 4 ⎛ 7 ⎞ 4 × (−7) 28 #D ×⎜− ⎟ = =− 5 ⎝ 3⎠ 5×3 15. gIqkwfh. 8 3 8× 3 8 2 ×4 4 × = = = = 3 14 3 × 14 14 2 × 7 7. ສດິ. $D. @D 9eo;oxUo0v’9eo;omuj8jk’l6o dD oypk,G lv’9eo;o9txUodaod+8+g,njvzqo76o0v’r;d,aogmqjk !D 4hk. a. gxao9eo;o8jk’l6oS. 1 a. c,jo9eo;oxUo0v’. 8q;1jk’ !D. 9eo;oxUo0v’ @ c,jo. @D. 9eo;oxUo0v’ WF( c,jo. a.. 1 2 1 10 = 0,7 7. 0D 9eo;oxUo0v’g]dlj;o .sh. a b. c,jog]dlj;og-yj’. 9eo;oxUo0v’. a b. c,jo. a ≠ 0;b ≠ 0. b a. grkt;jk. ¾ \kpgsfG W [+,u9eo;oxUo. a b ab × = =1 b a ba. 24. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. @D.

(31) 8q;1jk’D. 9eo;o 9eo;oxUo. 2 3 3 2. −6. −. #D. [qfg/ydsaf. !D. 9qj’7yfw]j c]h;7af9hvo X4hk7af9hvowfh? 2 6 × 3 5 4 1 d= × 3 6. 1 2 × 2 3 5 2 e= × 2 5. 3 5 × 7 11 3 44 d= × 44 5. 1 1000 × 100 55 100 3 × e= 7 100. a=. 7D. L. c=. 2 " 8 × = 9 " 27 17 " × =1 19 ". 0D ’D. 9qj’7yfw]jzqo76o]j5,oUG 5 × 16 8 3 21 4 D= × × 7 8 9. 9qj’7eo;og]d]5j,oUG dD 7. 7 4 1 + × 3 3 4 ⎛7 4⎞ 1 ⎜ − ⎟× ⎝3 3⎠ 4. 1 24 × 2 7 5795 2 × f= 2 5795. c=. 2,5 " 10 × = 7 " 77 3 7 7 × = 5 " 10. 7 6 1 2 3 × C= × × 12 35 2 3 5 1 15 4 E = × × ×6 18 14 5. B=. ⎛7 4⎞ 1 ⎜ + ⎟× ⎝3 3⎠ 4 7 4 1 − × 3 3 4. 0D. ’D. 25. ສດິ. b=. A=. &D. 0. 7 16 × 8 14 4 f = ×0 3. b=. 9qj’8nj,9eo;omujg\ktlq,.lj[jvo9Eg,afD dD. $D. −7. 4 5 5 − 4 −. 9qj’7yfw]j c]h;7af9hvo X4hk7af9hvowfh?D a=. #D. 1 6. 1 7. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. @D. −.

(32) 9qj’7eo;og]d]5j,oUG dD 7D. ’D. 40 + 5 25 − 5 × 35 + 5 80 − 5 3 ⎛ 4 7 ⎞ 12 ×⎜ + ⎟× 2 ⎝ 15 5 ⎠ 9. 9qj’-vd9eo;oxUo0v’9eo;o]j5,oU: a = 40. b = −0,8 3 e= 4. d = 0,016. )D. 0D. c = −3,2 1 f =− 7. 9qj’8nj,9eo;omujg\ktlq,.lj8k8t]k’]5j,oUG dD 4 9. × 3 2. ×. ×. =. =. × 9 4. ×. 8 27. ×. ×. 5 3. =. =. 0D 3 4. 2 3. 5 6. 4 9. =. 7 8. =. ສດິ. 7D. 9 + 1 13 − 1 × 8 + 1 16 − 1 3 4 7 12 × + × 2 15 5 9. 7 8. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. *D. 5. _D. gryjo.sh. a=. 1 1 1 , b= , c= 8 4 2 ab, ac, ab + ac, a + bc, a (b + c).. dD 9qj’7yfw]j 0D la’gdfgsaoco;.fL !WD gryjo.shlv’leo;o 9qj’7yfw]jleo;o. A=. 1⎞ 4 ⎛1 B = ⎜ × x − ⎟× 8⎠ 3 ⎝4 7 5 4 x= ; x= ; x= 2 4 3. 1 1 4 ×x− × 4 8 3. A c]t B le]a[. 26. c]t.

(33) [qfmu ) dkoskog]dlj;o dyf9tde dyf9tdemu ! dD I6h;jk c]t. 7 5 1 7 ×1 7 7× = = 5 5 5 7 ÷5 =. gsao;jkzqosko0v’ ( .sh & gmqjkda[zqo76o0v’ ( da[9eo;oxUo0v’ &D { 9qj’xjPodkosko]j5,oUgxaodko76oD 4 ÷ 3; 3, 5 ÷ 7, 5; 4,8 ÷ 5; 4 ÷ 1, 3. 0D 9qj’-vdglAozjko.9dk’0v’;q’,qomuj,u];’Iv[ @ m, I6h;jk. 3 3 3 " = ÷2 = × 8 4 4 ". 7D 9qj’8nj,9eo;omujg\ktlq,.lj[jvo9Eg,afD 4 4 " ÷2 = × ; 5 5 " 9 9 " ÷3 = × 2 3 " 3 " " 4 ÷ = 4× = ; 5 " " 2 " " 8 ÷ = 8× = 3 " " 7 2 7 " " ÷ = × = ; 3 5 3 " ". ≈ 0,31. 0, 5; 0, 25; 0,1; 0, 2. dyf9tdemu @ 8k,I6[c8h,F 9qj’8nj,xt3spd7eg;Qk]5j,oUG dD rkdlj;omklu0v’I6[ X!? dq’da[g]dlj;o DDDDDDDD s^n DDDDDDDF 0D rkdlj;omklu0v’I6[ X@? dq’da[g]dlj;o DDDDDDDD s^n DDDDDDDF ltcf’;jk. π. ສດິ. 7D .sh a gxao9eo;o|bj’8jk’l6oD 9qj’sko a .sh. 1. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !D. (1). (2). 2 8 2 " " ÷ = × = 3 5 3 " ". 27.

(34) @D. .97;k, oypk,G dD .shlv’9eo;o a c]t da[9eo;oxUo0v’ b. b. g-yj’. b ≠ 0 . dkosko a .sh b. c,jodko76o a. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. 1 a ÷b = a× ; b ≠ 0 b 1 81 8q;1jk’ !D 81 ÷ 4 = 81 × = 4 4 1 48 × 1 48 48 ÷ 1,2 = 48 × = @D = 1,2 1,2 1,2 1 4,5 × 1 4,5 4,5 ÷ 2,4 = 4,5 × = #D = 2,4 2,4 2,4 0D .shlv’g]dlj;o a c]t c g-yj’ a, b, c c]t d 8jk’l6oF b d gIqkwfhG a ÷ c = a × d c]h;7af9hvozqowfhIa[ X4hk7af9hvowfh?D b d b c 5 2 5 7 5 × 7 35 8q;1jk’ !D = ÷ = × = 3 7 3 2 3× 2 6 2 2 1 2 ×1 2 @D = ÷ 11 = × = 9 9 11 9 × 11 99 4 13 5 × 13 65 #D 5 ÷ = 5× = = 13 4 4 4 6 7 0× 6 0 $D 0 ÷ = 0× = = =0 7 6 6 6 6 3 6 5 2 × 3/ × 5 10 &D =− − ÷ =− × =− 17 5 17 3 17 × 3/ 17. #D. [qfg/ydsaf. !D. 9qj’7yfw]jG a=. @D. 4 3 4 2 42 ÷ ; b = 1 ÷ ; c = ÷ 2 S d = −7 ÷ ; 7 5 3 3 5. 9qj’7yfw]j c]h;7af9hvoG a=. 2 −7 ÷ ; 3 5. b = −36 ÷. 30 − 12 − 28 ;c= ; ÷ 7 9 27. 28. 3 3 e= ÷ ; 4 4 d=. f =. 34 38 ÷ 51 57. 8 4 ÷ 21 7.

