Kibalirampuyibbiri(Integers)

Mu kibalangulo , "Kibalirampuyibbiri"(integers )zirimu:

(a)Kiddamaaso(Positives)
(b)Kiddannyuma(negatives)

Bivudde eri Muwanga Charles

Namba za Kibalirampuyibbiri (Integers)
(i)	Enjatula ya Kibalirampuyibbiri (Integers)

Kiddannyuma(Negatives)	Kiddamaaso(Positives)
-1	1 eya kiddannyuma	+1	1  eya kiddamaaso
-2	2 eya kiddannyuma	+2	2 eya kiddamaaso
-3	3 eya kiddannyuma	+3	3 eya kiddamaaso
-4	4 eya kiddannyuma	+4	4 eya kiddamaaso
-5	5 eya kiddannyuma	+5	5 eya kiddamaaso
-6	6 eya kiddannyuma	+6	6 eya kiddamaaso
-7	7 eya kiddannyuma	+7	7 eya kiddamaaso
-8	8 eya kiddannyuma	+8	8 eya kiddamaaso
-9	9 eya kiddannyuma	+9	9 eya kiddamaaso
-10	10 eya kiddannyuma	+10	10 eya kiddamaaso
-11	11 eya kiddannyuma	+11	11 eya kiddamaaso
-12	12 eya Kiddannyuma	+12	12 eya kiddamaaso

(ii)	Amakulu g’Omulamwa gwa Kibalirampuyibbiri

Yintegya ziba namba za kibalirampuyibbiri, ekitegeeza nti ziriko oluuyi olwa kiddannyuma (negatiivu ) n’oluuyi olwa kiddamaaso(pozitiivu). Namba eza kibalirampuyibbiri n’olwekyo zibaako oluuyi olwa kiddamaaso (pozitiivu) n’oluuyi olwa kiddannyuma (negatiivu).

Mu namba za kirumirampuyibbiri (yintegya), amateeka g’okugabanyaamu ge gamu ne    ag’okukubisaamu ate ag’okugatta ge gafuga okwawuza. 

Singa tutwaala ebintu ebirungi okuba n’akalaga k’ekibalo aka kiddamaaso + ate ebibi ne biba n’akalaga aka kiddannyuma, engeri y’okujjukiramu ebigoberero by’ekibalo eri mu bino wammanga: 

 ”Ebintu ebirungi (+) bwe bituuka ku bantu abalungi   (+) ekivaamu kiba kirungi (+). (Ebirungi + abalungi + = + Mu kugatta oba okukubisaamu kibalirampuyibbiri  n’olekyo kiddamaaso bw’ogattako oba n’okubisaamu kiddamaaso  ofuna ansa eri mu kiddamaaso.

Kyokka ebintu ebibi  (-) bwe bituuka ku bantu abalungi (+) ekivaamu kiba kibi (-). 
 Ebibi - abalungi + = -  Mu kugatta n’okukubisaamu kiddannyuma  bw’ogigatta oba n’okubisaamu ne kiddamaaso , ansa eba mu kiddannyuma

Ate era ebintu ebirungi (+) bwe bituuka ku bantu ababi  (-) kiba kibi , (-).
(ebirungi  + ababi - = - 

So ng’ate ebintu ebibi (-) bwe bituuka ku bantu ababi (-) ekyo kiba kirungi (+).
ebibi - ababi - = + 

(iii)	Okukubisaaamu n’Okugabiza mu kibalirampuyibbiri

Nnyonnyoka ebigoberero bino:

(a) Bw’oba okubisaamu oba ogabiza mu  namba bbiri ezirina akalaga  ke kami ekivaamu kiba mu kya kiddamaaso.Ebyokulabirako: 

(6)(8)    =  48
(-6)(-8) = 48

Ebikunizo  bino byombi birina namba ezikubisibwamu  ezirina obubonero bwe bumu , n’olwekyo  zivaamu emiwendo gya kiddamaaso. 

