kompletni postupak i rješenja - matematika.bapotencije)_operaci… · 8 m-1- univerzalna zbirka...

M-1- rješenja i kompletni postupakPOTENCIJE

Rješenja svih zadataka s kompletnim postupkom i uputama 1. 2 3

puta

2 se ponavlja petputa pa u eksponent pišemo

Koristimo pravila:

Uputa: Prebrojite koliko se puta ponavlja isti faktor i taj broj stavite u eksponent:

, , ...

1) 2 2 2 2 2

n

n

a a a a a a a a a a a a−

−

⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =

⋅ ⋅ ⋅ ⋅

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

5

5

5

3

6 3 2 6 3 2

3

4

grupiramo iste faktore...

čitamo: dva na petu2

1) 2 2 2 2 2 22)3)

4)

5)

6)

x x x xx y x y x z x y x x z x x x x x x y y y z z x y z x y z

xy xy xy xy

x y x y x y x y x y

x x xy y y

= →

⋅ ⋅ ⋅ ⋅ =

⋅ ⋅ =

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ =

⋅ ⋅ =

+ ⋅ + ⋅ + ⋅ + = +

⋅ ⋅

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

4

3 2

3

7)

8)

x xy y

x y x y x y x y x y

x y x y x y x y x y x y x y

a b a b a b a bc c c c

⋅ =

− ⋅ + ⋅ − ⋅ − ⋅ + =

= − ⋅ − ⋅ − ⋅ + ⋅ + = − ⋅ +

− − − − ⋅ ⋅ =

2.

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( )( )

na eksponent

na

2 3

puta

2 3

2

parni3

neparni4

Zaključak:

Koristimo pravila:

1 1

1

, , ...

1) 5 5 5 25 2) 2 2 2 2 8 3) 3 3 3 3 3 81

4) 1 1 1 1

5) 1 1 1 1 1

6) 1 1 1 1 1 1

n

n


− =

−

= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

= ⋅ = = ⋅ ⋅ = = ⋅ ⋅ ⋅ =

− = − ⋅ − =− = − ⋅ − ⋅ − = −

− = − ⋅ − ⋅ − ⋅ − =

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

eksponent

3

4

Imamo paran broj "minusa" pa je produkt pozitivan broj

Imamo neparan broj "minusa" pa će i umnožak biti negativno tj. imati će predznak minus

1

7) 2 2 2 2 8

8) x x x x x

↓

= −

− = − ⋅ − ⋅ − = −

− = − ⋅ − ⋅ − ⋅ − 4x=

4

7

8

M-1- univerzalna zbirka zadataka M-1- univerzalna zbirka zadataka

M.I.M-Sraga centar za [email protected]

tel-01-4578-431

2 3

puta

2

3

5

2

3

Koristimo pravila: , , ...

1 1 1 19)3 3 3 9

3 3 3 3 2710)5 5 5 5 125

2 2 2 2 2 2 3211)3 3 3 3 3 3 243

2 2 2 2 2 412)3 3 3 3 3 9

2 213)3

n

n


= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

= ⋅ =

= ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

⋅ − = − ⋅ − = = ⋅

− = −

4

2

3

4

2 2 2 2 2 83 3 3 3 3 3 27

3 3 3 3 3 8114)4 4 4 4 4 256

4 4 4 4 4 1615)5 5 5 5 5 25

4 4 4 4 4 4 4 6416)5 5 5 5 5 5 5 125

4 417)5 5

⋅ ⋅ ⋅ − ⋅ − = − = − ⋅ ⋅

= ⋅ ⋅ ⋅ =

⋅ − = − ⋅ − = + = ⋅

⋅ ⋅ − = − ⋅ − ⋅ − = − = − ⋅ ⋅

− = −

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

2

2

3

3

2

4 4 4 4 4 4 4 2565 5 5 5 5 5 5 625

18) 0,2 0,2 0,2 0,04

19) 0,2 0,2 0,2 0,2 0,2 0,04

20) 0,2 0,2 0,2 0,2 0,04 0,2 0,008

21) 0,2 0,2 0,2 0,2 0,2 0,2 0,2 0,008

22) 2,5 2,5

⋅ ⋅ ⋅ ⋅ − ⋅ − ⋅ − = + = ⋅ ⋅ ⋅

= ⋅ =

− = − ⋅ − = + ⋅ =

= ⋅ ⋅ = ⋅ =

− = − ⋅ − ⋅ − = − ⋅ ⋅ = −

− = − ( ) ( )2,5 2,5 2,5 6,25⋅ − = + ⋅ =

2.


2.

( ) ( ) ( ) ( ) ( )

) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

2 3

puta

3

4

2 3 4 5

Prebrojimo min

Koristimo pravila: , , ...

23) 2,5 2,5 2,5 2,5 2,5 2,5 2,5 15,625

24 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 39,0625

25) 1 1 1 1 1 1 1 1

n

n


= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

− = − ⋅ − ⋅ − = − ⋅ ⋅ =

− = − ⋅ − ⋅ − ⋅ − = + ⋅ ⋅ ⋅ =

− + − + − + − = + ⋅ − ⋅ + ⋅ −

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

use -imaih paran broj pa će umnožakbiti pozitivan broj

20 30 45

1 2 3 4

5 2

4 5

1 1

26) 1 1 1 1 1 1 1

27) 2 2 2 2 2 4 8 16 4 16 2 8 10

28) 2 3 2 2 2 2 2 3 3 32 9 23

29) 3 2 3 3 3 3 2 2 2 2 2 81 32

= + =

− + − + − = + ⋅ + ⋅ − = −

− + − + − + − = − + − + = + − − =

− = ⋅ ⋅ ⋅ ⋅ − ⋅ = − =

− = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ = − =

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

3 5

2 3

2 22 3

2 2

49

30) 2 2 2 2 2 2 2 2 2 2 8 32 40

1 1 1 1 1 1 1 1 1 1 2 131)4 2 4 4 2 2 2 16 8 16 16

32) 0,2 0,2 0,2 0,2 0,2 0,2 0,2

0,04 0,008 0,032 0,001024

− + − = − ⋅ − ⋅ − + − ⋅ − ⋅ − ⋅ − ⋅ − − − = −

− + − = ⋅ + − ⋅ − ⋅ − = − = = −

− + − = − ⋅ − + − ⋅ − ⋅ − =

= − = =

9

10



tel-01-4578-431

3. ( ) ( )

( ) ( )

( )

( )

je potpuno isti izraz...

ili taj isti zadatak na malo duži ali sigurn

Koristimo pravila:

51) 2 3 2 3 5 5 ili

5

2) 2 4 1 2 4 7

n n n n n n

c a d a c d a c a d a c d ac a d a c d a c a d a c d a

xx x x x x

x

x x x x x

⋅ + ⋅ = + ⋅ ⋅ − ⋅ = − ⋅⋅ + ⋅ = + ⋅ ⋅ − ⋅ = − ⋅

⋅ + = + ⋅ = ⋅ =

+ + = + + ⋅ =

( )iji naćin:

Dakle:

Praksa je pokazala da velika većina đaka radi istu grešku: uzimate da je: što nije točno!!

dakle vi kada računate u glavi grešite na ovaj način:

2 4 1 2 4 1 2 4 7 1

0

2

x x x x x x x x x x

x x

x x

+ + = + + = + + ⋅ = =

=

+ ( )

( )

( )

( )

ili što nije točno!!!

