RST UVW

100 Elementarnih zadataka
**** MLADEN SRAGA ****
POTPUNO RIJEŠENI
ZADACI
PRIRUČNIK ZA SAMOSTALNO UČENJE
100 ELEMENTARNIH ZADATAKA
koje bi svatko trebao znati riješiti
I
b
ALGEBARSKI IZRAZI
b g b g b g
a+ b = a+ b ⋅ a+ b = a + 2ab+
b
2 2
ba+ bg = bb+
ag
b g b g b g
a− b = a− b ⋅ a− b = a − 2ab+
b
2 2
ba− bg = bb−ag
2 2
b−a− bg = ba+
bg
b
b
a+ b = a + 3a b+ 3ab + b
a− b = a − 3a b+ 3ab − b
a − b = a− b ⋅ a + ab+
b
a
2 2 2
2 2 2
2 2
ba− bgb ⋅ a+ bg
= a − b
3 3 2 2 3
g
3 3 2 2 3
g
b i
3 2 2
= ba+ bgd ⋅ a − ab+
b i
2 2 2 2
g
2 2 2
m + n = p
= x + m ⋅ x + n
RST m⋅ n = q
UVW
a f a f
Rm + n = bU
ST V
m⋅ n = a⋅c
W
3 3 2 2
3
+ b
a + b + c = a + b + c + ab + ac + bc
2
x px q
+ + =
2 2
ax + bx + c = = ax + mx + nx + c = ...
a
a
b
n
m
POTENCIJE
a ⋅ a = a
n m n+
m
a : a = a
n m n−
m
= a : a = a
g
abc
da
i
a
n m n−
m
n n n n
a
a
= a b c
= a
n m nm ⋅
n n
F aI a
n
HG
b K J =
b
F aI b
HG
b K J F =
H G I
a K J =
− n n n
a
0
1
−1
− n
= 1
= a
1
=
a
1
=
n
a
b
a
n
α
TEL. 01-4578-431
©
M.I.M.-SRAGA 1999./2004.
©
M I M SRAGA 1999 / 2004
1

100 Elementarnih zadataka
Autor:
MLADEN SRAGA
Grafička obrada
MLADEN SRAGA
Procijena i ocijena znanja
ako se spremate :
Za prijemne na sve Za prijemne na sve
EKONOMSKE fakultete TEHNIČKE fakultete
0 − 50 = 0 0 − 50 = 0
51 − 75 = 1 51 − 81 = 1
76 − 85 = 2 82 − 86 = 2
86 − 91 = 3 87 − 90 = 3
92 − 95 = 4 91 − 94 = 4
96 − 100 = 5 95 − 100 = 5
Moj komentar:
ako ste dobili ocjenu 0 -
hitno tražite instruktora
ako ste dobili ocjenu 1 - trebali bi ozbiljno poćeti vježbati
ako ste dobili ocjenu 2 - Vi znate nešto ali ima tu puno rupa
ako ste dobili ocjenu 3 - dobro možete krenut na pripreme
ako ste dobili ocjenu 4 - vi ste spremni za zadatke sa prijemnih
ako ste dobili ocjenu 5 - odlično- vama zadaci s prijemnog
nebi trebali predstavljat problem
Prvo riješite zadatke pa tek onda gledajte u rješenja jer su tamo kompletno riješeni....
Potpunu garanciju na zadatke i rješenja daje: centar za dopisnu poduku M.I.M.-SRAGA -dakle sve što vam se
čini nejasno krivo ili sumnjivo - zovite 01-4578-431 ili 01-4579-130 i tražite dodatne upute i objašnjenja ...
Zadatci su u potpunosti naš autorski rad , ako se neki zadatak već pojavio u nekoj drugoj zbirci ...nije namjerno ponovljen s naše strane.
Centar za poduku M.I.M.-Sraga zadržava sva prava na komercijalno iskorištavanje ovih zadataka.
Svako kopiranje umnažanje za vlastite potrebe i potrebe poduke i nastave je dozvoljeno i dobrodošlo.