(b) Bw’oba okubisaamu oba ogabanya mu namba bbiri ezirina obubonero obw’enjawulo ekivaamu kiba mu negatiivu. Ebyokulabirako: 
(8) (-5) = -40
(-8)(5) = -40

Ebibuuzo bino byombi birina namba ezirina obubonero obw’enjawulo nga zikubisibwaamu, n’olwekyo ebizivaamu biba mu negatiivu. 

(iv)	Okukubisa oba okugabanya namba ezisukka mu bbiri 

(a)  Bwe tukubisa oba okugabiza yintegya  ebbiri eza negatiivu, tuba tufuna pozitiivu.Kino kitegeeza buli mugogo (pair) gwa yintegya  eziri mu negatiivu gutondekawo pozitiivu. N’olwekyo akabonero k’ekivaamu okafuna ng’obala  obubonero obwa negatiivu obuli mu kibuuzo. 

(b) Omuwendo gwa negatiivu bwe guba nga gw  kigabanya (even) ekivaamu kiba mu pozitiivu. Kyoka omuwendo gwa negatiivu bwe guba nga gwa kigaanira (odd), ekiddibwaamu kiba mu negatiivu. Ebyokulabirako:

(-5)(-4)(-3)(-2)(-1) = -120. Ekibuuzo kino kirina obubonero bwonna bwa negatiivu n’olwekyo ekivaamu kiri mu negatiivu.

(-4)(2)(3)(-1)(3) = 72. Ekibuuzo kino kirina obubonero bwa negatiivu 2 n’olwekyo ekivaamu kiba mu pozitiivu (= 24)

Kino era kye kimu ne mu kugabanyaamu(division) oba omugattiko (combination) gw’okukubisaamu n’okugabanyaamu. Kyokka  mu kugatta n’okwawuza  kino tekikola. 

(-4)(-8)(-2)  = 2
(-16)(2)

Ekibuuzo kirina obubonero obuli mu negatiivu 4; kino kitegeeza ekivaamu kiba mu pozitiivu.

(v)	Okubaza Emifunzo egya Kibalirampuyibbiri

Bw’oba ofundiza (powering) ekikolo  ekiri mu kiddannyuma   n’obufunza (powers), namba gy’ekubiseemu  emirundi egiragibwa akafunza olwo otunuulire akafunza. Akafunza bwe kaba ka   kigaanira (odd number) ansa eba  mu kiddannyuma. Akafunza bwe kaba ka kyegabanya (even number) ansa eba mu kiddamaaso. 

Okugeza: 

(a) (-4)4 = (-4)(-4)(-4)(-4) = 256

Ansa eri mu kiddamaaso  kubanga akafunza/ ka kyegabanya (even); kitegeeza waliwo namba y’obubonero bwa kiddannyuma  eya kigabanya (even) mu kikunizo. 

(a) (-2)7 = (-2)(-2)(-2)(-2)(-2)(-2)(-2) = - 128

Wano ekivuddemu kiri mu negatiivu kubanga akafunza si  ka kyegabanya n’olwekyo omuwendo gw’obulaga  obwa kiddannyuma  mu kibuuzo si gwa kagabanya; gwa kigaanira.

Manya: Singa ekibuuzo kiwandiikibwa n’akalaga aka negatiivu kyokka nga tekuli bukomera( parentheses), ansa erina kubeera mu kiddannyuma . Okugeza:

-33 = - (3) (3) (3) = - 27
-44 = - (4) (4) (4) (4) = - 256

Wano oba okiraba nti waliwo akalaga  aka negatiivu kamu kokka n’olwekyo.                                     
Bw’oba ofundiza (powering)  enfunzo eri mu negatiivu  n’obufunza (powers), namba gy’ekubiseemu  emirundi egiragibwa mu kafunza olwo otunuulire akafunza. Akafunza bwe kaba ka   kigaanira (odd number) ansa eba  mu negatiivu. Akafunza  bwe kaba ka kyegabanya (even number) ansa eba  mu pozitiivu. 

Okugeza: 

(a) (-4)4  =   (-4)(-4)(-4)(-4) = 256

Ansa eri mu pozitiivu kubanga akafunza/ ka kigabanya (even); kitegeeza waliwo namba y’obubonero bwa negatiivu eya kigabanya (even) mu kibuuzo. 