Jednom zauvjek treba zapamtiti pa to u zadatku treba izgledati ovako:

ili taj

4 6 2 4 0 2 4 6

1

2 4 1 2 4 1 2 4 7

3) 7 2 7 2 5 5

4) 9 2 3 9 2 3 1 9 9

x x x x x x x

x x

x x x x x x x x

a a a a a

y y y y y y y

+ = + + = + + ⋅ =

=

+ + = + + = + + ⋅ =

− = − ⋅ = ⋅ =

− + − = − + − ⋅ = ⋅ =

( )

( ) ( )

( )

( )

isti zadatak na malo duži ali sigurniji naćin:

9 2 3 9 2 3 1 9 2 3 1 9

5) 2 3 5 7 22 7 3 5 2 2 1 7 3 5 2 8 6 8 6

6) 2 3 2 3 1 6

7) 2 4 2 4 1 1

y y y y y y y y y y

x a x a x ax x x a a a x a x a x a

xy xy xy xy xy

ab ab ab ab ab

− + − = − + − = − + − ⋅ =

+ − + + − =

= − + + + − = − + ⋅ + + − ⋅ = ⋅ + ⋅ = +

+ + = + + ⋅ =

− + = − + ⋅ = − ⋅

( )

( ) ( ) ( )

2 2 2 2 2 2

2 2 2

2 2 2

2

2

2Ovo je potpuno isti izraz

8) 4 2 9 4 2 9 3 3

9) 3 4 5 2 8 3 82 3 8 4 3 5 8

1 2 3 8 4 3 5 8

4 4 64 4 6

ab

xy xy xy xy xy xy

y x y z x y y z y x yy y y z z x y x y x y

y z x y

y z x yy z x y

= −

+ − = + − ⋅ = − ⋅ = −

+ − − − + − + =

= − − + − + − + =

= − − ⋅ + − ⋅ + − + ⋅ =

= − ⋅ + ⋅ + ⋅ =

= − + +

11


( ) ( )( ) ( )

( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

3 3 3 3 3 3

2 2 2 2 2

ili taj isti zad

Koristimo pravila:

10) 7 2 4 7 2 4 9 9

11) 2 3 2 3 1

12) 3 4 3 4 1 6

n n n n n n

c a d a c d a c a d a c d ac a d a c d a c a d a c d a

xy xy xy xy xy xy

x y x y x y x y x y

x y x y x y x y x y

⋅ + ⋅ = + ⋅ ⋅ − ⋅ = − ⋅⋅ + ⋅ = + ⋅ ⋅ − ⋅ = − ⋅

− + = − + ⋅ = ⋅ =

− − + = − ⋅ + = − ⋅ + = − −

+ + + − + = + − ⋅ + = +

( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )

( ) ( )

( ) ( )

2 3 2 3 2 3

2 3 2 3 2 3

2 3

2 3

2 3

2 2

atak na malo duži ali sigurniji naćin:

3 4 3 4 1

3 4 1

6

13) 5 2 2 7 3 35 2 3 2 75 2 3 2 7

6 96 9

14) 3 3 5 4 5 1

3

x y x y x y x y x y x y

x y

x y

x y z x y z x y zx y x y x y z z

x y z

x y zx y z

x x x x

+ + + − + = + + + − + =

= + − ⋅ + =

= +

+ − + + − =

= − + + + =

= − + ⋅ + + ⋅ =

= ⋅ + ⋅ =

= +

− + − − + =

= ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( )

2

2

2

ili taj isti zadatak na malo duži ali sigurniji naćin:

4 3 5

1 3 5

3 5

15) 3 1 6 1 1 3 6 1 1 2 1

3 1 6 1 1 3 1 6 1 1 1

3 6 1 1

2 1

x x

x x

x x

x y x y x y x y x y


x y

x y

− ⋅ − + =

= − ⋅ − + =

= − + −

+ − − + − + + − = − + ⋅ + − = − + −

+ − − + − + + − = + − − + − + + − =

= − + ⋅ + − =

= − + −

3.

12



tel-01-4578-431

5 3 5 3 8

2 7 2 7 9

2 2 3

2 5 2 2 5 2 2 5 2 3 7

2 6 4 8 2 6 4 8 20

4 7 4 7 4 7 11

Koristimo pravila: : :

1) 2 2 2 2

2) 5 5 5 5

3) 3 3 3 3

4) 7 7 7 7 7

5) 2 2 2 2 2 2

6) 3 2 3 2 6 6

nn m n m n m n m n m n m

m

x x x x x

m m m m m m m

aa a a a a a a a aa

a a a a a a

+ −

+

+

+

− − − + − + − − −

+ + +

+

⋅ = = = =

⋅ = =

⋅ = =

⋅ = =

⋅ = = =

⋅ ⋅ ⋅ = =

⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ =

2 3 2 3 5

3 5 1 3 5 1 3 5 9 1

2 4 6 2 4 6 12

2 2 5 2 2 2 6 5 3 4 12 15 315 223 3 2 6 6 62

2 2 2 3 2 2 2 3 2 2 2 3 2 5

3 6 3 6 9

4 2

Pazi

7)

8)

9)

10)

11)

12)

13)

a a a a a a a

x x x x

x x x x x x x x x x

x x x x x

x x x x x x x

x x x x x x x

a a a a

a a

+

+ +

+ +

⋅ + ⋅ + ⋅ + ++ +

+ − + + + − + + + + − + +

+

⋅ = =

⋅ ⋅ = ⋅ ⋅ = = =

⋅ ⋅ = =

⋅ ⋅ = = = =

⋅ ⋅ ⋅ = = =

⋅ = =

⋅ ⋅ 4 2 1 4 2 1 7 1

4 2 4 2 61

4 2 4 2 0 6

2 5 7 3 2 5 7 3 17

3 2 2 3 2 2 6 2

2

Pazi

Dosta često radite ovakve greške:

što nije točno jer je:

14)

15)

16)

x x x x x x x

x y x x

a a a a a a a a

a a a a aa a

a a a a a

a a a a a a

a a a a a

a a a

+ +

+

+ +

+ + +

+ + + + +

+

= ⋅ ⋅ = = =

⋅ ⋅ = = =⋅ ⋅ = =

⋅ ⋅ ⋅ = =

⋅ ⋅ = =

⋅ ⋅ 3 2 3 2 3 4 4

3 4 1 3 4 1 3 4 817) 2 6 3 2 6 3 36 36

y x y x x y x x x y y x ya a a

x x x x x x x x

+ + + + + + + + + +

+ +

= = =

− ⋅ ⋅ ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ ⋅ ⋅ = − ⋅ = −

4.

−


4.