Cijena ovih zadataka je 0 kn tj. besplatni su ...i tako treba i ostati...
Sve kritike i dobre i loše rado ćemo saslušati ...
javite nam se na tel. 01-4578-431 ili e-mailom : mim-sraga@zg.htnet.hr
2
©
M I M SRAGA 1999 / 2004

STO ELEMENTARNIH ZADATAKA
KOJE BI SVI TREBALI ZNATI RIJEŠITI
Koristeći ove formule trebali bi riješiti zadane zadatke:
100 Elementarnih zadataka
a
a
n
m
( abc)
POTENCIJE
a ⋅ a = a
n m n+
m
a : a = a
n m n−m
= a : a = a
m
n
( a )
m
n
(( a ) )
⎛a⎞
⎜
b
⎟
⎝ ⎠
1
−
n
n m n−m
n n n n
0
1
−1
1
n
n
z
1
n⋅
m
n⋅m⋅z
−n n n
m
n
−n
n
n
n
1 1
= =
1 n
a
n
a
m
n
n
a b c
a
a
a
b
⎛a⎞ ⎛b⎞
b
⎜
b
⎟ = ⎜ =
a
⎟
⎝ ⎠ ⎝ ⎠ a
a
a
−
a
a
a
a
a
a
=
=
=
=
=
=
a
1
=
a
1
=
n
a
=
=
a
a
m
=
1
=
1
n
a
a
m n m
n
1)
2)
3)
2 3
4) x : x
2 5
3 2 2
4 2
5) x : x : x
7 2 3
7 8 9
6) x ⋅ x : x =
7)
8)
9)
10)
8
3
3
2
( x ) =
4
5
3
(( x ) ) =
2 2
(( x ) ) ( x )
2 4
(( x ) ) ( x )
3
2
( ab ) =
11) :
12) 2
13)
⎛2
⎜ x y
⎝3
2
6 5
3
5 5
2 3
4
3
2 3⎞
2
⎛2
14) ⎜ ab ⎟ : 2
⎝3
⎠
15)
16)
x
⋅ x =
2 4 6
x ⋅x ⋅ x =
x ⋅x ⋅ x =
x
x
⎛
⎜
⎝
x
y
=
⎞
⎟
⎠
−1
=
=
=
−2 3
⋅ =
⎞
⎟
⎠
=
( ab )
⎛ x⎞ ⎛ y⎞
⎜ ⎟ ⋅ ⎜ ⎟ =
⎝ y⎠
⎝ x⎠
=
2
=
2
3 2 2
2
⎛a ⎞ ⎛ y ⎞ ⎛ y ⎞
17) ⎜ ⋅ : =
3 ⎟ ⎜ 3 ⎟ ⎜ 4 ⎟
⎝ y ⎠ ⎝a ⎠ ⎝a
⎠
18)
−1 −2 −3
⎛2⎞ ⎛1⎞ ⎛1⎞
⎜ ⎟ + ⎜ ⎟ − ⎜ ⎟ =
⎝3⎠ ⎝3⎠ ⎝2⎠
−1
0 0 0 1 ⎛1
⎞
19) 27 ⋅x
⋅y
⋅2
⋅ ⎜ ⎟ =
⎝3
⎠
1 −1
0 −1 ⎛1⎞ ⎛1⎞
−3
20) y ⋅2 ⋅⎜ ⎟ ⋅⎜ ⎟ ⋅ 2 =
⎝2⎠ ⎝5⎠
© M I M SRAGA 1999 / 2004
3

100 Elementarnih zadataka
a
a
n
m
( abc)
POTENCIJE
a ⋅ a = a
n m n+
m
a : a = a
n m n−m
= a : a = a
m
n
( a )
m
n
(( a ) )
⎛a⎞
⎜
b
⎟
⎝ ⎠
1
−
n
n m n−m
n n n n
0
1
−1
1
n
n
z
1
n⋅
m
n⋅m⋅z
−n n n
m
n
−n
n
n
n
1 1
= =
1 n
a
n
a
m
n
n
a b c
a
a
a
b