(35) #D. 9qj’-vd9eo;oxUo0v’9eo;o]5j,oUG 1 8 6 ; 5; ; 7 3 19. 0, 25; 0,1; 0, 01; 0, 02;. $D. 9qj’7yfw]j.o.9 a = 7 ÷ 0, 5;. b = 12 ÷ 0, 25;. c = 100 ÷ 0,1 3 e = 0, 25 ÷ 0, 01; f = ÷ 0, 02 5. d = 0, 6 ÷ 0,1;. &D. 9qj’7yfw]j c]h;7af9hvoD. 7 12 9 7 13 7 ÷ ; ÷ ; ÷ 10 100 100 10 100 100 13 9 47 9 64 32 0D ÷ ; ÷ ; ÷ 10 100 100 10 100 100 9qj’0Po − 27 rkp.8hI6[c[[8jk’M]5j,oUG 8 dD − 27 gxaozqo76o0v’lv’g]dlj;oD 8 − 27 gxaozqosko0v’lv’g]dlj;oD 0D 8 7D − 27 gxaozqo76o0v’lk,g]dlj;ogmqjkdaoD 8 ’D − 27 gxao9eo;omqfltoypq,D 8. (D )D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. *D. ສດິ. dD. 9qj’7af9hvog]dlj;o]5j,oUG 19 ; 95. 16 ; 65. 36 ; 64. 11 ; 110. 72 ; 27. 13 325. 9qj’8nj,9eo;omujg\ktlq,.lj8k8t]k’]j5,oU −1 3 ÷ −8. ×. ÷. = ×. 4 × −1 2 =. 1 4 ×. = ÷. = = −8. = =. 29. ÷. ÷. = × 1. ÷. =. = ×. -8. = =.

(36) _D 9qj’8nj,g7njv’\kp0v’dko7eo;o.ljcmog7njv’\kp < ∗ <G dD 0D 7D. 2 6 5 ∗ = ; 3 5 9 3 ∗ 3 1 14 ; ∗ = 2 6 3 3∗3 1 1 ∗ = ; 2 6 2. 5 5 3 5 6 ∗ = 3; ∗ = 4 12 7 14 5 3 ∗ 3 1 17 ∗ = 2 6 6 3∗3 1 3 ∗ = 2 6 4. !WD 4a’oE,ao4a’|bj’[ao95oE,aog8a,S mevyfgryjogvqkvvd 1 0v’4a’S gmnjvmulv’gvqkvvd 6. 1 0v’4a’ 5. c]t oE,aopa’gs^nv &(W ]ufD. 9qj’-vd[=]y,kf[ao950v’4a’oE,ao|j;poAoD !!D 4a’oE4a’|bj’,uoE1j6 1 0v’4a’S gryjo8nj,oE.lj @) ]uf g;]koAooE.o4a’,u 6. 4k,;jk[=]y,kf[ao950v’4a’,ugmqjk.fL. 0v’9eo;o0k0v’[Q’dnD 4k,;jk[Q’dn,u9ad0k. ສດິ. I6h;jkc,’,6,,u ) 0kL. 4 21. !#D dtc-afgra’d=h|bj’lk,kfvafwfh !@W okmuD |hk A 4ndvafwx. 7 2 ; |hk B 4ndvafwx . 12 5. 4k,;jkc8j]t|hkpa’gs^nvg;]kgmqjk.fL !$D 9qj’c8h,I6[]j5,oU7no.\j c]h;mklurkdlj;omujltcf’g4y’ dD. !&D .shleo;o. 2 ÷3 5. A=. 9qj’7yfw]jleo;o. a+b S c. A, B. 0D. B=. c]t. 0v’4a’D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !@D 9eo;o0k0v’c,’,6,gmqjkda[. 3 4. 4 ÷5 7. a−b a×b C= c c 2 3 2 C 3fp.sh a = ; b= ; c= 3 7 7. 30.

(37) [qfmu _ 9eo;oxqddt8y !D dyf9tde dyf9tdemu ! 9qj’skog]d]5j,oUG 14 5 23 31 109 ; ; ; ; . 35 3 99 90 50. @D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. dyf9tdemu # mhk; 7e,u c]t \6j @ 7qorkdaowxdyog0Qk16jIhkovkskocsj’|bj’7yfgxaog’yo &*WWW du[D dD g0qkg9Qk8hv’9jkp7jkvkskozh6]tgmqjk.fL 0D .o8q;9y’8hv’9jkpz6h]t9addu[L. .97;k, !D 9eo;oxqddt8y oyypk, 9eo;oxqddt8y c,jo9eo;omuj0Po.oI6[Ijk’ b. 8jk’l6o (b ≠ 0) D. 8q;1jk’ !D @D. a b. g-yj’ a F. ສດິ. dyf9tdemu @ mhk; cdh; -N\kd[u _ |j;pgxaog’yo @)&W du[D 4k,;jk\kd[u|j;p|bj’]k7kgmqjk.fL. b. gxao9eo;o4h;o c]t. 2 8 17 12 5 ; − ; ; 1 = ; −1 = − 13 4 12 5 5 6 1 17 15 4 6= ; − ; ; 3= ; 3 8 5 9 1. @D dkoxjPog]dlj;o gxao9eo;omqfltoypq, m5dM9eo;omuj0Po.oI6[Ijk’g]dlj;o. a b. g-yj’. a, b. gxao9eo;o4h;o c]t b 8jk’l6o. (b ≠ 0) D lk,kf0Pogxao9eo;omqfltoypq,wfh3fp.-hdkosko. 8q;1jk’ !D. 9 = 2,250…; 4. 5 = 1,666… 3 31. a. .sh b ..

(38) @D. −. 7 = −0,6363 11. 23 = 0,2555… 90. ;. faj’oUG. !D @D #D $D &D *D. 11 = 2,75000… 4 5 = 0,555… 9 7 = 0,4666… 15 50 − = −1,515151… 33 11 = 0,297297297… 37 83 = 0,30181818… 275. s^n. 0,750. Iv[;Poc,jo. 0. s^n. 0, 5. Iv[;Poc,jo. 5. s^n. 0,46. Iv[;Poc,jo. 6. s^n. − 1, 51. Iv[;Poc,jo. 51. s^n. 0, 297. Iv[;Poc,jo. 297. s^n. 0,3018. Iv[;Poc,jo 18. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. gryjo9af9eo;omqfltoypq,Iv[;Pogxaolv’dj5,7nG dD 9eo;omqfltoypq,Iv[;PolYol5f XIv[;Pol6o? gvUo;jk 9eo;omqfltoypq,D .od5j,oUgryjo[+oypq,0PoIv[;Pol6o0v’,aoD 8q;1jk’D 0,3 0 0Pogxao 0,3 2,65 0 0Pogxao 2,65 − 1,8352 0 0Pogxao − 1,8352 0D 9eo;omqfltoypq,Iv[;Po[+lYol5f XIv[;Po8jk’l6o? gvUo;jk 9eo;oxqddt8yD 8q;1jk’ !D 4, 7 ; − 3, 05 ; − 0,137 @D 1,1 8 ; − 2,06 3 ; − 2,1327 .od5j,oUgryjo[+lk,kf0Po7jkmujcojovo0v’,ao c8jgryjolk,kf0v[0Ao7jkxt,ko 0v’,aowfhD. ສດິ. 9eo;omqfltoypq,7n0hk’gmy’ gvUo;jk 9eo;omqfltoypq,Iv[;Po g-yj’gryjooypq,0Po. 707 = 7,141414 99 gIqkwfh 7 < 707 < 8 99 707 7,1 < < 7,2 99 707 7,14 < < 7,15 99. 8q;1jk’D. s^n. 7,14. fh;p7;k,-afg9oIvf ! fh;p7;k,-afg9oIvf WF! fh;p7;k,-afg9oIvf WFW! 32.