( )

2 5 2 3 6 3 2 2

2 5 2 1 3 1 1 6 3 2 2

7 2 1 3 1 1 6 3 4

7 7 3 4 3 4


18) 2 3 4 22 3 4 22 3 4 22 4 3 22 4


m

aa a a a a a a a aa

x x x y x y x x x y yx x x y y x x x yx x y x x yx x x y x y

+ − −

+ +

+ + +

⋅ = = = =

⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅ =

= ⋅ + ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ =

= ⋅ + ⋅ ⋅ − ⋅ + ⋅ ⋅ =

= ⋅ − ⋅ + ⋅ ⋅ + ⋅ ⋅ =

= − ( )7 3 4

7 3 4

7 3 4

2 3 4 2 3 4 9

2 1 3 2 2 1 3 2 2 3 1 2

To je potpuno isti izraz... uobičajeni zapis je ovaj zadnji...

3 2

2 52 5

1 1 1 1 119)2 2 2 2 2

20) m m m m m m

x x y

x x yx x y

x x x x

+ +

+ + + + + + + +

⋅ + + ⋅ ⋅ =

= − ⋅ + ⋅ ⋅ =

= − +

⋅ ⋅ = =

⋅ = = = 5 3

2 4 3 2 2 4 3 2 2 3 2 4 5 2

33 3 3 3 2 2 72 91 1 2 7 2 1 8 9 92 7 22 2 2 2 2

2 3 2 2 3 2 2 3 2 6 2

2 3 2 2

21)

22)

23)

24)

m

m m m m m m m

mm m m m m m mm

m n m n m n m n m n m n m m m n n n m n

m n m n m

x

x x x x x

x x x x x x x

x x x x x x

x x x

+

− + − + + + + − −

+ ⋅+ ⋅ ++ + + − + + + ⋅ + + −

+ − + + + − + + + + + − + +

− − −

⋅ = = =

⋅ = = = = =

⋅ ⋅ = = =

⋅ ⋅ 2 3 2 2 2 3 2 2 7 4

2 3 3 5 2 3 3 5 2 3 3 5 5 2

3 1 4 7 3 1 4 7 4 3 1 7 5 2 6 5 2 6

2 2

25)

26) 2 3 2 3 6 6 6

27) 2 5 2 5 10

n m n m n m n m m m n n n m n

m n m n m n m n m m n n m n

m n m n m n m n m m n n m n m n

x y x y x y x y

x x x

a a a a a

a a a a a a

a a a

− + − + − + + − − − −

+ − + + − + + − −

− + + − − + + + − + + − + − − − − −

+ + + + +

= = =

⋅ = = =

⋅ = ⋅ ⋅ = ⋅ = ⋅ =

⋅ = ⋅ ⋅ =

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 3 2

3 2 2 4 3 2 2 4 3 2 2 4 3 2 2 4 5 2

2 2 1 2 1 3

3 4 3 4 7

10

3 32 9 2 9 2 3 328)3 4 3 4 3 2 2 2 2

29)

30)

x x y y x y

m n m n m n m n m n m n m m n n m n

a a

a a a a a a a


x y x y x y x y

+ + + +

+ − + − + + − + + − −

+

+

⋅ =

⋅⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ = ⋅ =

⋅

− ⋅ − = − ⋅ − = − = −

+ ⋅ + = + = +

13


4.

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

1 2 2 3 4 1 2 2 3 4

2 3 1 2 4 6 1

1 1 1 1 22 2 2 2 2 2

2 3


31) 1 1 1 1

1 1

32)

33)


m

m m m m m m

m m m m

m n m n m n m n m n m n

aa a a a a a a a aa

x y x y x y x y

x y x y


x y x y x

+ − −

+ + − + + + + −

+ + + + − −

+

⋅ = = = =

+ − ⋅ + − ⋅ + − = + − =

= + − = + −

− ⋅ + = + ⋅ + = + = +

+ ⋅ − ⋅ +( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )

2 1 3 2 2 1 3 3

2 2 1 3 3

2 2 1 3 3

2 1

3 2 4 3 2 3 2 4 3 22 2 2 2 2

32

34)

m m m m

m m

m m

m m

x y x y x y y x x y x y x y y x

x

y x y x y x y x y x y

x y x y

x y x y

x y x y

ab ab ab ab abc c c c c

abc

− − − −

+ − + −

+ − + −

+

− − − − − + − + − + −

+

⋅ − = + ⋅ + ⋅ − ⋅ − =

= + ⋅ − =

= + ⋅ − =

= + ⋅ −

⋅ ⋅ ⋅ = =

=

( ) ( )

4 2 2 3

62

4 2 2 4 2 2 1 4 2 2 1 6 3 6 3

56 4 6 4 5 6 5 4 1 11 5

2 3 4 3 4 2 2 3 3 4 4 2 2 3 3

1 1 135) 9 9 3 3 3 33 3 3

136) 25 5 5 5 55 5

37) 2 3 2 3 6

x x x y y y y

x yabc

x y x y x x y y x y x y x y

yxx y x y y x x y x y

x y z x y z x x y y z z x y

+ − − − − +

−

+ +

+ +

+ +

=

=

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ =

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ =

⋅ − = ⋅ − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = − ⋅ ⋅

( ) ( ) ( )

( ) ( )

4 4 2 5 7 6

2 4 3 2 2 4 2 3 2 2 2 2

2 4 2 3 2 2 2 1 2

6 5 4 3 2

2 2 3 3 2 2 2 3 2 3

1 2 2 1

6

38) 3 2 7 3 2 7

3 2 73 2 7

1 139) 2 2 2 22 2

2

z x y

x x x x x x x x x x x x x x

x x x x xx x x x x

xy x y xy x y xy x y xy xy xy x y

x x y y

+

+ + + +

⋅ = −

⋅ − − + − = ⋅ − ⋅ + ⋅ − ⋅ + ⋅ ⋅ + ⋅ − =

= − − ⋅ + ⋅ − ⋅ =

= − − + −

− ⋅ − + = − ⋅ + − ⋅ − − ⋅ =

= − ⋅ ⋅ ⋅ ⋅ 1 1 2 3 1 3 2 1

1 2 2 1 1 1 2 3 1 3 2 1

3 3 2 5 4 3

12 22

2 22 2

x x y y x x y y

x y x y x yx y x y x y

+ + + + + +

+ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ =

= − ⋅ ⋅ + ⋅ − ⋅ ⋅ =

= − + −

z

14 M.I.M-Sraga centar za [email protected]

tel-01-4578-431

15


4

( )

9 5 9 5 4 4

7 5 7 5 2

5 3 5 3 2

3 2 23 2 2 3 2 2 3 2 2 2

U slijedećim zadacima koristimo pravilo:

Pazi: drugi eksponent se

. :

40) 2 : 2 2 2 2 2 2 2 2 1641) 13 : 13 13 13 16942) 3 : 3 3 3

43) 5 : 5 5 5 5 5

n m n m

x x x x x

m mm m m m m m m

a a a −

−

−

−

− − −− − − − + − + −

=

= = = = ⋅ ⋅ ⋅ =

= = =

= =

= = = =

↓

3 2 2 3 2 2 3 2 2 2 4

mora staviti u zagradu...vrlo često radite ovakvu grešku:

što nije točno!

