m m
m
+ + +
K
1442443
márte
n
4)
− +
− +
−
(3
) (3
) (3
);
b a
b a
b a
8)
;
K
144244
3
márte
k
b+ b+ + b
Kóbeymeni dáreje túrinde jaz
(123125):
123.
1)
⋅ ⋅ ⋅ ⋅
2 2 2 2 2;
2)
⋅ ⋅ ⋅ ⋅
1 1 1 1 1 ;
3 3 3 3 3
3)
( ) ( ) ( )
⋅
⋅
3
3
3
4
4
4
;
4)
−
⋅ −
⋅ −
⋅ −
( 2,7) ( 2,7) ( 2,7) ( 2,7).
124.
1)
⋅ ⋅ ⋅ ⋅
;
x x x x x
3)
⋅
⋅
(2 ) (2 ) (2 );
a
a
a
2)
⋅ ⋅ ⋅ ⋅
;
m m m m m
4)
−
⋅ −
⋅ −
⋅ −
( 3 ) ( 3 ) ( 3 ) ( 3 ).
b
b
b
b
125.
1)
− ⋅ − ⋅ −
(
) (
) (
);
x y
x y
x y
3)
⋅
3
3
2
2
;
x
x
2)
+ ⋅ +
(
) (
);
a b a b
4)
⋅ ⋅ ⋅ ⋅
.
m m m m m
n n n n n
Kóbeymeni dáreje túrindegi jazwnan paydalanp, a
latpan
ápiwaylastr
(126
128):
126.
1) 2
.
2
.
2
.
15;
3) 5
.
5
.
8
.
8
.
8
.
2
.
2;
2) 4
.
4
.
4
.
4
.
21;
4) 6
.
6
.
7
.
7
.
3
.
3
.
3.
127.
1) 1,2
.
1,2
.
2
.
2
.
5
.
5;
2) 0,5
.
0,5
.
0,5
.
2
.
2
.
4
.
4;
3)
⋅
⋅ ⋅ ⋅ ⋅
1 1 1 1
7 7 7 7
0,3 0,3
;
4)
⋅ ⋅ ⋅
⋅
2 2 2
3 3 3
2,3 2,3.
128.
1) 9
.
9
.
9
.
a
.
a
.
a
;
3)
−
⋅
−
⋅ ⋅
(
) (
)
;
x x x
y y y
x y
x y
2)
x
.
x
.
x
.
x
.
3
.
3;
4)
− ⋅
− ⋅
−
⋅ ⋅
(8
) (8
) (8
)
.
a a
b b
a b
a b
a b
57
A
latpan ápiwaylastr
(129130):
129.
1)
p
.
p
.
p
.
p
+
q
.
q
;
3)
a
.
a
+
a
.
a
+
a
.
a
;
2)
a
.
a
+
b
.
b
.
b
.
b
;
4)
x
.
x
.
x
+
x
.
x
.
x
.
130.
1)
;
c c c c
c c
⋅ + ⋅ + + ⋅
K
144424443
márte
k
3)
;
a a
a b b
b
⋅ ⋅ ⋅ + ⋅ ⋅ ⋅
K
K
14243 14243
márte
márte
k
m
2)
;
a a a a a a
a a a
⋅ ⋅ + ⋅ ⋅ + + ⋅ ⋅
K
1444442444443
n
márte
4)
5 5
5
a a
a
⋅ ⋅ ⋅ + ⋅ ⋅ ⋅
K
K
14243 14243
k
márte
17 márte
131.
A
latpan oq
, dárejeni tiykarn, dáreje kórsetkishin ayt
:
1) 3
2
;
3)
( )
−
41
2
9
;
5)
+
15
(4
) ;
m n
2)
( )
3
3
8
1
;
4)
−
39
( 1,2) ;
6)
( )
7
2
3
.
a
b
Esapla
(132
139)
:
132.
1) 2
3
;
2) 3
2
;
3) 4
4
;
4) 5
3
.
133.
1) 1
5
;
2) (
−
1)
7
;
3) 0
15
;
4) 0
5
.
134.
1)
( )
3
2
3
;
2)
( )
2
3
5
;
3)
( )
2
2
7
1
;
4)
( )
3
1
3
2
.
135.
1) (2,5)
2
; 2) (1,7)
2
; 3) (
−
0,2)
3
;
4) (
−
0,2)
4
.
136.
1) (
−
5)
3
; 2)
−
5
3
;
3)
( )
−
2
1
4
;
2
4)
( )
−
2
1
4
.