(b)  (-2)7  = (-2)(-2)(-2)(-2)(-2)(-2)(-2) = - 128

Wano ekivuddemu kiri mu negatiivu kubanga kipowesa si ya kagabanya n’olwekyo omuwendo gw’obubonero  obwa negatiivu mu kibuuzo si gwa kagabanya. 

Manya: Singa ekibuuzo kiwandiikibwa n’akabonero aka negatiivu kyokka nga tekuli bukomera (no parentheses), ansa erina kubeera mu negatiivu.  

Okugeza:

-33   = -(3)(3)(3) = - 27

-44  = -(4)(4)(4)(4) = - 256

Wano oba okiraba nti waliwo akabonero ka negatiivu kamu kokka n’olwekyo.                                

(vi)	Akafunza bwe kaba mu Negatiivu 

Akafunza bwe kaba  mu negatiivu kitegeeza nti namba eri wakati wa 0 ne 1. (nkutulemu/fraction). N’olwekyo namba erina akafunza akali mu negatiivu yenkana n’ensulike (reciprocal) yayo nga erina akafunza aka pozitiivu. 
Okugeza:

3-4= 1/(34)=1/81
Kikwetaagisa okunnyonnyoka emifunza  okusobola  okusobola okukola  okubalanguza okw’enjawulo naddala mu  namayingo (polynomials), omuli nnyingemu (monomials), nnyingobbiri (binomials), ne nnyingossatu (trinomials).
Ebyokulabirako:
(a)	Sonjola (-3)3    
Eky’okukola: 
Laga nti kino kitegeeza kye kimu ne  (-3)(-3)(-3)  =   -27

Lwaki? Laba nti waliwo obukomera. Ekitegeeza nti ekikunizo kiri  (-3) eya kyesatuza (cubed). Akafunza “3”  kalaga nti entobo (-3) erina okwekubisibwamu yo yennyini emirundi esatu..

(b)	Sonjola   -43 - (-4)2 + (-3)2

Ekibazo :  

     -(4)(4)(4) - (-4)(-4) + (-3)(-3) = 
     -64 - 16 + 9 =
     - 39
Okusonjola omweyoreko ogulimu obufunza  ky’okola kwe kusooka okulaba oba nga:  
	Mulimu namba eya negatiivu. Ky’olina okujjukira:
	Omweyoreko gulimu obukomera. Tandikanga na kusonjola ekintu kyonna ekiri munda mu bukomera. Eky’okulabirako:  
        -54   = -(5)(5)(5)(5)            = -625
        (-5)4 = (-5)(-5)(-5)(-5)         = 625

Okusonjola emyeyoreko egirimu obufunza kyangu nnyo. Ky’okola kwe kusooka okulaba oba mulimu namba eza negatiivu n’obulaga obwa negatiivu  kubanga buno bukyusa ekibazo.  Eky’okulabirako:
3.  Sonjola:   yx4z2
                 y = 2, x =3, z = 5

Ekibazo:  

  	 Sikiza:   (2)(3)2(5)3                     
                  (3)(9 )(75)
    	       =  2,025
Ebibuuzo bino byombi birina namba ezirina obubonero bwe bumuezikubisibwaamu, n’olwekyo zivaamu emiwendo gya pozitiivu. 
(ii) Bw’oba okubisaamu oba ogabanya mu namba bbiri ezirina obubonero obw’enjawulo ekivaamu kiba mu negatiivu. Ebyokulabirako: 
(8) (-5) = -40
(-8)(5) = -40
Ebibuuzo bino byombi birina namba ezirina obubonero obw’enjawulo nga zikubisibwaamu, n’olwekyo ebizivaamu biba mu negatiivu. 