⎛a⎞ ⎛b⎞
b
⎜
b
⎟ = ⎜ =
a
⎟
⎝ ⎠ ⎝ ⎠ a
a
a
−
a
a
a
a
a
a
=
=
=
=
=
=
a
1
=
a
1
=
n
a
=
=
a
a
m
=
1
=
1
n
a
a
m n m
n
21)
22)
23)
24)
1
2
1
−
2
2
3
3
−
4
25) 36
26) 16
27 )
28)
29)
30)
x
x
x
x
1
2
1
4
=
=
=
=
=
=
⎛25
⎞
⎜ ⎟
⎝49
⎠
⎛16
⎞
⎜ ⎟
⎝81⎠
1
2
1
4
⎛64
⎞
⎜ ⎟
⎝49
⎠
⎛ 9 ⎞
⎜ ⎟
⎝16
⎠
1
−
2
3
−
2
=
=
=
=
KORIJENI
a⋅ b = a⋅b
n n n
n
a⋅
n
a
b = n b
31) 10 ⋅ 10 =
⋅ =
3 3
32) 3 9
⋅ =
4 4
33) 2 8
34)
x
⋅ x =
5 2 5 3
35) x :
x
6 5 6 2
=
©
M I M SRAGA 1999 / 2004
4

Koristeći formule koje smo upravo obradili i ove nove koje vam dajem ovdje riješite sljedeće zadatke:
100 Elementarnih zadataka
36) x + 3 =
37) 2x+ 3y
=
38) 3x− 4y
=
39)
40)
( )
2
( )
( )
⎛1 2
⎜ x
⎝2 3
( x y)
2
2
2
3 ⎞
+ y =
⎛2 3 3 4⎞
⎜ x − y ⎟ =
⎝3 4 ⎠
−2
41) 5 − 6 =
( a y)
⎟
⎠
42) −2 − 3 =
2
2
( ) ( ) ( )
2 2
( a+ b) = ( b+
a)
( ) ( ) ( )
2 2
( a− b) = ( b−a)
( −a− b) = ( a+
b)
( ) ( )
( ) 3 3
( ) 3 3
( )
ALGEBARSKI IZRAZI
2 2 2
a+ b = a+ b ⋅ a+ b = a + 2ab+
b
2 2 2
a− b = a−b ⋅ a− b = a − 2ab+
b
2 2
a−b ⋅ a+ b = a −b
2 2
3 3 2 2 3
a+ b = a + a b+ ab + b
3 3 2 2 3
a− b = a − a b+ ab −b
( ) ( )
( ) ( )
a − b = a−b ⋅ a + ab+
b
3 3 2 2
a + b = a+ b ⋅ a − ab+
b
3 3 2 2
2 2 2 2
a+ b+ c = a + b + c + 2ab+ 2ac+
2bc
( x ) ( x )
43) 2 −5 ⋅ 2 + 5 =
44)
⎛2 2 4 3⎞ ⎛2 2 4 3⎞
⎜ x − y ⎟⋅ ⎜ x + y ⎟ =
⎝3 5 ⎠ ⎝3 5 ⎠
2 3 4 5 6 2 3 4 5 6
45) ( 3x y z −4a b ) ⋅(
3x y z −4a b ) =
( x )
46) + 3 =
3
2
( x ) ( x )
47) + 2 ⋅ + 2 =
( y z)
48) 2 − 3 =
3
49)
3
⎛1 1 ⎞
⎜ + x ⎟ =
⎝2 3 ⎠
50)
⎛2 2 3 3⎞
⎜ x − y ⎟ =
⎝3 2 ⎠
3
( )
2 3 4
51) 3x y − 2z
=
3
52)
( x y) ( x y)
3 3
+ − − =
© M I M SRAGA 1999 / 2004
5

Primjer:
rješenje:
x
− y =
3 6
27 8 ?