2
137.
1)
−
4
5
( 0,2)
(0,1)
;
2)
−
3
4
(0,3)
( 0,1)
;
3)
2
2
(3,2)
(1,6)
;
4)
2
2
(2,6)
(1,3)
.
138.
1) 2
.
(
−
3)
2
;
2)
−
5
.
(
−
2)
3
;
3)
− ⋅ −
2
1
2
( 4) ;
4)
− ⋅ −
2
2
3
( 3) .
139.
1)
( )
−
⋅ −
2
3
5
( 5)
;
2)
( )
−
⋅ −
3
2
3
( 3)
;
3)
− −
⋅
2
3
( 3) 2 ;
4)
− −
⋅ −
2
3
( 3) ( 2) .
58
140.
−
−
−
2
2
3
; ( ) ; ( )
x
x
x
a
latpan mánisin
=
−
1
2
1 ;
5
x
de tab
.
141.
x
2
a
latpasn mánisin
x
ti kestede kórsetilgen mánisleri
ushn esapla
:
142.
x
3
a
latpasn mánisin
x
ti kestede kórsetilgen mánisleri
ushn esapla
:
143.
Tómendegi pikirlerdi qays biri durs, qays biri nadurs?
Sebebin túsindiri
. Pikir nadurs dep aytsa
z, on biykar
etiwshi msal tab
.
1) eki sann kvadratlar te bolsa, bul sanlard ózleri de
te
;
2) eki sann kublar te bolsa, bul sanlard ózleri de
te
;
3) eger teris san®a on kvadrat qoslsa, o san payda
bolad;
4) eger teris sannan on kvadrat alnsa, teris san payda
bolad;
5) eger o sannan on kvadrat alnsa, o san payda
bolad.
Tómendegi pikirlerdi qays biri durs, qays biri nadurs?
Sebebin túsindiri
. Sáykes msallar dúzi
(144145)
:
144.
1) natural sann kvadrat qálegen san menen tamamlanw
múmkin;
2) natural sann kub qálegen san menen tamamlanw
múmkin.
145.
1) natural sann tórtinshi dárejesi tek 0; 1; 5; 6 san-
larnan biri menen tamamlanw múmkin.
2) natural sann besinshi dárejesi sol san qays san me-
nen tamamlan®an bolsa, sol san menen tamamlanad.
x
x
2
0 1
−
1 2
−
2 3
−
3 4
−
4 5
−
5 6
−
6
x
x
3
0 1
−
1 2
−
2 3
−
3 4
−
4 5
−
5 6
−
6
59
10-
Natural kórsetkishli dárejeni qásiyetleri
Dárejege kóteriw birneshe áhmiyetli qásiyetlerge iye.
1-qásiyet.
+
⋅
=
.
m
n
m n
a a
a
Tiykarlar birdey dárejelerdi kóbeytiwde tiykar ózgerissiz
qalad, dáreje kórsetkishleri bolsa qoslad
.
Natural kórsetkishli dárejeni anqlamasna muwapq
{
2
3
2 2
(2 2) (2 2 2)
⋅ =
⋅
⋅ ⋅ ⋅ =
1
424
3
2 márte 3 márte
(
) (
)
m
n
a a
a a a
a
a a a
a
⋅
= ⋅ ⋅ ⋅ ⋅ × ⋅ ⋅ ⋅ ⋅
K
K
1442443 1442443
márte
márte
m
n
kóbeytiwdi gruppalaw nzamna muwapq
2 2 2 2 2
= ⋅ ⋅ ⋅ ⋅ =
14243
5 márte
a a a
a
= ⋅ ⋅ ⋅ ⋅ =
K
14
4244
3
márte
(m+n)
natural kórsetkishli dárejeni anqlamasna muwapq
= 2
5
.
=
a
m
+
n
.
Solay etip,
2
2
· 2
3
= 2
2+3
.
a
m
·
a
n
=
a
m
+
n
.
2-qásiyet.
−
=
>
≠
:
,
,
0
m
n
m n
a
a
a
m n a
.
Tiykarlar birdey dárejelerdi bóliwde tiykar ózgerissiz
qalad, dáreje kórsetkishleri bolsa alnad.
Shártke muwapq
5 > 3.
m
>
n
,
a
≠
0.
Dárejeni birinshi qásiyeti boynsha
2
5
3
.
2
3
= 2
5
.
a
m n
.
a
n
=
a
m
.