(39) #D. [qfg/ydsaf. !D. 9qj’0Pog]dlj;o8+wxoUgxao9eo;omqfltoypq,D 2 3 35 3) 24. 15 8 6 4) − 11 15 6) 37 75 8) − 6 2) −. 1). 5 132 21 7) 15. 5) −. .o[aofk9eo;o]j5,oU 9eo;o.fgxao9eo;oxqddt8yL 11 5 2 5 + 3) 3 7. 1). 2). 3,2 1,3. 4) − 3,56 − 4,81 3 5 × 7 2 8) − 3,5 ÷ 6. 5) − 4,6 × 0,7. 7) − 5,6 ÷ 1,1 4 3. #D. 9qj’sko $ .sh # c]h;0v[0Ao. $D. 9qj’sko @& .sh ( c]h;0v[0Ao. &D. 9qj’sko #&& .sh !!# c]h;0v[0Ao. *D. c,j-N\kdok; # |j;p8+ !WWW du[D xhk-Nwfh !WW |j;p8+ ##WWW du[D 4k,;jkc,jo.z-N \kdok;wfh4ndd;jkdaoL c,jc]tok’ 4k;vo wxIhko7hkcsj’|bj’grnjv-Ncx’46c0h;Ia[.-h7v[7q;D cx’46c0h; vao|bj’]k7k !$&W du[F |bj’3s^]k7k !&)WW du[D c,jo c]t 4k;vo9t8aflyo.9-N cx’46c0h;c[[.fL phvosap’L cdh;F dhvp c]t d5h’ vvdg’yodao-N0v’0;ao;aogduf.sh\j6 ];,gxaog’yo #@&WW du[D 4k,;jkG dD c8j]t7qo9t9jkpg’yo7jk0v’0;aogmqjk.fL 0D .o8q;9y’c8j]t7qo9t9jkpg’yolqfwfh8k,9eo;o.o0=h dD [+L phvosap’L. (D. )D. fh;p7;k,-afg9oIvf WFWW!. ສດິ. 6). ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. @D. 25 7. fh;p7;k,-afg9oIvf WFW! 355 113. fh;p7;k,-afg9oIvf WFWWWWW!. 33.

(40) _D. 7D c8j]t7qo8hv’9jkpxt,kogmqjk.fL 9qj’0Pog]dlj;o s^n 9eo;omqfltoypq,mujdq’da[muj8A’0v’g,af A; B; C c]t D A. C. B. 0. !WD dD 9qj’[vdva[-yf0v’g,af. 6 3 1 4 4 A; B c]t C.. A. D 2. C. B. 1 1 2 0D 9qj’;k’g,af D; E c]t F muj,uva[-yf 1 ; 2 6 3 dD 9qj’[vdva[-yf0v’g,af P; Q c]t R. 0. Q. L. L. R 1. 0D 9qj’;k’g,af M; N; K; c]t L muj,uva[-yf 8k,]efa[D. 34. 8k,]efa[D. L. 1 3 3 ; ; 1+ 5 10 10. 11 10. ສດິ. 0. P. 7 6. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !!D. c]t. c]t.

(41) [qfmu !W 9eo;oxqddt8y X8+? !D dyf9tde dyf9tdemu ! 9qj’xjPo9eo;omqfltoypq,]j5,oUgxaog]dlj;omuj[+lk,kf7af9hvowfhD 4,6 ; 1,04 ; 0,168 ; 0,054 ; 1,275 ; 0,0196. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. dyf9tdemu # gryjo.sh9eo;o x = 0,545454… dD 9qj’7yfw]j 10 0 x ; 10 0 x − x 0D 9qj’0Po9eo;o x rkp.8hI6[c[[g]dlj;omuj[+lk,kf7af9hvowfhD. ສດິ. dyf9tde mu @ gryjo.sh9eo;o x = 0,777… dD 9qj’7yfw]j 10 x ; 10 x − x 0D 9qj’0Po9eo;o x rkp.8hI6[c[[g]dlj;omuj[+lk,kf7af9hvowfhD. dyf9tdemu $ gryjo.sh9eo;o x = 0,2333… dD 9qj’7yfw]j 10 0 x ; 10 x ; 100 x − 10 x 0D 9qj’0Po9eo;o x rkp.8hI6[c[[g]dlj;omuj[+lk,kf7af9hvowfh. @D. .97;k, ¾ dkoxjPo9eo;omqfltoypq, gxaog]dlj;o { m5dM9eo;omqfltoypq,Iv[;Pol6o lk,kf0Pogxaog]dlj;owfh 8q;1jk’. 11 100. 1.. 0,11 =. 2.. − 0,137 = −. 3.. 2,19 =. 137 1000. 219 100. { le]a[9eo;omqfltoypq,Iv[;Po8jk’l6o lk,kf0Pogxaog]dlj;owfh 35.

(42) 3fpdkoxt8y[affaj’oUG 8q;1jk’ !D 0Po 0,6 gxaog]dlj;o N = 0, 6 = 0,666… 76oma’lv’rkdda[ !W gryjo;k’ gIqkwfhh 10 N = 10 × 0,666 = 6,666… faj’oAo 10 N − N = (6,666 ) − (0,666…). ltcf’;jk. −. s^n. 6, 666... 0, 666... 6, 000.... 2 3. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 8q;1jk’ @D 0Po 0,725 gxaog]dlj;o gryjo;k’ N = 0,725 = 0,7252525… ............(1) 76oma’lv’rkd0v’ X!? da[ !WWW gIqkwfh 1000 N = 1000 × 0,7252525… = 725,252525… ............(2) 76oma’lv’rkd0v’ X!? da[ !W wfh 10 N = 10 × 0,7252525 = 7,252525… ............(3) gvqk X@? ]q[vvd X#? wfh 1000 N − 10 N = (725,252525 ) − (7,252525…). ສດິ. 9bj’wfh. 9N = 6 6 N= 9 6 0, 6 = 9. 725, 252525... 7, 252525... 718, 000000..... 990 N = 718 faj’oAo N = 718 990. ltcf’;jk. 0, 725 =. −. 718 359 = 990 495. lts^6[G gIqklk,kf0Pog]dlj;o.shgxao9eo;omqfltoypq,Iv[;Powfh c]t 0Po9e o;omqfltoypq,Iv[;Po.shgxaog]dlj;owfh faj’oAogIqklk,kfg;Qkwfh;jkG 9eo;oxqddt8y c,jo9eo;omuj0Pocmowfhfh;p9eo;omqfltoypq,Iv[;Po s^n g]dlj;o. a b. g-yj’ a ,. b. gxao9eo;o4h;o c]t b 8jk’l6o (b ≠ 0) D. 36.

(43) 9kdzqowfhIa[0hk’gmy’gIqkgsao;jk9eo;oxqdt8y c,jo9eo;o-toyf8jk’Mfaj’ czo;kf]j5,oUG 9eo;oxqdt8y. 9eo;o4h;o. l6o. 9eo;o4h;o[;d. [qfg/ydsaf. !D. 9qj’0Po9eo;omqfltoypq,8+wxoUgxaog]dlj;omuj[+lk,kf7af9hvowfh dD 3, 6; 1, 08; 0, 231 0D 0, 076; 1, 025; 0, 2318 .o[aofk9eo;oxqdt8y8+wxoU9eo;o.fgxao9eo;omqfltoypq,L. @D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. #D. 60 90 ; 140 48 35 6600 250 99 ; ; 2400 28 225. dD 14 ; 0D. #D $D. &D. 9qj’[vdIv[;Po0v’c8j]t9eo;o]j5,oUG dD 0,222…; 5,141414…; 3, 8 ; 41,29 0D 6,21555…; 0,452; 28,373737 ; 8, 209 9qj’xjPo9eo;omqfltoypq,Iv[;Po8+wxoUgxaog]dlj;omuj[+lk,kf7af9hvowfh dD 0,777…; 0, 52; 0,48 ; 0,236236236… 0D 0,215 ; 0,453; 0,24838383…; 0,290 la’gdfgsao;jk9eo;o x = 2,999… lk,kf0Powfh.oI6[c[[ x = 2,999… = 2 + 0,999…. *D. ສດິ. 9eo;o4h;o]q[. g]dlj;omuj[+c,jo 9eo;o4h;o. 9qj’0Po9eo;o x rkp.8hI6[c[[g]dlj;omuj[+lk,kf7af9hvowfh 9qj’xjPo9eo;omqfltoypq,Iv[;Po8+wxoUgxaog]dlj;omuj[+lk,kf7af9hvowfh 37.