100 Elementarnih zadataka
( )
3 2
( ) ( ) ( ) ( ) ( ) ( )
3 3 2
3 6 3 3 3 2 2 2 2 2
27x − 8y = 3 x − 2 y = 3x − 2y = 3x−2y ⋅ 3x + 3x⋅ 2y + 2y
=
Prepoznaj:
↓ ↓
3
( 3 ) ( 2 )
3 3 2
2
( ) ( )
a − b = a−b ⋅ a + a⋅ b+
b
3 3 2 2
↓ ↓
a = x b = y
a = 3x b = 2y
3
2 2 2 4
( 3x 2y ) ( 9x 6xy 4y
)
= − ⋅ + +
Koristi gornji primjer i ove formule riješi sljedeće zadatke:
ALGEBARSKI IZRAZI
Rastavi na faktore:
( ) ( )
( ) ( )
a − b = a−b ⋅ a + ab+
b
3 3 2 2
a + b = a+ b ⋅ a − ab+
b
3 3 2 2
53
3
) 8
x
− =
54
3
) 8 27
y
− =
55
6
) 125
x
+ =
56)
x
− y =
6 6
57
1
y
27
3
) 1
+ =
9 6 3
58) 64xy
+ 125 z =
©
M I M SRAGA 1999 / 2004
6

100 Elementarnih zadataka
Koristeći se formulom
riješi sljedeće zadatke:
( ) 2 2 2 2
a + b + c = a + b + c + 2ab + 2ac + 2bc
( x y )
59) + 2 + 3 =
2
( x y z)
60) 2 + 3 + 4 =
2
( x y )
61) 3 + − 3 =
2
Koristeći se formulom:
rastavi na faktore:
( ) ( )
− = − ⋅ +
2 2
a b a b a b
2
62) x − 16 =
2 2
63) 4y
− 36x
=
64)
1 4 25 6
x − y =
4 49
65)
x
− y =
4 4
66) 4xy
− z =
2 4 8
( ) ( )
2 2
67) x+ y − x− 2y
=
( ) ( )
2
68) 36 x+ y −16
x− y =
2
©
M I M SRAGA 1999 / 2004
7

100 Elementarnih zadataka
Koristeći se formulama:
( ) ( )
2 2
2
2 2
2
a + 2ab+ b = a+ b a − 2ab+ b = a−b
slijedeće zadatke zapiši u obliku kvadrata binoma:
2 2
69) 9x + 6y+ y =
70)
2 2
4x − 20xy+ 25y
=
6 3 4 8
71) 9x − 12x y + 4y
=
4 6 2 3
72) x + y + 2x y =
2 4 8
73) − x + 2xy − y =
1 2 2 4
74) a − 3ab + 9b
=
4
Kvadratni trinom
⎧m+ n = p⎫
+ + = ⎨ ⎬ = + ⋅ +
⎩m⋅ n = q ⎭
( ) ( )
2
x px q x m x n
⎧
m+ n = b
2 2
ax + bx + c = ⎨ ⎬ = ax + mx + nx + c = ...