Ovdje je pogreška u predznaku kod zadnjeg člana, jer prvom promjenite predznak,

a drugom

5 : 5 5 5 5m m m m m m m− − − − − − − − −= = = →

↓

4 2 4 2 2

6 2 6 2 4

7 2

ne, to se događa zbog toga što radite napamet... tj. preskaćete korake...

Preporuka: čim imamo višečlane eksponente koristite zagrade i ne preskačite korake...

44) :

45) :46) : :

x x x x

x x x xx x x

−

−

= =

= =3 7 2 3 2

7 3 4 7 3 4 6

2 1 2 1 2 1

1 1 2 2 1 4 1 322 2 2 2 2 2

7 1 7 1 7 1 3 7 3 49 3 9 3 9 9 9

2 2 1 2 2 2 10 1 5 4 20 5 211 225 5 2 10 10 102

4 2 4 2 2

47) :148) : :

49) :

50) :

51) : :52) :x x x x

x xx x x x x

x x x x x xx

x x x x x x

x x x x x x

x x x x x x xa a a a

− −

+ −

− −

⋅ − −−

− ⋅ −−

⋅ − ⋅ − ⋅ − −− − −

−

= =

⋅ = =

= = = =

= = = =

= = = =

= = = =

= =8 3 2 8 3 2 3

2 3 1 2 3 1 2 3 44

7 2 3 7 2 3 8

55 2 5 2 3

2

88 3 8 3 5

3

33 32 112 2

ili

U slijedećim zadacima koristimo pravilo:

53) : :154) : : : :

55) :

:

56) :

57) :

58) :

x

nn m n m

m

a a a a a

a a a a a a a aa

a a a a a

a a a aa

x x x x xxx x x x xx

x x x x xx

− −

− − −

− +

−

−

−

−

= =

= = = =

⋅ = =

= =

= = =

= = =

= = =3 2 1

2 2

71 7 17 7 3 1 4 21 4 1743 4 34 12 12 12

13

59) :

x

x x x x x x xx

−

⋅ − ⋅ −−

=

= = = = =


4.

( )

( )

11 2 21 2 2 1 2 2 2 1 2 1

2 2

3 73 7 3 53 7 3 5 3 7 3 5 3 3 7 5 2

3 5

8 71 8

3 2

U slijedećim zadacima koristimo pravilo: :

60) :

61) :

62) :

nn m n m

m

mm mm m m m m m m

m

mm mm m m m m m

m

mm m

m

a a a aa

x x x x x x xxx x x x x x xx

x x xx

−

++ − ++ + + − − − + − − −

+

++ − ++ + + − − − + −

+

+−

+

= =

= = = = =

= = = = =

= ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( )

8 7 3 2 17 3 2 1 8 7 3 2 1 8 3 7 2 1 4 6

9 119 11 2 3 7 107 10 9 11 2 3 7 10 9 11 2 3 7 10

2 3

9 2 7 11 3 10 4

7 4 7 4 3

2 3

: :

63) : : :

64) :

65) 2 3

m m mm m m m m m m m m

mm m mm m m m m m m

m

m m m

x y

x x x x x x

a a a a a a aa

a a

x y x y x y x y

x y

+ − + − −+ + − + − − − + − − + − + +

++ − − − ++ + − + + − + − −

−

− − + + −

−

+

= = = =

= = = =

= =

− − = − = −

− ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

2 3 2 3 2 3 2 3 2 3 0

6 4 6 4 22 2 2 2

3 3 3 3

2 4 4 2 4 2 4 4 2 4 2 4 4 2 4

2 4 4 4 2 5

: 2 3 2 3 2 3 2 3 1

66) :

67) 2 2 : 2 2 2

2 2

x y x y x y x y x y

m m m m m m m m m

m m m m

x y x y x y x y

a b a b a b a bc c c c

x y x y x y x y x y

x y x y

+ + − + + − −

−

− − − − + − − − + − − − +

+ − + − −

− = − = − = − =

= =

+ ⋅ + + = + = + =

= + = +( )

2

2 3 3 4 4 2 3 3 4 4 3 2 3 4 42 2 2 2 2

3 3 3 3 3

3 4 4 2 3 12 2 3

3 3 2

U ovom slučaju možete ostaviti i ljevo rješe

ili

68) :x x x x x x x x x

x x x

a b a b a b a b a bc c c c c

a b a b cc c a b

−

− − − − + − − − + − − − +

+ − + − − −

⋅ = = =

= = =

( ) ( )

nje...jer još nismo obradili pravilo:"na minus prvu"

2 4 2 2 4 2 2 2 4 2

2 4 2 2 2 4 2

Množimo svaki član prve zagrade sa svakim članom druge zagrade...

69) 1 1 1 1 1 1 1

1

x x x x x x x x x x

x x x x x+ +

− ⋅ + + = ⋅ + ⋅ + ⋅ − ⋅ − ⋅ − ⋅ =

= + + − − − =

( ) ( ) ( ) ( )

6 4 2 4 2

6 4 4 2 2

6

2 2 2 1 2 2 2 2

2 1 2 2 2 2

3 4 2 2

4 3

11

1

70) 1 1 1 1 1 1 1

11

1

x x x x xx x x x xx

x x x x x x x x x x

x x x x xx x x x xx x x

+ +

= + + − − − =

= + − + − − =

= −

+ ⋅ − + = ⋅ + ⋅ − + ⋅ + ⋅ + ⋅ − + ⋅ =

= − + + − + =

= − + − + + =

= − + + +


tel-01-4578-431

17


( ) ( ) ( ) ( )( )( )

Koristimo pravila:

vidi zadatke: 7),13),14),15),16),17),18),19),23),24)...

vio nam se jedan problem a taj je 2.

: ogledajmo opet u zada kako smo k tt oa

zm mn nn n n n n n n m n n m z

n

ab a b abc a b c a a a a

x

⋅ ⋅= = = =

−

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( )( )

( ) ( ) ( ) ( )

na eksponent

na eksponent

2

parni3

neparni4

3

Imamo neparan broj "minusa" pa će i umnožak bi

1 1Zaključak:

1 1

tamo rješili:

. 4) 1 1 1 1

5) 1 1 1 1 1

6) 1 1 1 1 1 1

7) 2 2 2 2 8

2

↓

− =

− = −

− = − ⋅ − =− = − ⋅ − ⋅ − = −

− = − ⋅ − ⋅ − ⋅ − =

− = − ⋅ − ⋅ − = −

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

4 4

Imamo paran broj "minusa" pa je produkt pozitivan broj

53

53 puta bi to trebali na

ti negativno tj. imati će predznak minus

8)

vlja ovakav problem: ...

x x x x x x

x x x x

− = − ⋅ − ⋅ − ⋅ − =

− = − ⋅ − ⋅ ⋅ −

( )( )

pisatia to je previše pisanja pa ćemo rađe gledati eksponent

53 53

54je logički je pitanje koliko je eksponent je 54, 54 je paran broj

u eksponentu je 53 ,

an broj pa je:

?

x x

x

=

− = −

− ( )

( )

( )( )

( )( )

na eksponent

na eksponent

54 54

55 55

56 56

2 parni 2parnu

2 1 neparni neparnu

pa je:

Zaključak:

jer je eksponent 55, a 55 je neparan broj

jer je eksponent 56, a 56 je paran broj

ilin n

n

x x

x x

x x

x x

x x

x x

x +

− =

− = −

− =

− = − =

− = =

− =

−

( )( )

2 1

4 4

3 3

2 je oznaka za parni broj

2 1 je oznaka za neparan broj

Sada to primjenimo u zadatcima:

.8) zato što je eksponent 4, a 4 je paran broj

7) 2 2 8 zato što je eksponent 3, a 3 je neparan broj

Po

2

n

n

n

x

x x

+

+

= −

− = =

− = − = − =

( ) ( ) ( )( ) ( )

( )

( ) ( )

( )

( ) ( ) ( )

3 3 3 3

3 3 3 3

Svaki negativan broj da se zapisati u obliku:

1

4 4 4 4 4 4 II na

stoji i drugi način rješavanja ovakvih zadataka:II način

7) 2 1 2 1 2 1 8 8

2 1 2 1 2 1 8 8

8) 1 1 1

x

x x

x x x x x

−

− = − ⋅

− = − ⋅ = − ⋅ = − ⋅ = −

− = − ⋅ = − ⋅ = − ⋅ = −

− = − ⋅ = − ⋅ = ⋅ = čin

5.

⋅

Poja

P

Sada se ja

53 je nepar

dal

( )

18



tel-01-4578-431

( ) ( ) ( ) ( )( )( )

) ( )) ( )

) ( ) ( )

2 2 2 2 2

2 2 2 2 2

2 2 22 22 3 2 3 2 2 3 2 4 62

Koristimo pravila:

Postupili smo prema prvom pravilu...

1 2 2 4 4

2 3 3 9 9

3 3 3 934 4 164


n n n


ab a b

x x x x

x x x x

x y x y x y x y

ab

⋅ ⋅

⋅ ⋅

= = = =

=

= ⋅ = ⋅ =

= ⋅ = ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

( ) ( )

) ( )

) ( ) ( )

) ( )

) ( ) ( )

3 3 332 3 2 3 2 3 3 63

4 4 44 42 3 2 3 2 4 3 4 8 124

22 2 2 4

2 22 2 2 2 4

Postupili smo prema drugom i trećem pravilu...

I

2 2 2 843 3 273

1 1 1 152 2 162

6

7

mn n n n n n mc a b c a a

xy x y x y x y

x y x y x y x y

x x x

x x x x

⋅

⋅

⋅ ⋅

⋅

⋅

= =

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= =

− = = =

) ( ) ( ) ( ) ( )) ( )

) ( )

( )

) ( ) ( ) ( )

) ( ) ( ) ( )

) ( ) ( )

2 2 222 2 2 2 2 4

32 2 3 6

52 2 5 10

2 2 23 4 2 3 4 3 2 4 2 6 8

3 3 33 4 3 3 4 3 3 4 3 9 12

4 43 4 4 3

način

II način:Prema pravilu:

7 1 1 1

8

9

10 2 2 4 4

11 2 2 8 8

12 2 2

mn na ax x x x x

x x x

x x x

x y x y x y x y

x y x y x y x y

x y x

⋅⋅

⋅

⋅

⋅ ⋅

⋅ ⋅

=

− = − ⋅ = − ⋅ = ⋅ = = == =

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ( )

) ( ) ( )) ( ) ( ) ( ) ( )

44 3 4 4 4 12 16

3 32 2 2 3 6

3 3 3 332 2 2 2 2 2 2 3 6II način:

16 16

13

13 1 1 1

y x y x y

y y y y

y y y y y y y

⋅ ⋅

⋅

⋅

⋅ = ⋅ ⋅ =

− = = =

− = − ⋅ = − ⋅ = ⋅ = = =

5.

⋅

m

19


5.

( )( )

( ) ( )

na eksponent

na eksponent

parni parnu

neparni neparnu

3 32 2 2 3 6

Koristimo pravilo: prilikom računanja načinom u

14), 15), 16),17), 18), 19)...

to je bio I način rješavanja

I

14)

II način

1

x x

x x

y y y y⋅

− =

− = −

− = − = − = −

) ( ) ( ) ( ) ( )

( ) ( )( ) ( )

) ( ) ( ) ( ) ( )

( ) ( )

3 3 332 2 2 2 3 6 6

3

2

2 2 223 3 3 3 2 6 6

2 23 3 3 2 6

Prema pravilu:

Prema pravilu:

to je bio II način rješavanja

to je bio I

4 1 1 1 1

1 1 1 1

1 1 1 1 1

15 1 1 1 1

15)

neparnu

parnu

y y y y y y

y y y y y y

y y y y

⋅

⋅

⋅

− = − ⋅ = − ⋅ = − ⋅ = − ⋅ = −

↓

− = − − = −

− = + − = + =

↑

− = − ⋅ = − ⋅ = ⋅ = ⋅ =

− = = =

( ) ( )

) ( ) ( )( ) ( )

) ( ) ( ) ( ) ( ) ( ) ( )

) ( ) ( )

3 3

3 34 4 4 3 12 12

3 3 334 4 4 4 3 12 12

2 25 5 5 2 10

način rješavanja


II način


16

16 1 1 1

17

1

parnu parnuy y

y y y y y

y y y y y y

x x x x

⋅

⋅

⋅

− =

− − = − − = − − = + =

− − = − − ⋅ = − − ⋅ = − − ⋅ = − − =

− − = − = − = −

) ( ) ( ) ( ) ( ) ( ) ( )

) ( ) ( ) ( ) ( )

) ( ) ( )( ) ( )

2 2 225 5 5 5 2 10 10

2 2 2 25 5 5 5 5 2 10

2 25 5

22 5

II način

U 18) pokazat ćemo tri načina rješavanja:


7 1 1 1

18

18 1

1 1

x x x x x x

x x x x x x

x x

x

⋅

⋅

− − = − − ⋅ = − − ⋅ = − ⋅ = − = −

− − = + − = − = = =

− − = − ⋅ − =

= − ⋅ − = = ( ) ( ) ( )) ( ) ( ) [ ]

2 225 5 5 2 10

2 225 5 5 5 2 10

II način

III način ...u min us i minus daju plus

U ovom 18) zadatku najbrži ne III način rješavanja... dok je u 19) definitivno II način najb

1 1 1

18

x x x x

x x x x x

⋅

⋅

⋅ − ⋅ = − ⋅ = ⋅ =

− − = + = = =

olji....

20



tel-01-4578-431

5.