(44) (D )D. 23 ; 91. 77 ; 40. 221 ; 90. 21 317 ; 407 37. dD 0D 7D ’D 9D. 9 9 9 9 + + + + 10 100 1000 10000 23 23 23 + + + 100 10000 1000000 2 8 8 8 + + + + 10 100 1000 10000 3 54 54 54 + + + + 10 1000 100000 1000000 3 3 3 3 + + + 2+ + 10 100 1000 10000. 38. ສດິ. w0jwdjc8t|bj’,u #W |j;p ]k7k @_WWW du[D 4k,;jkG dD w0j|j;p|bj’]k7kgmqjk.fL 0D 4hk0kppjvpgryjo8hv’0kp|j;p]t9addu[L !WD 9qj’7yfw]jzqo[;d]j5,oUG. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. _D. dD 1,888…; 5,12; 2,34 ; 4,2393939… 0D 8,215 ; 7,152; 9, 2 ; 23,40 9qj’ltcf’.shgsao;jk 1,999… = 2 9qj’-vdIv[;Po0v’9eo;oxqddt8y]j5,oUG.

(45) [qfmu !! 9eo;ovtxqddt8y !D. dyf9tde 9qj’[vdIv[;Po0v’c8j]t9eo;o]j5j,oUG dD 0,151515…; 0,63917256…; − 1,8555…; 3,14168306701… 0D 4,089201537…; − 2,543626262…; 0,666…; 0,758130165…. @D .97;k, oypk, 9eo;omuj[+lk,kf0Pogxao9eo;omqfltoypq,Iv[;Po s^n g]dlj;o gxao9eo;o4h;o c]t. b. a b. g-yj’ a , b. 8jk’l6o (b ≠ 0) gvUo;jk 9eo;ovtxqddt8yD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !F@#$&*()_!!!@!#!$DDD #F$#@#@@#@@@DDD !*F(_((_(((_((((_DDD {$F#__##__###__DDD 9kd8q;1jk’0hk’gmy’ la’gdfgsao;jk gIqk[+lk,kf9af8q;g]dsâ’95f.shgxaoIv[;PoF vud8q;1jk’|bj’0v’9eo;ovtxqddt8y7n π g-yj’,u7jkgmqjkda[ #F!$!&_@*&#&)_(_#@#)$*@DDD π c,jovaf8klj;o]ts;jk’];’Iv[0v’;q’,qo c]t glahozjko.9dk’0v’,aoD g,njvgIqk-vdskgonhvmuj0v’czjo,qo3fp.-hl6f π r 2 s^n -vdsk];’Iv[0v’;q’,qo 3fp.-hl6f 2π r g-yj’ r c,jo]aflt\u0v’;q’,qo gryjo,ad.-h7jkxt,ko0v’ π gxao. ສດິ. 8q;1jk’D. 22 7. s^n #F!$!* s^n #F!$@ g-yj’gxao9eo;oxqddt8yma’\qfD. 9eo;omujgxao9eo;oxqddt8y sn^ 9eo;ovtxqddt8y gvUo;jk 9eo;o9y’ 8q;1jk’G. 5 7 ; − ; 6,3; 0; − 15, 02; 8,999...; −1, 40555; 2 8 2,121221222...; −0, 2534970128...; 28,346. 7; − 4;. 39.

(46) czo;kfltcf’9eo;o-t|yf8jk’M. 9eo;o9y’ 9eo;oxqddt8y 9eo;o4h;o l6o. g]dlj;omuj[+c,jo9eo;o4h;o 9eo;o4h;o[;d. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ;ymultcf’9eo;ovtxqddt8y.ljcdo9eo;o zjko,kgIqklk,kfltcf’9eo;oxqddt8yc8j]t9eo;o.ljcdo9eo;owfhD le]a[9eo;ovtxqddt8y gIqkd=,u;ymultcf’c8j]t9eo;o.ljcdo9eo;owfh faj’]5j,oUG 8q;1jk’G le]a[ π ];'Iv[0v';'q,qo gonjv’9kd π = gloAzjko.9dk' faj’oAo 4hkgIqkc8h,;q’,qo|bj’muj,uglAozjko.9dk’ ! sq;|j;pF ];’Iv[0v’;q’ ,qod=9tgmqjkda[ π , ltcf’;jk π c,jo];’Iv[0v’;q’,qomuj,uglAozjko.9dk’ ! sq;|j;pD faj’oAo gIqklk,kfltcf’9eo;o π .ljcdo9eo;o3fpdY’;q’,qo muj,uglAozjko.9dk’ ! sq;|j;p9kdg,af W wxg[Nv’0;k,n.shwfh|bj’Iv[D. ສດິ. 9eo;o4h;o]q[. 9eo;ovtxqddt8y. 40.

(47) [qfg/ydsaf 9eo;o8+wxoU 9eo;o.fgxao9eo;oxqddt8y c]t 9eo;o.fgxao9eo;ovtxqddt8yL 1) 3) 5) 7) 9). #D $D &D. 7 3 − 2 + 0,2 1,366666 − 5,9326483264832648 − 0,1666676869. 2) − 4) 6) 8) 10). dD 9qj’cmd0t|kf0v’g9hp A 4 . 0D 9qj’-vdvaf8klj;o]tsj;k’];’pk; c]t ];’dh;k’0v’,aoD fh;pg7njv’7yfw]jg]dF 9qj’-vdsk 2 ; 3 c]h;.sh0=hla’gdfD 9qj’ltcf’9eo;ovtxqddt8y π c]t − π .ljcdo9eo;ofP;daoD ,u9eo;o9y’.f[+ mujgxaoma’9eo;oxqddt8y c]t 9eo;ovtxqddt8yL. 7jk0v’ π. ສດິ. @D. 9 7 0 0,842 2,4313113111 2,137137137. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. #D !D. dko7eo;o 7jk0v’ π .ovtfufgryjo.-h;ymumujc8d8jk’dao g-ajoG { lhk’I6[s^kpc9ltg\uco[.o c]t 8yfovd;q’,qo { 7yfw]j];’Iv[0v’I6[s^kpc9ma’lv’I6[ { 7yfw]j7jkxt,ko0v’];’Iv[0v’lv’I6[;q’,qo { -vdvaf8klj;o]ts;jk’];’Iv[0v’;q’,qo c]t glAozjko.9dk’0v’;q’,qo 7jkmujwfh9tgxao7jkxt,ko0v’ π dko7eo;o 7jk0v’ π fh;p;ymuoU0Boda[9eo;o0hk’0v’I6[s^kpc9ltg\upyj’9e o;o0hk’0v’I6[s^kpc9ltg\upyj’s^kp0Bogmqjk.f7jkmujwfhpyj’.dh7P’da[7jk0v’ π gmqjkoAoD. 41.