*Manya: Amateeka g’okugabanyaamu ge gamu ne ag’okukubisaamu ate ag’okugatta ge gafuga okwawuza. Singa tutwaala ebintu ebirungi okuba n’akabonero k’ekibalo aka pozitiivu + ate ebibi ne biba n’akabonero aka negatiivu -, engeri y’okujjukiramu amateeka g’ekibalo gano eri mu bino wammanga: 

”Ebintu ebirungi (+) bwe bituuka ku bantu  abalungi  (+) ekivaamu kiba kirungi (+). Ebirungi + abalungi + = + Kyokka ebintu ebibi  (-) bwe bituuka ku  ombi abalungi (+) ekivaamu kiba kibi (-).   Ebibi – abalungi + = - 

Ate era ebintu ebirungi (+) bwe bituuka ku  ombi ababi  (-) kiba kibi, (-).
(ebirungi  + ababi - = - 

So ng’ate ebintu ebibi (-) bwe bituuka ku  ombi ababi (-) ekyo kiba kirungi (+).
ebibi – ababi - = + 

Okukubisa oba okugabanya namba ezisukka mu bbiri 
Bwe tukubisa oba okugabiza entegere ebbiri eza negatiivu, tuba tufuna pozitiivu. Kino kitegeeza buli mugogo(pair) gwa ntegere eziri mu negatiivu gutondekawo pozitiivu. N’olwekyo akabonero k’ekivaamu okafuna ng’obala  obubonero obwa negatiivu obuli mu kibuuzo. 
Omuwendo gwa negatiivu bwe guba nga gw  kigabanya(even) ekivaamu kiba mu pozitiivu.Kyoka omuwendo gwa negatiivu bwe guba nga gwa kigaanira(odd), ekiddibwaamu kiba mu negatiivu.Ebyokulabirako: (-5)(-4)(-3)(-2)(-1) = -120. Ekibuuzo kino kirina obubonero bwonna bwa negatiivu n’olwekyo ekivaamu kiri mu negatiivu.

(-4)(2)(3)(-1)(3) = 72. Ekibuuzo kino kirina obubonero bwa negatiivu 2 n’olwekyo ekivaamu kiba mu pozitiivu (=24)

Kino era kye kimu ne mu kugabanyaamu(division) oba omugattiko(combination) gw’okukubisaamu n’okugabanyaamu. Kyokka  mu kugatta n’okwawuza  kino tekikola. 

(-4)(-8)(-2)  = 2
(-16)(2)

Ekibuuzo kirina obubonero obuli mu negatiivu 4, kino kitegeeza ekivaamu kiba mu pozitiivu .
1.	-(- 8) = 8 Lwaki?
Ansa: Kiddannyuma, negatiivu, oba kikontana ya -(-x) = x, n’olwekyo negatiivu ya -(-8) = 8. Kino  kiri kityo ku namba yonna x: -(-x) = x. Negatiivu ya – x eba x ate negatiivu ya x eba –x. Kino kitegeeza buli lw’olaba omweyoreko oguli nga –(-x) manya nti kivaamu “x” bwe kiba –(-7) = 7 

2.	Balanguza:

a)	–(–15) =			e) – (5 – 8) = 
b)	–(–y) =			f) –(3 + 5 – 12) = 
c)	–(–a) = 			g) – (–10) =
d)	–(–5) = 			h) – (–k ) = 

3.	Namba yonna x egendana na namba emu yokka – x eyitibwa negatiivu yayo. Bw’ogatta namba eno x ku – x oba ofuna 0. Balanguza:
i)	y + (–y) = 
ii)	y + (–x) = 
iii)	113 + (-113) = 
iv)	15 + (–15) = 

4.	Bw’oba olina “kavu” wa siringi emitwalo esatu (30,000=) n’owandiika kyeeke ya mitwalo ena kitundu ( 45,000=). Oba osigaza sente meka ku akawunta yo?
Ekigoberero: Ekitono bw’oyawulako ekinene(ekinji), osigaza kya negatiivu (ekiddannyuma).
N’olwekyo 30,000 – 45,000 =  –15000 
Kino(-15,000) kitegeeza banka eba ekubanja siringi 15,000 kuba osabye sente ezisinga kwezo z’olina ku akawunta yo.