) ( ) ( ) ( ) ( ) ( )( )

) ( ) ( )( ) ( )

2 22 2 2 2 4 45 5 5 5 5 5 4 20

2 22 2 25 5

I način

I način u ovom zadatku pomalo zbunjuje pa je bolje to rješavati na II način izlučivanjem 1 :

II način:

1 1

x x x x x x x

x x x

⋅ ⋅

−

− − = + − = − = − = = =

− − = − − ⋅ = − − ⋅ ( )( )( )

( ) ( ) ( )

) ( ) ( ) ( )

) ( ) ( )

) ( ) ( ) ( )

) ( ) ( ) ( )

( ) ( )

225

25 2

210

22 210 10 10 2 20

25 5 2 10 10

3 32 3 3 2 3 2 3 3 6

4 4 42 3 2 3 2 4 3 4 8 12

2 223 3 3 2 6

2 2

1

1

1 1 1

2 2 8 4

2 2 4 4

2 4 jer je: 2

x

x

x x x x

x x x x

ab a b a b a b

a b a b a b a b

y y y y

⋅

⋅

⋅

⋅

⋅ ⋅

⋅

=

= − ⋅ =

= − ⋅ =

= − ⋅ = − ⋅ = ⋅ =

− = − = − = −

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ = ⋅ =

− = − ⋅ = ⋅ =

↓

− = − = −( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )

( )( )

na eksponent

na eksponent

2 2 2

3 3 3 3 3

4 4 4 4

2

3

parni parnu

neparni neparnuoristimo pravilo:

1 2 1 2 1 4 4

2 8 jer je: 2 1 2 1 2 1 8 8

2 1 2 1 2 1 16 16 itd.

ili ovako: 2 2 2 4

2 2 2 2 8

ili kx x

x x

− =

− = −

⋅ = − ⋅ = ⋅ =

− = − − = − ⋅ = − ⋅ = − ⋅ = −

− = − ⋅ = − ⋅ = ⋅ =

− = − ⋅ − =

− = − ⋅ − ⋅ − = −

( )( )

( )

) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2

3 3

3 3

3 332 2 3 2 3 6

3 3 3 33

pa je:

ili

2 2 4

2 2 8

3 3 27

24 3 3 3 27

3 1 3 1 3 1 27 27 3 3 3 3

y y y y⋅

− = =

− = − = −

− = − = −

↑

− = − ⋅ = − ⋅ = −

↓

− = − ⋅ = − ⋅ = − ⋅ = − − = − ⋅ − ⋅ − = −

19

19

20

21

22

23

27


5. 25 25 25 26 26 27

) ( )

) ( )

) ( ) ( )

2 2 2 222 2 2 2 2 42

2 2 222 2 2 2 42

2 2 2 2222 2 2 2 2 42

25) zadatak na tri nači

2 2 2 2 43 3 3 93

2 2 2 43 3 93

2 2 2 2 41 1 13 3 3 93

x x x x x

x x x x

x x x x x

⋅

⋅

⋅

− = = ⋅ = ⋅ =

− = − ⋅ = + ⋅ =

− = − ⋅ ⋅ = − ⋅ ⋅ = ⋅ ⋅ =

) ( )

) ( )

) ( )

) ( )

) ( ) ( ) ( )

) ( ) ( ) ( )

3 3 3 332 2 2 2 3 63

3 3 332 2 2 3 63

2 2 2

3 3 3

4 4 4 4 4 4 4

na

I način

II način

2 2 2 2 83 3 3 273

2 2 2 83 3 273

28

29

30

m m m

n n n

m n m n m n m n

m m mm n m n

x x x x x

x x x x

x x x

y y y

x y x y x y x y

x y x y

⋅

⋅

⋅

⋅

⋅ ⋅

− = − = − ⋅ = − ⋅ = −

− = − ⋅ = − ⋅ = −

= =

= =

= ⋅ = ⋅ =

= ⋅ =

) ( ) ( ) ( )

) ( ) ( ) ( )

) ( ) ( )

) ( ) ( )

) ( )

) ( ) ( ) ( ) ( ) ( )

2

2

2

2 2 23 2 3 2 3 2 2 2 6 4

3 2 3 2 3 2 3 2

2 2 2 2

3 3 3 3

2 2 2 2 2 2 2 2 2

31

32

33 2 2 2 2 4

34 3 3 3 3 27

35 2 2 2

36 2 3 2 3 2 3 2 3 2 3 4

m m n m m n m

x y x y x y x y

x x xx y x y x x y x x xy

xx x x x

xx x x x

xx x x x

m nm n m n m n m n m

x y x y

a b a b a b a b

a b a b a b a b

⋅ ⋅ ⋅

⋅ ⋅

⋅ ⋅

⋅ ⋅

⋅ ⋅

⋅

⋅ ⋅ ⋅ ⋅

⋅ =

= ⋅ = ⋅ =

= ⋅ = ⋅ =

= = = =

= = = =

= =

= ⋅ = ⋅ = ⋅ = ⋅ = 9 4 9n m n⋅ =

21


5. 37 38 39

) ( )( ) ( )( )

) ( )( )

) ( )( )

) ( )( )

) ( )( ) ( )

) ( ) ( )( )

) ( )( ) ( )

432 2 3 4 24

543 3 4 5 60

42 2 4 8

3 3 3

26 52 3 2 6 2 3 5 24 15 24 15 39

24 33 5 3 4 5 3 2 12 30 12 30 42

35 72 4 2 5 3 4 7

Prema pravilu:

40

41

42

43 : :

zmn n m z

x x x

yx x y xy

x x x a a

y y y

x x x

y y y

x x x x x x x x

y y y y y y y y

x x x x

⋅ ⋅ ⋅ ⋅

⋅ ⋅

⋅ ⋅

⋅ ⋅

⋅ ⋅ ⋅ +

⋅ ⋅ ⋅ +

⋅ ⋅ ⋅

= = =

= =

= =

= =

⋅ = ⋅ = ⋅ = =

⋅ = ⋅ = ⋅ = =

=

) ( )( ) ( )( )

) ( ) ( ) ( ) ( )

( )

( )

30 28 30 28 2

8 56 43 2 3 6 8 2 4 5 144 40 144 40 104

3 32 3 3 22 3 2 2 3 2 2 2

32 3 3 3 2 2 2

3

6 9 2 4

6 2 9 4

6 2 9 4

:

44 : : :

2 245 : 2 : 23 3

2 : 438 : 4

278 :4 :

272

27

x x x x

a a a a a a a a

a b ab a b a b

a b a b

a b a b

a a b b

a b

−

⋅ ⋅ ⋅ ⋅ −

⋅ ⋅ ⋅

− −

= = =

= = = =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ =

= ⋅ ⋅ =

4 5227a b=


tel-01-4578-431


23

( ) ( ) ( ) ( )( )

) ( ) ( ) ( ) ( ) ( )

( )

( )

3 33 3 3 3 32 3 3 2 2 3 3 3 2

32 3 3 3 3 3 2 3

3

6 9 9 6

6 9 9 6

6 9 9 6

Koristimo pravila:

1 14 42 2

1 6421 6481 648648

zm mn nn n n n n n n m n n m zab a b abc a b c a a a a

x y x y x y x y

x y x y

x y x y

x x y y

x y

⋅ ⋅

⋅ ⋅ ⋅ ⋅

+ +

= = =

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ =

=

) ( ) ( ) ( ) ( ) ( )

( )

( )

) ( )