(48) vkd-uc,f XArchimèdesG xt,ko @)({@!@ xu djvo7iy8tladdtikf? oad7toyflkf c]t ;ymtpklkf -k;gdiadltcf’.shgsao;jk π ,u7jk]ts;jk’ vkd-uc,f. 223 71. c]t. 22 7. • -695’9u. XTsu Ch’ung chih: xt,ko 7DlD $#W{&W!) ;ylt;tdvo-k; 9uowfh7jkxt,ko0v’ π c,jo 355 s^n xt,ko #F!$!&_@_ 113. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. • m=g],u XPtolemy: xt,ko 7DlD @WW) wfh7jkxt,ko0v’ π c,jo #D!$!* •.o7iyf8tlt8t;afmuj !) ,udko.-h8q;vadlvogdiad7n π cmovaf8klj;ofaj’djk; c]t ,udko7eo;osk7jk0v’ π fh;ps^kp;ymudkomujc8d8jk’daogIaf.shwfh7jk0v’ π c8d8jk’daogIaf.shwfh7jk0v’ π mujgxao9eo;omqfltoypq,s^kp8ec\j’ g-ajoG { .o 7D lD !)(# c-’ (Shanks) oad7toyflkf-k;va’dyf wfh7eo;o7jk0v’ π gxao9eo;omqfltoypq,g4y’ (W( 8ec|j’3fp.-hg;]k7eo;os^kpdj;k !& xuD { .o 7D lD !_$_ g7njv’7v,ru;g8ulk,kf7eo;o7jk0v’ π gxao9eo;omqfltoypq, wfhg4y’ @WWW 8ec|j’ { .o 7D lD !_*! π gxaomqfltoypq,wfhg4y’ !WW@*& 8ec|j’ { .o 7D lD !_*( π gxaomqfltoypq,wfhg4y’ &WWWWW 8ec|j’ { .o 7D lD !___ g7njv’-5[gxu7v,ru;g8umuj,tsk;ymtpkw]38dP;7eo;owfhg4y’ @W* !&) $#W WWW 8ec|j’ c]t .oxt95[aowfh9eo;omqfltoypq,s^kpd;jklv’rao]hko 8ec|j’D. 42.

(49) rkdmu @ dkog7yj’7n [qfmu !@ dkog7yj’7nmP[.ljg,af|bj’ dyf9tdemu ! .shglAo-nj Δ F g,af A, B, C C c]t D. A { 9qj’lhk’g,afg7yj’7n A' , B', C' D c]t D' 0v’ A, B, C c]t D mP[.ljglAo-nj Δ. la’gdfgsao co;.fL B { 9qj’8nj,xt3spd8+wxoUfh;pG glAo-nj Δ F A, A' , glAodk’lkdF Δ mjvo-nj [AA'] D i g,af DDDDDDDDDDDDDDD c]t DDDDDDDDDDDDDDDg7yj’7ndao mP[.lj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD i DDDDDDDDDDDDDDDDDDDDDDDc,jo DDDDDDDDDDDDDDDDDDDDDDDDDDD0v’ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD. ສດິ. dyf9tde. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !D. dyf9tdemu @ .shg,af O, A, B c]t C. A { 9qj’lhk’g,af A', B' c]t C' g7yj’7n da[g,af A, B c]t C mP[.ljg,af O. B { g,af A' gvUo;jk g,afg7yj’7n0v’ A O mP[.ljg,af O. { 9qj’lhk’g,afg7yj’7n O' 0v’ O mP[ C .lj8q;,aogv’ c]h;.sh0=hla’gdfD A' { 9qj’8nj,xt3spd8+wxoUfh;pG g,afg7yj’dk’F A, O, A' c]t [AA'] . i g,af DDDDDDDDDDDDDDDDDD c]t g,af DDDDDDDDDDDDDDDDDDDDg7yj’7ndaomP[.ljg,af DDDDDDDDDDDDDDDDDDDD i g,af DDDDDDDDDDDDDDDDDDc,jo DDDDDDDDDDDDDDDDDDDDDDDDDDDDD 0v’mjvo-nj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD { 9qj’xP[mP[w]pt OA' , OB' c]t OC' da[ OA, OB c]t OC 8k,]efa[D 43.

(50) { I6[lk,c9 A'B'C' c,joI6[g7yj’7n0v’I6[lk,c9 ABC mP[.ljg,af O. 9qj’xP[mP[lv’I6[lk,c9faj’djk;D { g,af O gvUol6odk’dkog7yj’7nD. .97;k, !D dkog7yj’7nmP[.ljglAo-nj|bj’D { g,af A’ c,jog,afg7yj’7n0v’g,af A mP[.lj glAo-nj Δ. { glAo-nj Δ c,joglAodk’lkd0v’mjvo-nj [AA’]. { g,afmuj16jglAo-nj Δ ,ug,afg7yj’7nc,jo8q;,aogv’D { glAo-nj Δ gvUo;jkG cdog7yj’7nD. POP' = QOQ' = ROR' = 180. Δ. A'. R. ສດິ. @D dkog7yj’7nmP[.ljg,af|bj’D dD P’, Q’, R’ c,jog,afg7yj’7n0v’ P, Q, R 8k,]efa[mP[.ljg,af O. { O c,jog,afg7yj’dk’0v’mjvo-nj [PP’], [QQ’] c]t [RR’]. { g,afg7yj’7n0v’ O c,jo8q;,aogv’D { g,af O c,jo l6odk’g7yj’7nD 0D lv’I6[g7yj’7ndaomP[.ljg,af|bj’ Q' 9tg8a’daor=fu4hkgIqkxyjoI6[|bj’vhv, l6odk’g7yj’7nwfh 180 D. A. 180. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. @D. Q. P. O. P' R'. { lv’I6[g7yj’7nmP[.ljg,af|bj’ gmqjkdaoD ΔPQR = ΔP'Q'R'. #D dkolhk’g,afg7yj’7nmP[.ljg,af|bj’D lhk’g,af M’ g7yj’7nda[g,af M mP[.ljg,af O. d=]toumu ! • g,af M g8a’da[g,af O, g,afg7yj’0v’ M c,jo8q;,aogv’D ma’lk,g,af O, M c]t M' g8a’dao16jg,affP;D 44. M O M'. D.

(51) ສດິ. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. d=]toumu @ • g,af M [+g8a’da[g,af OD O M' M { 0ufg7yj’glAo-nj [MO) { 0ufmjvodqj’muj,ug,af.9dk’16j O c]t ]aflt\u OM. { mjvodqj’8af [MO) 16jg,af M’. M’ c,jog,afg7yj’7n0v’ M mP[.lj O. 8q;1jk’G lhk’I6[g7yj’7n A’B’C’D’ A C 0v’I6[ ABCD mP[.ljg,af O. B ;ymulhk’G { lhk’g7yj’glAo-nj [AO), [BO), [CO) c]t [DO). O D' D { 0ufmjvodqj’muj,ug,af.9dk’ O c]t ]aflt\u OA, OB, OC, OD 8k,]efa[D B' { [aofkmjvodqj’8af [OA), [OB), C' [OC) c]t [OD) 16jg,af A' , B' , C' 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[g7yj’7nda[ ABCD mP[.ljg,af O muj8hv’dkolhk’D. #D. [qfg/ydsaf. !D. 9qj’lhk’I6[ B' c]t B'' g7yj’7n da[I6[ B XI6[ !? mP[.ljglAo -nj d1 c]t d 2 8k,]efa[D 9qj’lhk’g,afg7yj’7nda[g,af A, B c]t C mP[.ljg,af D muj .shdjvo XI6[ @?D. @D. A'. d1 C B. B A. (I6[1). I. 45. d2. D (I6[2).