m⋅ n = a⋅c
⎩
⎫
⎭
Koristeći se gornjim formulama rastavi na faktore slijedeće kvadratne trinome:
2
75) x 2x
15
+ − =
2
76) 3x
x 2
+ − =
77)
x
2
− x− 12 =
©
M I M SRAGA 1999 / 2004
8

100 Elementarnih zadataka
Skrati razlomke koristeći sve što smo izvježbali do sada:
78)
4x−
6y
6x−
9y
=
79)
2 2
x − y
x + 2xy+
y
2 2
=
80)
4 4
a − b
a − 2ab+
b
2 2
=
81)
( x−
y)
y
− x
2 2
2
=
Pogledaj u formule za ove brojnike u 81) i 82)
82)
( y+
x)
x
− y
4 4
2
=
83)
3 3
x − y
3x
− 3y
2 2
=
84)
x
3
+ y
3
( x+
y)
3
=
85)
2
3x
+ 18x+
27
2
2x
18
−
=
86)
2 2
2x − xy+
3y
3x + 6xy+
3y
2 2
=
©
M I M SRAGA 1999 / 2004
9

100 Elementarnih zadataka
87)
x + 3 x − 2 x
− + =
2 3 4
87)
x
1 2
− =
+ 2 x − 2
88)
2 3 3x
− 2
+ −
2
x − 1 x + 1 x − 1
=
89)
1 1
− =
2
y − 3 3
( y − )
90)
2
1 y
− =
2 2 4 4
x + y x − y
91)
y y
− =
2 2 2
x − y y x
( − )
Izračunaj:
92)
2 2 2 2
x − y 4x + 4xy+
4y
⋅ =
3 3 2
x − y y+
x
( )
93)
4 4 6 3 3 6
x − y x + 2x y + y
⋅ =
3 3 4 2 2
x + y x + x y
94)
4
1 2 ⎞ x −1
⎟ 2
+ 1 −1 9 −1
⎛
⎜ + ⋅ =
⎝ x x ⎠ x
©
M I M SRAGA 1999 / 2004
10

100 Elementarnih zadataka
2 2
3x y+ 18xy+ 27y x −9
2 2
6 24 4 4
95) :
x y− y x + x+
=
( )
( − )
2 2 2
x+ y − y yx + 2y
96) :
x y −z
xz − yz −z
2 2
2
=
3 2
⎛ 1 1 3x− 2⎞
x + 2x + x
97 ) ⎜ − +
2 ⎟⋅
=
2
⎝ x− 1 x+ 1 x −1⎠
6x −9x
98)
1
1+ y − 1 =
1
1
y 1
− +
+
99)
1
1 − x
1 − 2 x
1+
x
=
1+ 1−
2 x
100)
x− y x+
y
−
x+ y x−
y
2 3
4x y − 4xy
2
xy x y y
− + −
=
©
M I M SRAGA 1999 / 2004
11

1)
2)
3)
2 3 2+
3 5
2 4 6 2+ 4+
6 12
Kompletna rješenja s postupkom
2 5 2 5 22 ⋅ + 26 ⋅ + 53 ⋅ 31
+ 2+
3 2 2 3 2 3⋅2
6
4) x : x = x = x
4 2 4−2 2
5) x : x : x = x = x
7 2 3 7−2−3 2
6) x ⋅ x : x = x = x
7 8 9 7+ 8−9 6
8
x
7) = x : x = x = x
3
x
8)
9)
10)
x ⋅ x = x = x
x ⋅x ⋅ x = x = x
x ⋅x ⋅ x = x = x = x
8 3 8−3 5
3
2 2⋅3 6
( x ) = x = x
4
5
3 345 ⋅ ⋅ 60
(( x ) ) = x = x
6
2
5
2 2 2⋅6⋅
(( x ) ) ⋅ ( x ) = x
5
3
5
(( ) ) ( )
3 3
( ) ( )
2 2⋅ 5 24 10 24+
10 34