Son ushn
2
5
3
= 2
5
: 2
3
.
a
m n
=
a
m
:
a
n
.
60
Solay etip,
2
5
: 2
3
= 2
5
3
.
a
m
:
a
n
=
a
m n
,
m
>
n
,
a
≠
0.
=
≠
1,
0
n
n
a
a
a
ekenin aytp ótemiz.
3-qásiyet.
=
( )
m n
mn
a
a
.
Dárejeni dárejege kóteriwde tiykar ózgerissiz qalad,
dáreje kórsetkishleri bolsa óz ara kóbeytiledi.
Natural kórsetkishli dárejeni anqlamasna muwapq
=
⋅
=
3 2
3
3
(2 )
2 2
(
)
m n
m
m
m
m
a
a
a
a
a
=
⋅
⋅
⋅ ⋅
K
14442444
3
márte
n
dárejeni birinshi qásiyeti boynsha
= 2
3 + 3
=
m m
m
a
+ + +
=
=
K
6447448
márte
n
kóbeytiwdi anqlamasna muwapq
= 2
3
.
2
.
=
a
mn
.
Solay etip,
(2
3
)
2
= 2
3
.
2
.
(
a
m
)
n
=
a
mn
.
4-qásiyet.
=
( )
n
n n
ab
a b
.
Kóbeymeni dárejege kóteriwde hárbir kóbeytiwshi us
dárejege kóteriledi.
3
(2 3)
(2 3) (2 3) (2 3)
⋅
= ⋅ ⋅ ⋅ ⋅ ⋅
14442444
3
3 márte
( )
( )( ) ( )
n
ab
ab ab
ab
=
=
K
1442443
n
márte
kóbeytiwdi gruppalaw hám orn almastrw nzam boynsha
(2 2 2) (3 3 3)
= ⋅ ⋅ ⋅ ⋅ ⋅
1
424
3 1
424
3
3 márte 3 márte
(
)(
)
a a
a b b
b
= ⋅ ⋅ ⋅
⋅ ⋅ ⋅ =
K
K
14243 14243
n
n
márte
márte
natural kórsetkishli dárejeni anqlamasna muwapq
=
2
3
· 3
3
.
=
a
n
·
b
n
.
Solay etip,
(2·3)
3
=
2
3
· 3
3
.
(
ab
)
n
=
a
n
b
n
.
61
5-qásiyet.
( )
=
≠
;
0
n
n
n
a
a
b
b
b
.
Bólshekti dárejege kóteriwde on alm hám bólimi
tap us dárejege kóteriledi.
Natural kórsetkishli dárejeni anqlamasna muwapq
⋅ ⋅
14243
3
2
2 2 2
=
=
3
3 3 3
3 márte
⋅ ⋅
14243
3
2
2 2 2
=
=
3
3 3 3
3 márte
bólsheklerdi kóbeytiw qa®ydasna muwapq
2 2 2
3 3 3
⋅ ⋅
⋅ ⋅
=
=
678
123
3 márte
3 márte
a a
a
b b
b
⋅
⋅
⋅
⋅
=
=
K
K
6474
8
1424
3
márte
márte
n
n
natural kórsetkishli dárejeni anqlamasna muwapq
=
3
3
2
3
.
=
n
n
a
b
.
Solay etip,
( )
=
3
3
3
2
2
3
3
.
( )
=
≠
,
0
n
n
n
a
a
b
b
b
.
1-másele.
Esapla
:
⋅ ⋅
⋅ ⋅
7
3
4
6
4
11 7 3
11 7 3
.
7
3
4
7 6
3 1
6
4
11 7 3
11
7
1 11 49 539.
11 7 3
−
−
⋅ ⋅ =
⋅
⋅ = ⋅
=
⋅ ⋅
2-másele.
Jaqtlqt tarqalw tezligi 3
.
10
8
m/s qa jaqn.
Jerden Quyashqa shekemgi ortasha aralq 1,5
.
10
11
m. Jaqtlq
nur Quyashtan Jerge shekemgi aralqt qansha waqtta ótedi?
Te ólshewli qoz®alsta® jold formulasna tiykarlanp:
1,5 · 10
11
=
3 ·10
8
·
t
,
bul jerden
11
3
8
1,5 10
0,5 10
500( ).
3 10
⋅
=
=
⋅
=
⋅
s
t
Juwab: 500 s = 8 min 20 s.
( )
·
:
(
)
( )
·
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