15 15

15 15

3 33 3 3 3 32 3 3 2 2 3 3 3 2

32 3 3 3 3 3 2 3

3

6 9 9 6

6 9 9 6

6 9 9 6

3 3

33

3

3

2

88

1 1: 4 : 42 2

1 : 6421 : 6481:64 : :8

18 64

1512

1 1512

512

48 3

x yx y

x y x y x y x y

x y x y

x y x y

x x y y

x y

x y

yx

yx

a b

⋅ ⋅ ⋅ ⋅

− −

−

⋅ ⋅ =

=

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ =

= ⋅ ⋅ =⋅

= ⋅ ⋅ =

= ⋅ ⋅ =

=

( ) ( ) ( ) ( )( )( )

) ( ) ( )

2 2 23 2 3

2 2 3 2

2 2 6

2 2 6

2 2 6

4 7

1 3 13 3 3

3

3

3

333

49

x xx x x x x x

x x x x x

x x x x x

x x x x x

x x x x x

x x x

m m m

a b a b a b

a b a b

a b a b

a a b ba ba b

x x x

⋅ ⋅ ⋅

+ +

+ ⋅ + +

⋅ = ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ =

=

= =

5. ⋅= 46 47


5. 49 50 51 52 53

( ) ( ) ( ) ( )( )

) ( ) ( )

) ( ) ( ) ( ) ( )

) ( ) ( ) ( ) ( )

1 3 13 3 3

2 1 1 3 2 1 2 13 2

6 3 2 2

6 3 2 2

6 2 3 2

8 5

2 1 1 3 2 1 2 15 3

6 3 2 2

6

Koristimo pravila:

: :

:


m m m

m m m m

m m

m m

m m

m

m m m m

m m

m


x x x

x x x x

x xxxx

x x x x

x x

x

⋅ ⋅

+ ⋅ + +

− − ⋅ − ⋅ −

− −

− + −

+ − −

−

− + ⋅ − ⋅ −

− −

−

= = =

= =

⋅ = ⋅ =

= ⋅ =

= =

= =

=

= =

= =

= ( )

) ( ) ( )

( )

) ( ) ( ) ( )

( )

3 2 2

6 3 2 2

6 2 3 2

4 1

3 22 3 2 3 3 2

6 6

6

6

4 6 33 2 4 3 4 2 6 4 3

12 12 12

12

12

12

2 3 2 3

2 32 3

5

3 4 3 4

3 41 3 4

22

m

m m

m m

m

xxx

x x x x

x xx

x

a a a a a a

a a aa

aa

− −

− − +

− − +

−

⋅ ⋅

⋅ ⋅ ⋅

=

= =

= =

=

+ = ⋅ + ⋅ =

= ⋅ + ⋅ =

= + ⋅ =

=

− + = − ⋅ + ⋅ =

= − ⋅ + ⋅ =

= − + ⋅ =

= ⋅ =

=

⋅=


tel-01-4578-431


6. ( )

( ) ( )( ) ( )

22 2 4 4 42 4

3 3

Koristimo pravilo:

1) 2 5 2 5 10 2) 3 4 3 4 12

3) 4 6 4 6 24 4) 2 4 2 4 8

5) 6) 1 1

3 8 3 87)4 3 4 3

nn n n

x xx x x x x x

a ya a a y y y

a b c abc

a cac b b a b a bab c b c b a b a

=

⋅ = ⋅ = ⋅ = ⋅ =

⋅ = ⋅ = ⋅ = ⋅ =

⋅ ⋅ = ⋅ = ⋅ = ⋅ = =

⋅ = ⋅

333

2 2 22 2

2 2

2

2 2

3 33

možemo ostaviti u ovom obliku

ili dalje:

nakon kraćenja

4 23 2 84 3

2 24 4 28)2 2 2

2 4

9)

x yxy xyx x y xx y x y

x x

xy x xy xz yz z yz

⋅ = ⋅ = =

⋅ ⋅ ⋅ = ⋅ = ⋅ = = ⋅ ⋅

= =

⋅ = ⋅ =

( )( )

( )( )

( )

( )

333 22 2 3 6

2 3 2 3 62

1 m+11

111 2 2 12 2 2

2 1 2 22 12

22 3 6

4 3 2

nakon kraćenja10)

11)

mm

mmm m m

m mm

xx x x x xz z z z zz

xy x xy xz yz z yz

xx x x x xz z z zzz

x y zz x y

⋅

⋅

++

+++ ⋅ + +

+ +⋅ +

⋅= = = = = ⋅

⋅ = ⋅ = =

⋅= = = = = ⋅

⋅

( )( )

22 2 2 2 1 4 22 3 6

4 3 2 4 2 1 2

22 22 2 2 4

1 2 1 2 21

2 2 1 4 22 3 6 2 3 6

4 3 2 4 3 2 4 2 1 2

nakon kraćenja

nakon kraće12)mm m m

x y y z zx y zz x y z x x y

zz z zx x xx

x y y z zx y z x y zz x y z x y z x x y

⋅

⋅

⋅ ⋅ ⋅= ⋅ = ⋅ = = ⋅ ⋅

= = = =

⋅ ⋅ ⋅⋅ = ⋅ = ⋅ = ⋅ ⋅

( ) ( )( )

( ) ( )( )

( )( )

( ) ( )( ) ( )

1 21 2 1 2 2

1 11

5 555 2 2 2

2 2 2 2

5

2

nja

.

1 113)1 12 2

1 11

1 11

m mm m m m m

m m m

y zy z y z y zx xx

x y x x y x y x x yx x y x x yx xy y x xy y

x xx y x yx x yx y

x y x x xx x y x y

⋅ ⋅

⋅

=

⋅ ⋅= = = =

− − + − − + ⋅ ⋅ = ⋅ ⋅ = + − + −+ + + +

− ⋅ +− + = ⋅ ⋅ = + −+

− − ⋅ += ⋅ ⋅

+ + ⋅ +( )( )

( )( )

5

55

5

pa kratimo

11

yx y

xxx y x y

+= = −

− −= = + +

x

25


7.

2 2

2

2 2 2

2

po pravilu:

primjenili smo pravilo:

Koristimo pravila:

Izračunaj:

2 2 41) 3 93

2 3 3 92)3 2 42

2uputa: 2)3

n n

n

n n nn n

n n

n n

n

a ab b

a a a b bb b ab a

a ab b

−

−

=

= = =

= = =

= = =

2 2 2

2

po pravilu

3 3

3

4 4

4

1 1



3 3 92 42

1 1 13) 2 82

2 2 164) 3 813

5 7 75)7 5

n na bb a

n n

n

n n

n

a ab b

a ab b

−

−

=

−

= = =

= = =

= = =

= =

11 1

2 2 2

2

3 3 3

mješoviti broj treba prvo pretvoriti u razlomak...

primjenili smo pravilo: 5

1 7 22 9 76) 17 7 7 9

4 5 5 257)5 4 164

2 3 38)3 2

n n n

n

a b bb a a

−

−− −

−

−

→

= =

⋅ + = = =

= = =

= = 3

22 2 2 2 2

2

22 2 2 2 2

2


2782

2 3 11 6 1 7 3 3 99) 23 3 3 3 7 497

1 7 22 7 2 9 7 7 4910) 17 7 7 7 9 9

−− − −

−− − −

→

=

⋅ + + = = = = = =

⋅ + + = = = = = = 81


tel-01-4578-431

27


33 3 3 3 3

3

2


Koristimo pravila:

1 2 11 2 1 3 2 2 811) 12 2 2 2 3 273

3 412)4 3

n n nn n

n n

a a a b bb b ab a

−

−− − −

−

= = =

⋅ + + = = = = = =

=

( )( )

na eksponent

na eksponent

2 2

2

2 2

2

parni parnu

neparni neparnu

2 2 22

2

način oristimo pravilo: pa je:

4 1693

13)

14) I k

14) jer je:parnu parnu

x x

x x

x xy y

x x x x x x xy y y y yy

− =

− = −

= =

=

− = = − = − = =

( )

( )

2

2 2 2 2 22

2 2

3 3 3

3

3 3 33

II način

način rastavili smo na:

I način

...