11) x : x = x : x = x : x = x = x
2 4 2⋅5⋅3 4⋅5 30 20 30−20 10
12) 2ab = 2 ⋅a ⋅ b = 8⋅a ⋅ b = 8a b
13)
15)
16)
⋅ x = x ⋅ x = x = x
2 3 3 2 3 2⋅3 3 6
4 4
4 4
2 3 2 3 8 12
4
⎛2 ⎞ 2 16
⎜ xy⎟
= ⋅ ( x) ⋅ ( y)
= xy
⎝3 ⎠ 3 81
( )
3 3
2 3 3 2
2 3⎞
2 2 3 2 2 2 6 9 2
−1
8⋅a ⋅b 1 4⋅2⋅a ⋅a ⋅b ⋅b 1 2⋅a ⋅b
= ⋅ = ⋅ =
⋅a ⋅b ⋅a ⋅b
6 9 2 4 4 5 4 5
2 4 2 4
27 4 27 4 27
−2 3 2 3 2 3 2+
3 5
3 2 3 2+
3 5
( a )
( ) ( )
( y )
( )
3 2
3 2
2 2 2 2 2 2 6
⎛a ⎞ ⎛ y ⎞ ⎛ y ⎞
y a
17) ⎜ ⎟ ⋅ ⎜ ⎟ : ⎜ ⎟ = ⋅ : =
⎝ y ⎠ ⎝a ⎠ ⎝a ⎠ y a a y
3 3 4 3 2 2
3 3 4
9
100 Elementarnih zadataka
⎛2 2 8
4
14) ⎜ a b ⎟ :( 2 ab ) = ⋅
3 ( a ) ⋅( b ) : 2 ⋅a ⋅ ( b ) = ⋅a ⋅b :( 4⋅a
⋅ b ) =
⎝3 ⎠
3 27
⎛ x⎞
⎜ ⎟
⎝ y⎠
=
y
x
⎛ x⎞ ⎛ y⎞ ⎛ y⎞
y y y y y
⎜ ⎟ ⋅ ⎜ ⎟ = ⎜ ⎟ ⋅ = ⋅ = =
⎝ y⎠
⎝ x⎠ ⎝ x⎠
x x x x x
y ⋅a a a
= = =
y ⋅y ⋅y y y
2 8 8 8
9 2 2 9+
2 11
−1 −2 −3 2 3
2 4 6 2 8
y y a y a
⋅ : = ⋅ ⋅ =
6 8 9 6 4
a a y a y
⎛2⎞ ⎛1⎞ ⎛1⎞ 3 ⎛3⎞ ⎛2⎞
3 3 3+
2 5
18) ⎜ ⎟ + ⎜ ⎟ − ⎜ ⎟ = + ⎜ ⎟ − ⎜ ⎟ = + 9 − 8 = + 1 = =
⎝3⎠ ⎝3⎠ ⎝2⎠ 2 ⎝1⎠ ⎝1⎠
2 2 2 2
©
−1
0 0 0 1 ⎛1⎞
3
19) 27 x y 2 1112 2 3 6
⋅ ⋅ ⋅ ⋅ ⎜ ⎟ = ⋅ ⋅ ⋅ ⋅ = ⋅ =
⎝3⎠
1
0 −1
⎛1⎞ ⎛1⎞
20) y ⋅2
⋅⎜
⎟ ⋅⎜
⎟
⎝2⎠ ⎝5⎠
1
12
M I M SRAGA 1999 / 2004
−1
1 1 5 1 1⋅5⋅1 5
⋅ = ⋅ ⋅ ⋅ ⋅ = =
−3
2 1 2 2 1 2
3 2 ⋅ 2 ⋅ 8 32

100 Elementarnih zadataka
21)
x
1
2
=
x
22)
23)
x
1
−
2
1 1
= =
1
2
x
x
2 1 1
2⋅
3 3 2 3 3 2
( )
x = x = x = x
24)
x
3
−
4
1 1 1
= = =
x
3 1 4 3
4 4 x
3
( x )
25
1
)
2
36 = 36 =
2
6 = 6
26
1
)
4 4 4 4
16 = 16 = 2 = 2
27 )
1
2
⎛25 ⎞ 25 25 5
⎜ ⎟ = = =
⎝49 ⎠ 49 49 7
28)
1
4
4
16 4 16 2 2 2
= 4 = 4 = 4 =
4
⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟
⎝81⎠ 81 3 ⎝3 ⎠ 3
29)
1 1
−
2 2
⎛64 ⎞ ⎛49 ⎞ 49 49 7
⎜ ⎟ = ⎜ ⎟ = = =
⎝49 ⎠ ⎝64
⎠ 64 64 8
30)
3 3 3
3
3
−
2 2 2 2 2
2⋅
3
2
2
3
⎛ 9 ⎞ ⎛16⎞ ⎛4 ⎞ ⎛⎛4⎞ ⎞ ⎛4⎞ ⎛4⎞
4 64
⎜ ⎟ = ⎜ ⎟ = ⎜ ⎟ = ⎜ ⎟ = ⎜ ⎟ = ⎜ ⎟ = =
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎜⎝ ⎠ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
2 3
16 9 3 3 3 3 3 27