(52) #D. .shI6[lk,c9 ABC c]t g,af I D 9qj’lhk’I[6lk,c9 A'B'C' g7yj’7nda[ I6[lk,c9 ABC mP[.ljg,af I D. $D. 9qj’lhk’I6[g7yj’7nda[I6[ A mP[da[g,af O muj.shdjvo. A A. O. B I. *D. (D. .shI6[lk,c9 ABC 8k,.9D 9qj’lhk’g,af I c]t J g7yj’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 .shI6[lk,c9 ABC 8k,.9D M c,jog,af|bj’muj16j.oI6[lk,c9 ABC. dD 9qj’lhk’G { g,af I g7yj’7nda[ M mP[.ljg,af A. { g,af J g7yj’7nda[ M mP[.ljg,af B { g,af K g7yj’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 lv’glAo-nj p c]t q 8A’lkddao16j O. A c,jog,af|bj’8k,.9[+1j6 p c]t q D dD 9qj’lhk’G { g,af A' g7yj’7nda[g,af A mP[.ljglAo-nj q. { g,af A" g7yj’7nda[ A' mP[.ljglAo-nj p. 0D 9qj’-Uc9’;jk g,af A'' c]t A g7yj’7ndaomP[.ljg,af O.. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. &D. ສດິ. C. 46.

(53) [qfmu !# dkog7yj’7nmP[.ljg,af|bj’ X8+? dyf9tde. B. d. A O. d 'D. dyf9tdemu @ B i .shmjvo-nj [AB] c]t g,af O. { 9qj’lhk’g,af A' c]t B' g7yj’7nda[ M g,af A c]t B mP[.ljg,af O. O A { 9qj’-Uc9’;jk mjvo-nj [AB] c]t [A'B'] 0t|kodao c]t gmqjkdaoD i .sh M c,jog,afg7yj’dk’0v’mjvo-nj [AB]. { 9qj’lhk’g,afg7yj’7n M' 0v’ M mP[.ljg,af O c]h;d;f7nog[yj’;jk g,af M' 16jmjvo-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. ສດິ. dyf9tdemu ! .shglAo-nj d F g,af O, A c]t B g-yj’;jkg,af A c]t B 16glAo-nj d. { 9qj’lhk’g,afg7yj’7n A' c]t B' 0v’ A c]t B mP[.ljg,af O. { 9qj’0ufglAo-nj d' zjkog,af A' c]t B'. { la’gdfgsaoco;.fdjP;da[lv’glAo-nj d c]t { I6[lujc9 ABA'B' gxaoI6[lujc9-toyf.fL. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !D. dyf9tdemu # gryjog;Qk;jk I6[|bj’,ul6odk’g7yj’7n s^n cdog7yj’7n g,njvI6[oAog8a’da[I6[g7yj’7n0v’ ,aogv’mP[.ljl6odk’g7yj’7n s^n cdog7yj’7n0v’,aoD { .o[aofkI6[]5j,oU 9qj’[vd9eo;ocdog7yj’7n c]t l6odk’g7yj’7n0v’,ao X4hk,u?D. 47.

(54) .97;k,. @D dkog7yj’7n0v’mjvo-nj c]t g,afg7yj’dk’ 0v’,aomP[.ljg,af|bj’D lv’mjvo-njg7yj’7ndaomP[.ljg,af|bj’ r;d,ao 0t|kodao c]t ,u];’pk;gmqjkdaoD { lv’mjvo-nj [AB] c]t [A'B'] g7yj’7ndaomP[.lj g,af O D gIqkwfh [AB]//[A'B'] c]t [AB] = [A'B'] D { M c,jog,afg7yj’dk’0v’mjvo-nj [AB] gIqkwfh M' c,jog,afg7yj’dk’0v’mjvo-nj [A'B'] D { I6[lujc9 ABA'B' c,joI6[lujc90hk’0t|koD 8q;1jk’G lhk’mjvo-nj [M'N'] g7yj’7nda[mjvo-nj [MN] mP[.ljg,af P muj.shdjvoD ;ymulhk’G 48. d. B. d'. O. A'. B' A B. k. O A' k '. ສດິ. !D dkog7yj’7n0v’glAo-njmP[.ljg,af|bj’D 4hklv’glAo-njg7yj’7ndaomP[.ljg,af|bj’ c,jolv’glAo-njoAo 0t|kodaoD A { glAo-nj d c]t d' g7yj’7ndaomP[.ljg,af O D gIqkwfh d // d' D 8q;1jk’G lhk’glAo-nj k ' g7yj’7nda[ k mP[.ljg,af O muj.shdjvoD ;ymulhk’G { ;k’lv’g,af A c]t B 8k,.916jglAo-nj k. { lhk’g7yj’glAo-nj [AO) c]t [BO) D { 0uflv’mjvodqj’muj,ug,af.9dk’ O c]t ]aflt\u AO c]t BO 8k,]efa[D B' { [aofkmjvodqj’8af [AO) c]t [BO) 16jg,af A' c]t B' D { 0ufglAo-njzjko A' c]t B' 9twfhglAo-nj k ' mujg7yj’7nda[glAo-nj k mP[.ljg,af O D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. #D. B. M. A'. O. A. M'. B' A'. B M A. O. B'.

(55) M' { 0ufg7yj’glAo-nj [MP) c]t [NP) D N { 0uflv’mjvodqj’muj,ug,af.9dk’16j P P fh;p]aflt\u MP M c]t NP 8k,]efa[D N' { lv’mjvodqj’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 gryjog;Qk;jkG <I6[\bj’,ul6odk’g7yj’7n< d=8+g,njvc8j]tg,af0v’I6[,ug,afg7yj’7n16j.oI6[ oAogv’D 8q;1jk’G. #D. [qfg/ydsaf. !D. 9qj’lhk’glAo-nj. d'. g7yj’7nda[. 0. 3. 4. 0. 1. 1. d. mP[.ljg,af O D. 0. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 9eo;ocdog7yj’7n 9eo;ol6odk’g7yj’7n. ສດິ. I6[. 1. d O. @D. #D. 9qj’lhk’glAo-nj g7yj’7nda[glAo-nj dD mP[.ljg,af A D 0D mP[.ljg,af B D. d1. c]t. d2 D. d1. B. A. 9qj’lhk’g7yj’glAo-nj [A'X') g7yj’7nda[g7yj’glAo-nj [AX) mP[.ljg,af .oc8j]td=]tou8+wxoUD. 49. d2. O.

(56) dD. 0D. O A. 7D. O. X. A. ’D. 7D. 0D. 7D. O •. O•. ’D. 9D. •O. •O • O. lD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 0D. • O. 9qj’[vd9eo;ocdog7yj’7n c]t l6odk’g7yj’7n0v’[aofkI6[]5j,oUD dD 0D. 50. ສດິ. 9qj’c8h,I6[g7yj’7n0v’I6[]5j,oUmP[.ljg,af O D dD. *D. X. 9qj’-vdskl6odk’g7yj’7n0v’I6[]5j,oUD la’gdfgsaoco;.fL dD. &D. O. X. ’D. O A. $D. A. X. 7D.

(57) [qfmu !$ dkog7yj’7n0v’,5,mP[.ljcdo c]t mP[.ljg,af dyf9tde dyf9tdemu ! .sh,5, xAy F glAo-nj Δ c]t g,af O D x dD { 9qj’lhk’g,af A' g7yj’7nda[ A mP[.ljglAo-nj Δ D { 9qj’lhk’g7yj’glAo-nj [A'x') c]t [A'y') g7yj’7nda[ [Ax) c]t [Ay) mP[.ljglAo-nj Δ D A { 9qj’xP[mP[,5, xAy da[ x'A'y' fh;pdko;afcmdD Δ 0D { 9qj’lhk’g,af A'' g7yj’7nda[ A mP[.ljg,af O D { 9qj’lhk’g7yj’glAo-nj [A"x") c]t [A''y") g7yj’7nda[ [Ax) c]t [Ay) mP[.ljg,af O D { 9qj’xP[mP[7jk;afcmd0v’,5, xAy da[ x''A''y'' fh;pdko;afcmdD. y. ສດິ. O. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !D. dyf9tdemu @ x .shlv’glAo-nj (xy) c]t (x'y') 8afdao1j6g,af O. O { 3fpmP[.ljl6odk’g7yj’7n O F 9qj’deoqfg7yj’glAo x' -njmujg7yj’7nda[ [Ox) F [Ox') c]t ,5,mujg7yj’7nda[ xOx' D { 9qj’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 ;jklv’,5,0hk,9v,D .o[aofk,5,]5j,oU ,u,5,.fc,jo,5, 0hk,9v,D. (1). (2). (3). 51. (4). y'. y.