5.	Kiriwa ekitali kituufu ku namba za kibalirampuyibbiri.
a)	Ziba namba yintegya (intergers)
b)	Ziba namba nzijuvu omuli ne ziro
c)	Zigenda ekiddannyuma ne ekiddamaaso ,ekitegeeza mu negatiivu ne positiivu.
d)	Kibalirampuyibbiri giba mikutule(fractions).
6.	Namba ki gy’ogatta ku 11 okufuna ziro(0) ? Koloboza ku ansa entuufu.
(a)– 1
(b)10
(c)–11
(d)11
7.	Namba ki gy’ogatta ku 1,751 okufuna ziro.
(a)751
(b) 1,751
(c )- 751
(d ) -1,751
8.	Negatiivu oba kiddannyuma ya – a eba namba ki? Lwaki?
(a) 5 , kubanga namba nzijuvu
(b)– 5,  kubanga namba ya kiddannyuvu
( c ) 15 , kubanga ya kiddamaaso
(d) a , kubanga – a + a = 0
9.	Singa x + y = 0, kakwate ki akali wakati wa x ne y.
x = y 
x = y2
x = -y 
x = x/y
10. Okulaga obukakafu nti y – x ye kiddannyuma (negatiivu) ya x – y okola otya?
Olaga nti x + y – y + x = 0
Olaga nti x – y + y – x = 0
Ekyokulabiriko: 5 – 8 + 8 – 5 
                          –3 + 3 = 0
Ebikunizo ku Okugatta n’okwawuza kibalirampuyibbiri
1.	Saza ku kitali kituufu:
(a)Namba za kibalirampuyibbiri ziba yintegya (integers)
(b)Namba za kibalirampuyibbiri ziba za kiddannyuma ne kiddamaaso.
(c)Namba za kibalirampuyibbiri ziba za negatiivu ne positiivu.
(d)Namba za kibalirampuyibbiri ziba za kigaanira zokka.

2.	Kiriwa ekikyamu ku bino:
(a)Bwe tugatta namba ya positiivu tufuna ennene okusingawo.
(b) Tekisoboka kugatta namba za positiivu.
(c) Bwetugatta namba ya negatiivu ekivaamu kiba kikendeevu (kitono) okusingako.
(d)x + y = x + y ate x + ( - y) = x – y ekitegeeza: x + y > x – y

3.	Mweyoreko ki ku gino omubage obulungi ku bubonero bw’ekinambirizo n’obubonero bwa aligebula(ekinyukuto)
(a) x + - y 
(b) x + (- y)

4.	Mu aligebula (ekinyukuto) tukozesa obukomera ( ) okwawula akabonero ak’ekinambirizo + ku kabonero aka algebula – . Ku myeyoleko gino, guli wa ogusaanidde(ogusinga okuba omubage obulungi) ?
(a) (8) + - 3 = 5
(b)8 + ( - 3 ) = 11
( c ) 8 + ( - 3 ) = 5 

5.	Omweyoleko 10 + (-20) + 30 (-40) gulimu ennyingo nnya: 10, -20, 30 ne -40. Saza ku kitali kituufu ku bigambululo bino wansi:
(a)Akabonero aka kiddannyuma kiba kitundu kya nnyingo. Ekyokulabirako,        – 40 (namba eno esomebwa n’akalaga kayo :negatiivu ana).
(b)Ennyingo bw’eba teriiko kabonero konna kalagiddwa, omanyirawo nti ya kiddamaaso (positiivu). Ekyokulabirako: 30 = + 30
(c)Ennyingo za negatiivu z’ezo ezirina akalaga aka +(positive sign).
6.	Mu buli mweyoleko ku gino wansi, laga buli nnyingo egulimu:
i)	1 + (-2) + 3 + (-4) + 6
ii)	5 – 6 + 8 – 7
iii)	–x – y + c + z
iv)	-13 – 7
7.	Saza ku kikyamu mu bigoberero(rules) by’ okugatta ennyingo:
a)	Singa ennyingo ziba n’akalaga (sign) ke kamu, gatta emiwendo gyazo egy’enkomeredde (absolute values),olekeko akalaga (sign) ke kamu.Eky’okulabirako:  - 5 + - 7 =  - 12  oba  5 + 7 = 12.
b)	Singa ennyingo ziba n’akalaga ke kamu gatta emiwendo okyuse akalaga.
c)	Singa ennyingo ziba n’obulaga obwa kikontana (opposite signs), yawuza oba yawula ennyingo entono ku nene (mu muwendo ogw’enkomeredde) olyoke olekeko akalaga ak’ennyingo esingako obunene. Ekyokulabirako: 3 + (- 5) = - 2, naye  – 3 + 5 = 2
d)	Bw’ofuna ekyewolo (loan) kya siringi 15, n’osasulako siringi 8 zokka, mu aligebula tukiraga nga: – 15 + 8 = - 7 ekitegeeza nti okyabangibwa siringi 7.
8.	Singa wewola siringi 500 okuva mu UCB n’oyongera okwewola siringi 150 okuva mu Tunakopesha, kino kiragibwa kitya mu aligebula?  Nga:
a)	500 + 150 = 650   oba 
b)	– 500 – 150 =  – 650 
Manya: mu aligebula tugamba nti “tugasse ennyingo yadde nga obulaga bwa kwawuza. Buli nnyingo tugyogera nga bw’eri ate ansa ne guba omugatte (sum). Waggulu mu (b), – 650 gwe mugatte gw’ennyingo -500 ne -150
9.	Gatta ennyingo z’emyeyoreko gino
i)	3 + 2 = 			vi) 10 + ( - 5) = 
ii)	– 3 – 2 =			vii) – 10 + 5 = 
iii)	7 + (– 3) = 			viii) 1 + ( - 1 ) = 
iv)	–7 + 3 = 			ix) – 1 – (–1) = 
v)	10 + (–5) = 