31
2
) 10 ⋅ 10 = 10⋅ 10 = 10 = 10
32
3 3 3
3 3 3
) 3 ⋅ 9 = 3⋅ 9 = 27 = 3 = 3
33
4 4 4
4 4 4
) 2 8 2 8 16 2 2
⋅ = ⋅ = = =
34)
x ⋅ x = x ⋅ x = x = x
5 2 5 3 5 2 3 5 5
35
6 5 6 6 5 2 6 5−2 6 3 23 ⋅ 3 2
) x : :
x = x x = x = x = x = x = x
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M I M SRAGA 1999 / 2004
13

( )
2 2 2
36) x+ 3 = x + 2⋅x⋅ 3 + 3 = x
2
+ 6x+
9
100 Elementarnih zadataka
( ) ( ) ( )
2 2 2 2 2 2 2
37 ) 2x+ 3y = 2x + 2⋅2x⋅ 3y+ 3y = 2 x + 12xy+ 3 y = 4x + 12xy+
9y
( ) ( ) ( )
2 2 2 2 2 2 2
2 2
38) 3x− 4y = 3x −2⋅3x⋅ 4y+ 4y = 3 x − 24xy+ 4 y = 9x − 24xy+
16y
2 2
2 2 2 2
3 3 3 2 3
2
⎛1 2 ⎞ ⎛1 ⎞ 1 2 ⎛2 ⎞ 1 2 2
39) ⎜ x+ y ⎟ = ⎜ x⎟ + 2⋅ x⋅ y + ⎜ y ⎟ = x + xy +
⎝2 3 ⎠ ⎝2 ⎠ 2 3 ⎝3 ⎠ 2 3 3
2
2
( y )
1 2 4
= x + xy + y
4 3 9
2
3 2 3 6
2 2 2 2 2
2 2
3 4 3 3 4 4 3 223 ⋅ ⋅ 3 4 3
2 ( ) 2 ( )
40)
⎛2 3 ⎞ ⎛2 ⎞ 2 3 ⎛3 ⎞ 2 3
⎜ x − y ⎟ = ⎜ x ⎟ −2⋅ x ⋅ y + ⎜ y ⎟ = x − ⋅x ⋅ y +
⎝3 4 ⎠ ⎝3 ⎠ 3 4 ⎝4 ⎠ 3 3⋅4
4
x =
4 6 3 4 9 8
= x − x y + y
9 16
41)
−2
1 1
1
( 5x− 6y)
= = =
2 2 2 2 2
25x − 60xy+
36y
( 5x−6y) ( 5x) −2⋅5x⋅ 6y+
( 6y)
2
2
( − a− y) = ( − ⋅ ( a+ y)
) = ( − )
42) 2 3 1 2 3 1
( ) ( ) ( )
2 2 2 2
43) 2x−5 ⋅ 2x+ 5 = 2x − 5 = 2 x − 25 =
( )
( 2a 3y) 1 ( 2a) 2 2a 3y ( 3y)
2 2 2 2
⋅ + = ⋅ + ⋅ ⋅ + = 4a + 12ay+
9y
2
4 25
x
−
2 2
44)
2 2 2 2
⎛2 2 2
2 4 3⎞ ⎛2 2 4 3⎞ ⎛2 2⎞ ⎛4 3⎞
2 2 4 3 4 16
⎜ x − y ⎟⋅ ⎜ x + y ⎟ = ⎜ x ⎟ − ⎜ y ⎟ =
2 ( x ) −
2 ( y ) = x − y
⎝3 5 ⎠ ⎝3 5 ⎠ ⎝3 ⎠ ⎝5 ⎠ 3 5 9 25
4 6
2
2
2 3 4 5 6 2 3 4 5 6 2 3 4
5 6
( x y z − a b ) ⋅( x y z − a b ) = ( x y z ) −( ab ) =
2 2 3 4 2 5 6
3 ( x ) ( y ) ( z ) 4 ( a ) ( b )
45) 3 4 3 4 3 4
( )
3 3 2 2 3 3 2
+
( ) ( ) ( ) ( ) ( ) ( )
( y z) ( y) ( y) z y ( z)
2 2 2 2 2
46) x+ 3 = x + 3⋅x ⋅ 3 + 3⋅x⋅ 3 + 3 = x + 9x + 3⋅9⋅ x+ 27 =
2 2 1 2 1 3 3 2 2 3
3 3 2 2
= − = 9x y z −16a b
+ 9 + 27 + 27
3 2
x x x
4 6 8 10 12
47 ) x+ 2 ⋅ x+ 2 = x+ 2 ⋅ x+ 2 = x+ 2 = x+ 2 = x + 3⋅x ⋅ 2 + 3⋅x⋅ 2 + 2 =
( ) 3 3 3 2 2 2 2 3 3