(58) dyf9tdemu # v .shglAo-nj (uv) 8aflv’glAo-nj0t|kodao y A (xy) c]t (x'y') 16jlv’g,af A c]t B D x { 9qj’;k’g,af I 16j6mjvo-nj AB 3fp;jk y' B glAo-nj (x'y') g7yj’7nda[ (xy) mP[.lj I. x' u { 3fpmP[.ljl6odk’g7yj’7n I F 9qj’deoqf g,afg7yj’7n0v’ A F g7yj’glAo-njmujg7yj’7nda[ [Ax) c]t ,5,muj g7yj’7nda[ xAB. { 9qj’-Uc9’;jk xAB = ABy' { 9qj’[vd[aofk,5,mujg7yj’7ndaomP[.ljg,af I. { 9qj’-Uc9’;jk ABy' = x'Bu; xAB = x'Bu; c]t vAy = ABy' D i 4hklv’glAo-nj0t|kodao4nd8affh;pglAo-nj|bj’9txtdv[gxaos^kp,5,gmqjkdao8k, c8j]txtgrffaj’oUG ,5,-hvo0t\ko. d''. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ 4 1 3. 1 2 4 3. d'. d. d'. d''. 1 2 4 3. d. ສດິ. ,5,ltsâ[.o. d. 2. 1 2. d''. 4. d''. 1 2 4 3. 1. 2 4. d'. d 3 1. 1 2 4 3. 3. 2 4. d' 3. { 9qj’oe.-h75o]adltot0hk’gmy’oUgrnjv0Po[vd[aofk,5,ltsâ[.o c]t ,5,-hvo0t \komujgmqjkdao ma’\qf.odyf9tdemu #D. 52.

(59) .97;k, !D dkog7yj’7n0v’,5,mP[.ljglAo-nj|bj’ c]t g,af|bj’D x. x. A. A. y. y. Δ. O. y'. y'. A'. A' x'. x'. dkog7yj’7n0v’,5,mP[.ljglAo-nj. dkog7yj’7n0v’,5,mP[.ljg,af. lv’,5,g7yj’7ndaomP[.ljglAo-nj|bj’ sn^ g,af|bj’gmqjkdaoD 8q;1jk’G ,5, xAy c]t x'A'y' g7yj’7ndaomP[.ljglAo-nj Δ s^n mP[.ljg,af gIqkwfh xAy = x'A'y'. O.. ສດິ. i. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. @D. @D ;ymulhk’,5,g7yj’7nD. dD mP[.ljglAo-nj|bj’D 8q;1jk’G lhk’,5, A'O'B' g7yj’7nda[ AOB mP[.ljglAo-nj d muj.shdjvoD ;ymulhk’G { 0ufg7yj’glAo-nj [Ox) F [Ay) c]t [Bz) d 8A’lkdda[ d 16jg,af I F L c]t M 8k,]efa[D A B { 0ufmjvodqj’muj,ug,af.9dk’16j I, L c]t M muj M B' ,u]aflt\u OI, AL c]t BM F mjvodqj’8af O z L [Ox), [Ay) c]t [Bz) 16jg,af O', A' c]t B' D I { 0uf8+ B' sk O' c]t O' sk A' 9twfh,5, A' A'O'B' g7yj’7nda[,5, AOB mP[.ljglAo-nj d D O' y x. 53.

(60) 0D mP[.ljg,af|bj’ 8q;1jk’G lhk’,5, x'O'y' g7yj’7nda[ xOy mP[.ljg,af R. ;ymulhk’G y x { lhk’g7yj’glAo-nj [OR), [xR) c]t [yR). { 0uflk,mjvodqj’muj,ug,af.9dk’16j R muj,u]aflt\u O' OR, xR c]t yR. [aofkmjvodqj’8af O R [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' g7yj’7nda[,5, xOy mP[.ljg,af R.. O x'. 8q;1jk’G xOy'. xOx'. c]t. c]t. x'Oy. yOy'. c,jo,5,0hk,9v, gIqkwfh. xOy' = x'Oy. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. x. dD oypk,G g,njvlv’glAo-nj (xy) c]t (x'y') y' 8afdao16jg,af O F ,5, xOx' c]t yOy' gvUo;jk <,5,0hk,9v,<D 0D 75o]adltotD y 7jk;afcmd0v’lv’,5,0hk,9v,gmqjkdaoD c,jo,5,0hk,9v, gIqkwfh xOx' = yOy' D. $D ,5,mujwfh9kddko8afdao0v’glAo-nj|bj’da[lv’glAo-nj0t|kodao lv’glAo-nj0t|kodao8afda[glAo-nj|nj’xtdv[gxao,5,faj’oUG dD ,5,ltsâ[.oD i oypk,G lv’,5,mujwfh9kddko8afdao0v’ z glAo-nj|nj’8afda[lv’glAo-nj0t|kodaogvuho;jkG x y A ,5,ltsâ[.o g,njvG { r;d,ao[+8yfcxtdaoD y ' { r;d,ao[+16jrkdfP;dao0v’glAo8afD B x' { r;d,ao16jrkp.og0fmujdeoqffh;plv’ z' glaho-njoAoD. 54. ສດິ. #D ,5,0v’lv’glAo-nj8afdaoD.

(61) xAz'. c]t. zBy'. z x. A. y B. z'. c,jo,5,ltsâ[.oD i 75o]adltotD 4hklv’glAo-nj0t|kodao,5,ltsâ[.og-yj’xt x' dv[fh;pglAo8afoAo8hv’gmqjkdaoD 8q;1jk’G xAz' = zBy ' c]t x'Az' = zBy i 75o]adltotxUo7noD y' 4hklv’glAo-njxtdv[da[glAo8afgxao,5,ltsâ[ .ogmqjkdao lv’glAo-njoAo8hv’0t|kodaoD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 8q;1jk’G 4hk;jk x'Az' = zBy gIqkwfh ( x'x ) // ( y'y ) 0D ,5,-hvo0t|koD i oypk,G lv’,5,mujwfh9kddko8afdao0v’ x z glAo-nj|bj’da[lv’glAo-nj0t|kodaogvuho;jkG ,5,-hvo0t|ko g,njvG A { r;d,ao[+8yfcxtdaoD y { r;d,ao16jrkdfP;dao0v’glAo8afD x' { ,5,|bj’16jrkp.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 4hklv’glAo-nj0t|kodao ,5,-hvo0t|kog-yj’ z xtdv[fh;pglAo8afoAo8hv’gmqjkdaoD 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-njoAo8hv’0t|kodaoD z' 8q;1jk’G 4hk;jk xAz' = yBz' gIqkwfh ( x'x ) // ( y'y ). ສດິ. 8q;1jk’G. 55.

(62) #D. [qdg/ydsaf. !D. 9qj’lhk’,5,g7yyj’7n0v’,5,. xAy. ]5j,oUmP[.ljcdo. ΔD. Δ. Δ. A. A x. y. x. y. Xd?. X0?. Δ. Δ x. y. X7? @D. 9qj’lhk’,5,g7yyj’7n0v’,5,. X’? AOB. ]5j,oUmP[.ljg,af. A. P. y. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. A. A. ສດິ. x. P.. A P. O B. P B. Xd?. X0?. 56. A. B. X7?.