10. Mu aligebula okugatta oba okwawuza ziro ku nnyingo yonna tekirina kye kigikyusa, naye okwawula namba yonna ku ziro osigaza namba eyo mu negatiivu. Eky’okulabirako: 0 – 8 = – 8 , 8 – 0 = 8 .Balanguza:
i)	– 4 + 0 = 
ii)	0 – 12 =
iii)	12 – 0 =

10.	Okwawuza namba ya negatiivu nga 1 – (-1) = 1 + 1 = 2.  Baza
i)	– 3 – (– 7) =
ii)	– x – (– y) =
iii)	x – (– y)  = 
iv)	20 – (– 4) = 
v)	– 20 – (– 4 ) 

11.	 Wandiika ennyingo zino nga ojjeeko obukomera (brackets). Eky’okulabirako:
a + ( - b) = a – b
a – ( - b) = a + b
i)	– 2 – (– 3) + (– 4) – 6 + 5 =
ii)	– 4 – (–8) + (–8 ) – 10 + (–9)=

12.	Kyusa  a + 7  efuuke  x – a 
Ekibazo: a + 7 = a – (– 7).Kyusa zino mu ngeri y’emu:
i)	a + 5 = 
ii)	a – 5 =
iii)	a + 3 = 
iv)	a – 3 = 
Ebikunizo ku kukubisa n’okugabiza mu kibalirampuyibbiri
1.	Saza ku kigoberero ekitali kituufu mu kukubisa n’okugabiza ennyingo: 
a)	Obulaga obutafaanagana butondeka namba ya pozitiivu.
b)	Obulaga obufaanagana (like signs) butondeka (buvaamu) namba ya pozitiivu ate obutafaanagana buvaamu namba ya negatiivu.
c)	– x / – y = x / y , - x / y = -( x/y), x/-y = – (x/y) 