48) 2 − 3 = 2 −3⋅ 2 ⋅ 3 + 3⋅2 ⋅ 3 − 3z = 2 y −9⋅2 ⋅y ⋅ z+ 6⋅y⋅3 ⋅z − 3 z =
= 8y − 36y z+ 54yz −27z
3 2 2 3
+ 6 + 12 + 8
3 2
x x x
3 3 2 2 3 3 3
2 3
3 3
49)
⎛1 1 ⎞ ⎛1⎞ ⎛1⎞ 1 1 ⎛1 ⎞ ⎛1 ⎞ 1 1 1 1 1 1
⎜ + x⎟ = ⎜ ⎟ + 3⋅⎜ ⎟ ⋅ x+ 3⋅ ⋅ ⎜ x⎟ + ⎜ x⎟
= + 3⋅ ⋅ ⋅ x+ 3⋅ ⋅ ⋅ x +
⎝2 3 ⎠ ⎝2⎠ ⎝2⎠ 3 2 ⎝3 ⎠ ⎝3 ⎠ 2 4 3 2 9 3
x =
1 1 1 2 1 3
= + x+ x + x
8 4 6 27
50)
3
3 2 2 3
2 3 2
⎛2 3 ⎞ ⎛2
⎞ ⎛2 2⎞ 3 3 2 2 ⎛3 3⎞ ⎛3
3⎞
⎜ x − y ⎟ = x 3 x y 3 x y y
⎝3 2
⎜
⎠ 3
⎟ − ⋅⎜ 3
⎟ ⋅ + ⋅ ⋅ − =
2 3
⎜
2
⎟ ⎜
2
⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
3 2 2 3
2 3 2 2 3
2 2 2 3 3 2 3 3 3 3
= ( x
3 ) −3⋅ 2 ( x ) ⋅ y + 2⋅x ⋅
2 ( y ) −
3 ( y ) =
3 3 2 2 2
8 6 4 3 9 2 6 27 9
= x − 2x y + x y − y
27 2 8
3 3 2 2 3
2 3 4 2 3 2 3 4 2 3 4 4
( x y − z ) = ( x y ) − ⋅( x y ) ⋅ z + ⋅ x y ⋅( z ) − ( z ) =
3 3 2
3 2 3 2 3 4 2 3 4 3 4
= 3 ( x ) ( y ) −3⋅2⋅9⋅( x ) ⋅( y ) z 9x y 4 ( z ) 2 ( z )
51) 3 2 3 3 3 2 3 3 2 2
= 27x y − 54x y + 36x y z −8z
6 9 4 6 2 3 8 12
2 2 3
⋅ + ⋅ ⋅ − =
3 3 3 2 2 3 3 2 2 3
( x y) ( x y) x x y xy y ( x x y xy y )
52) + − − = + 3 + 3 + − − 3 + 3 − =
x 3x y 3xy y x 3x y 3xy y
3 2 2 3 3 2 2 3
= + + + − + − + =
6x y+
2y
2 3
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M I M SRAGA 1999 / 2004
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100 Elementarnih zadataka
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100 Elementarnih zadataka
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100 Elementarnih zadataka
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100 Elementarnih zadataka
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100 Elementarnih zadataka
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100 Elementarnih zadataka
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100 Elementarnih zadataka
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M I M SRAGA 1999 / 2004
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