(63) #D. 9qj’lhk’glAo-njg7yj’7nda[. d1. c]t d2 mP[.ljg,af I .oc8j]td=]tou]5j,oU D. d1. d1. d2. d2. O. I O. 9qj’0Po,5,0hk,9v,c8j]t76jmujltcf’16jI6[]5j,oUG z. x'. x. B. x. C z'. (1). y'. .shglAo-nj d c]t d' 8afdao16jg,af O Xfaj’I6[?D 9qj’-vdsk7jk;afcmd0v’,5, 1; 2 c]t 3 D. 9qj’[vd7jk0v’,5,. a, b, c, d, e, f. 1 3. 55 d'. c]t. g. 16jI6[]5j,oUD. 60 a c b. c. g. d f. y'. ( 2). z'. d. x'. B. y. 2. *D. A. ສດິ. y A. &D. z. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. $D. ( 2). (1). I. g. e. f. (1). 57. e. d. ( 2). 120 a b.

(64) (D. .shlk,glAo-nj. d1; d 2. c]t. 4nd8affh;plv’glAo-nj. d3. d4. c]t. d5. Xfaj’I6[?D. 9qj’-vdsk 7jk;afcmd0v’,5, 1; 2; 3; 4; 5 c]t 6 D. d1. 5. 6. 110 d2 60. 4. d5. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. gryjo.shI6[lk,c9mjP’ ABC ( AB = AC ) . dD 9qj’-vdsk7jk0v’[aofk,5,.o0v’I6[lk,c9faj’djk;I6h;jk,5, B gmqjklv’ gmnjv,5, A D 0D 9qj’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 9qj’-vdsk7jk0v’,5, BDA c]t BDC D ’D .oI6[lk,c9mjP’ ABC ,uI6[lk,c9mjP’vnjo9adI6[L glAo-nj0t|koda[ AB ,u9adglAo-njL da[ BD ,u9adglAo-njL 9qj’.shgsfzqorhv,D. ສດິ. d4. )D. 2. 3. 1. d3. 58.

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

(66) 0D 9qj’0Po8nj,.lj8k8t]k’]5j,oU8k,8q;1jk’.oc4;muj|nj’D. . m. n. am. an. 5. 2. a5. a2. 4. 1. 3. 3. am an a5. m−n. 3. a m−n. a3. a2. am 9kd8k8t]k’ gIqkwfhG n = a.......−....... a m m 4hk m = n c]t a ≠ 0 F gIqkwfhG a n = a m a a. X.og’njvow0. m>n. c]t. a ≠ 0?. = a m−m = a...... =1. faj’oAo gIqkwfhG. a 0 = ......... ສດິ. a. m. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. vudfhko|bj’. am. 7D 3fpvu’.ljzqowfhIa[9kd0=h Xd? c]t X0?D 9qj’0Po8nj,.lj[jvo9Eg,afD i. ( a4 ). 2. = a4 × a4 = a......+....... i. ( a3 ). = a 4 ×...... 4. = a3 × ..... × ..... × ...... = a....+....+....+.... = a 3 ×...... faj’oAo gIqkwfhG ( a m ). n. = a....×..... 4. a ⎛a⎞ i ⎜ ⎟ = × ..... × ..... × ..... b ⎝b⎠ a × .... × .... × a = b × .... × .... × b a.... = .... b. 60.

(67) n. an ⎛a⎞ ⎜ ⎟ = n ⎝b⎠ b. faj’oAo gIqk9twfhG 3. ( ab ). i. = ab × ..... × ..... = a×a×a×b×b×b. = a...b.... faj’oAogIqk9twfhG ( ab )n = a...b.... .97;k, 8jk’l6o c]t 9eo;o4h;o[;d n D vjko;jk a 0Node]a’ n a n = a × a × a × ..... × a a gvUo;jk rNo c]t n gvUo;jk8q;de]a’D n gmnjv a. 8q;1jk’ !D 43 = 4 × 4 × 4 = 64 Xvjko;jk $ 0Node]a’ # gmqjk *$? @D (−5)3 = (−5) × (−5) × (−5) = −125 Xvjko;jk {& 0Node]a’ # gmqjkda[ {!@&? @D 75o]adltotD le]a[m5dM9eo;o a c]t gIqk9twfhG zqo76o0v’g]dde]a’. b. 8jk’l6oF. m. c]t. n. c,jo9eo;o4h;oleraf. a m × a n = a m+ n. 8q;1jk’ !D. 52 × 53 = 52+3. #D. @D. = 55 = 3125 5 6 x × x × x 7 = x 5+ 6 + 7. 23 × 24 × 2 −5 = 23+ 4+ ( −5). = 22. #D. =4 ( xy ) ( xy ) ( xy )11 = ( xy )9+10+11 9. = x18. am a. n. = a m−n. 10. = ( xy )30. zqosko0v’g]dde]a’ g,njv a ≠ 0 gIqkwfhG 4hk 61. m > n,. ສດິ. !D oypk,G .sh9eo;o. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. @D.

(68) am an am a. n. 45. 8q;1jk’ !D. 43. =. =. 1. 4hk. m < n,. 4hk. m = n.. a n−m am a. m. = a0 = 1. = 45 − 3. 73 × 96 × 85 × 7 4 9 85 7 4 1 = 9 7 − 6 × 1 × 4 −3 7 1 = 9× 7 9 = 7 dko76og]dlj;o a × c = a × c ? b d b×d. @D. = 42. = 16. X. 73 × 85 × 97. =. (22 ) 2 = 22 ×2. @D. ((−3) 2 )3 = (−3) 2 × 3. = 24. #D. = (−3)6. = 16 (a ) = a 4 × 5. $D. 4 5. =a. 85. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. (a m ) n = a m×n. 8q;1jk’ !D. × 6. ສດິ. de]a’0v’de]a’. 97. (x. = 729 ) × ( x 2 )1−a = x ( a +1) × 2 × x 2 × (1−a ). a +1 2. = x 2 a + 2 × x 2− 2 x. 20. = x (2 a + 2)+(2−2a ) = x 2 a + 2+ 2 − 2 a = x4. de]a’0v’zqo76o ( a × b) n = a n × b n. 62.

(69) 8q;1jk’ !D. @D [(−2)4 × 53 ]2 = (−2)4× 2 × 53× 2. (4 × 5) 2 = 42 × 52. #D. = 16 × 25. = (−2)8 × 56. = 400. = 256 × 3125. 3. $D. 3 3. (−4b) = (−4) b. = 800000 (2ab c ) = 24 a 4b 2 × 4c3× 4 2 3 4. = −64b3. = 16a 4b8c12. de]a’0v’zqosko n. an ⎛a⎞ ⎜ ⎟ = n b ⎝b⎠. 3. @D. 4 25. lyj’muj7;ogvqk.9.ljD i i i. i. ⎛ −3n3 ⎞ (−3)3 (n3 )3 ⎜⎜ ⎟⎟ = 43 ⎝ 4 ⎠ −27 n3× 3 = 64 27n9 =− 64. a1 = a 1 a −1 = . a −1 a 1 a−n = n . a−n a n 0 = 0, n ≠ 0. vjko;jk9eo;oxUo0v’ vjko;jk9eo;oxUo0v’ vjko;jkl6o0Node]a’. ສດິ. 2. 22 ⎛2⎞ = ⎜ ⎟ ⎝5⎠ 52. =. b≠0. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 8q;1jk’ !D. g-yj’;jk. a an. n. 8q;1jk’G i. (−5)0 = 1;. i. 81 = 8;. i i i. (−0,38)0 = 1;. 270 = 1;. (−0, 25)1 = −0, 25;. 1 ⎛a⎞ 5 = ; ⎜ ⎟ 5 ⎝b⎠ 1 1 7 −2 = 2 = ; 49 7 −1. 9eo;oxUo0v’. −1. 5. =. x1 = x;. (π )0 = 1 (−13)1 = −13. b a. 3−4 =. c,jo. 1 5. 1 1 = 34 81. s^n. 5−1 S. 63. 9eo;oxUo0v’. a b. c,jo. b a. s^n. ⎛a⎞ ⎜ ⎟ ⎝b⎠. −1. D.

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