2.	Sazaamu ekikyamu
a)	- a (-b) = ab
b)	a(–b) = –ab
c)	-a/-b = –(a/b)
d)	a /–b = -(a / b)
e)	–a(–b) = a + b

3.	Balanguza:
i)	– 3 (–4) =
ii)	3(– 4 ) = 
iii)	– 15 ÷ (–5 ) =-15/-15= 
iv)	15 ÷ ( - 5) = 15/-5=

4.	Okukubisa n’okugabiza namba kulimu ebigobererwa eby’enjawulo. Saza ku kikyamu ku bino :
a.	Obulaga obutafaanagana buvaamu ennyingo ya negatiivu. 
b.	Obulaga obufaanagana buvaamu ennyingo ya pozitiivu.
c.	Obulaga obutafaanagana buvaamu ennyingo ya pozitiivu. 
5.	Omweyoleko x-y (-z) gulimu ennyingo bbiri: x ne y (-2). Baza 
i)	5-7(-3)
Ekibazo: sooka osonjole ennyingo erimu obukomera: 7x-3= -21
Zzaako etaliiko bukomera: 5-(-21) = 5+21
Ofuna  = 26
ii)	(3-4) (-5) = 
iii)	a4-6   (-2) =
iv)	(2-3) (-7) =
v)	(-10)  (-4) =
6.	Mu aligebula x-y(-z)=x-yz. Balangul:
i)	5 – 3(-2) =
ii)	-5-3(-2) =
iii)	-5+3(-2)=

7.	Namba ya kyegabanya (even number) eya nambuluzo ya negabivu evaamu nnyingo ya positiivu ate namba ya kigaanira eya nambuluzo eya negatiivu evaamu nnyingo ya negatiivu.

8.	Nambuluzo z’okubisaamu bwe ziba nga obunji bwazo gwa kyegabanya (even) ekitondeko ekivaamu kiba mu pozitiivu,  kyokka obunji bwazo bwe buba bwa kigaanira, ekitondeko kiba kya kigaanira ekitondeko kiba kya negatiivu. Eky’okulabirako;
(-a)x(-a)x(-a)x(-a) = a4
(-a)x(-a)x(-a)x(-a)x(-a) = -a5
(-2) (-2) (-2) (-2) = 16
(-2) (-2) (-2) (-2) (-2) = -32

Balanguza:
i)	(-1) (-1) (-1) =
ii)	(-1) (-1) (-1) (-1)  =
iii)	(-1) (-1) (-1) (-1) (-1)  =
iv)	2(-5) (-2) (-4) =
v)	2(-5) (-2) (-4) (-3) =
vi)	(-1) (-2) (-3) =
vii)	(-1) (-2) (-3) (-4) =
viii)	(-3)4 (-6)6 =
ix)	3(-2)4 =
9.	Okukubisa emikutule(fractions), kubisa kinnawaggulu okubise ne kinnawansi.
Ekyekulabirako:    x/y x n/m  =  yn/ym	
Manya: yn  ye kinnawaggulu ate ym ye kinnawansi	

Balangula:
i)	½ x ¼ 
ii)	2/3 x 3/5 
iii)	5/7 x 1/3
iv)	2/5 x 2/9
Balanguza nga ojja obukomera ku myeyoleko gino wansi:
i)	d + (f – g + h) =
ii)	d – (f – g + h) = 
iii)	25 + (5 – 2 + 1) =
iv)	25 – (5 – 2 + 1) =
v)	(a – 2) – (b – 3) =
vi)	(g + 3) + (h + 5) =
vii)	(g + 3) – (h + 7) =
viii)	(a – 4) + (b + 7) = 
ix)	(a – 4) – (b + 7) =
x)	(a – 4) – (b – 7) =
xi)	– (–a + b) =
xii)	– (a – b) =
Nga bwe kisoboka okujja obukomera ku myeyoleko ate era kisoboka okuteeka obukomera ku myeyoleko. Tusobola okiwandiika h – k + m – n
Nga: h – (k – m + n) oba 
         (h – k) – (- m + n) oba 
        h – (k – m) – n
Ddamu owandiike buli mweyoleko guno nga oteekamu obukomera 
(i)	 – a + b = – (a – b)
(ii)	– a – y = – (a + y) 
(iii)	– x + y – z + m = - (x – y + 2 – m)
(iv)	– 5 – 7 =
(v)	– 5 + 7 =
(vi)	– 5 + 3  – 4 + 2 =
Jjukira nti  okusonjola omweyoleko ogulimu obukung’anyizo (braces), obuweto (brackets) n’obukomera (parenthesis), sooka ojjeko ekisinga okuba munda osembyeyo ekyo ekisembayo wabweru.Eky’okulabirako balanguza:
i)	– 5 + 3 – (x – 2)   = - 5 + 3 – (x – 2)
= - 2 – x + 2
= - x

Olupapula luno lwetaaga program ey'enjawulo okulutereeza !!!!!! Lindirira