14) 1 1 1

II 1 ...

15)

15) 1 1

y

x x x x xy y y y y

x xy y

x x xy y y

x x xy y y

=

− = − ⋅ = − ⋅ = ⋅ =

− − ⋅

− = − = −

− = − ⋅ = − ⋅ = −

( )( )

( )

na eksponent

na eksponent

3 3

3 3

parni parnu

neparni neparnu

4 4 4na parni eksponent parnu

4

4

II način

način oristimo pravilo: pa je:

II način:

1

16) I k

16) jer je:

16)

x x

x x

x xy y

x x x x xy y y

xy

− =

− = −

⋅ = −

− = = − =

− = −

( )4 4 4 4

4

4 4

22 2 2 2

2

1 1 1

1 2 11 2 1 3 317) 12 2 2 2 2

x x x xy y y y

⋅ = − ⋅ = ⋅ =

⋅ + + = = = =

7.


7.

22 2 2 2 2

2

22 2

I način

II način:

Koristimo pravila:

2 3 22 6 2 8 8 8 6418) 23 3 3 3 3 93

2 3 22 6 2 818) 23 3 3 3

n n nn n

n n

a a a b bb b ab a

− = = =

⋅ + + − = − = = − = − = = =

⋅ + + − = − = = − = −

( )2 2 2 2

2

2

33 3 3 3 3

3

33 3 3

I način

II način:

8 8 8 61 1 13 3 93

2 3 22 6 2 8 8 8 51219) 23 3 3 3 3 273

2 3 22 6 2 8 819) 2 13 3 3 3 3

= − ⋅ = − ⋅ = ⋅ =

⋅ + + − = − = = − = − = − = − = −

⋅ + + − = − = = − = − = − ⋅

( )

( )( )

( )( )

3 3 33

3

22 2 2 2

2

44 22 2 4 8

3 4 3 4 123

33 3 33 3 3 3 9

4 4 3 4 3 124

3

4

I način

8 8 51 13 23

3 5 44 15 4 19 19 36120) 35 5 5 5 255

21)

22)

22)

xx x xy y yy

xx x x xy y y yy

xy

⋅

⋅

⋅

⋅

= − ⋅ = − ⋅ = −

⋅ + + = = = = =

= = =

− = − = − = − = −

−

( )( )( )

( )( )

( )( )( )

33 3 33 3 3 93

4 3 4 3 124

22 2 33 3 3 2 6

4 4 2 4 2 84

22 2 33 3 3 2 62

4 4 2 4 2 84

II način

I način

II način

1 1 1

23)

23) 1 1 1

xx x xy y yy

xx x x xy y y yy

xx x x xy y y yy

⋅

⋅

⋅

⋅

⋅

⋅

= − ⋅ = − ⋅ = − ⋅ = −

− = = = =

− = − ⋅ = − ⋅ = ⋅ =

127

4


tel-01-4578-431


7.

( ) ( )( )

( )( ) ( )

( )

2 22 2 3 2 2 3 22 3 4 6

4 2 4 2 84

2 22 5 6

6 2 5

26

2 22 5

Koristimo pravila:

po pravilu:

po pravilu:

24)

25)

n n nn n

n n

n n n

n

n

a a a b bb b ab a

x y x yx y x yz z zz

x y z a b bb az x y a

za

x y

−

⋅ ⋅

⋅

− −

= = =

⋅ ⋅= = =

= = = =

= =⋅

( )( )

6 2

2 2 5 2

12

4 10

33 3 4 4 34 12

2 3 2 3 3 6 33 2 3

3 32 3 3 3 4

3 4 2

Najkrače objašnjenje:

Sve što ima minus u eksponentu

"skaće" iz brojnik

2 82 826)3 27 273

227)2

m n ma

zx yzx y

x xx xy z y z y zy z

x y y zz x

⋅

⋅

⋅ ⋅

⋅

⋅

−

− −

=

= =⋅

=

⋅ ⋅= = = ⋅ ⋅ ⋅ ⋅

= =

( ) ( )( )

33 4

2

3 33 3 4

32

3 3 4 3 9 12

2 3 6

2 3

3 4

2 3 3 3 4

3 4

a u nazivnik i obrnuto...

Sada dodatna uputa: Razlomak

Ima u eksponentu "minus" pa se prvo rješavamo tih "minusa"

8

8

512 512

2

22

y zx

y z

x

y z y zx x

x yz

x y y zz

⋅ ⋅

⋅

−

− −

−

− −

= =

⋅ ⋅= =

⋅ ⋅= =

= 2 22

3 4 3 4

U brojniku je bio: i on je "skočio" u nazivnik i sada je:

U nazivniku su bili i oni su "skočili" u brojnik i sada su:

: 2 2

x xx

i z i z

−

− −

29


7.

( )( ) ( )

2 24 2 4

3 2 3

22 4 4 2 8

2 2 2 2 3 2 4 62 2 3

2 24 2 4

3 2 3

22 3

4

2

Koristimo pravila:

2 228)5 5

2 4 425 255

2 229)5 5

52

5

n n nn n

n n

a a a b bb b ab a

x y xz y z

x x xy z y zy z

x y xz y z

y zx

−

−

⋅

⋅ ⋅

− −−

= = =

= =

⋅ ⋅= = =

⋅ ⋅⋅ ⋅

= =

= =

⋅=

( ) ( )( )

( )( ) ( )

2 22 3

22 4

2 2 3 2

4 2

4 6

8

33 3 3 4 4 34 2 4 12

3 2 3 3 3 2 3 3 33 2 3

1 1

2 3 2 3 2 3

3 2

2

254

254

2 82 2 830)5 5 125 1255

31)

32)

y z

x

y zx

y zx

x xx y x xz y z y z yy z

x y yy x x

x y y y y yy x x x x

⋅ ⋅

⋅

⋅−

⋅ ⋅

−

−

⋅=

⋅

⋅ ⋅= =

⋅

=

⋅ ⋅= = = = ⋅ ⋅ ⋅ ⋅

= =

⋅ = ⋅ = ⋅

( )( )

( )

( )( ) ( )

2 12 1

3 2 2 1 1

33 3

3

2 22

33 3 32 5 3 3 3 9

3 2 5 3 3 2 3 5 3 6 152 5

2 32 3 2 333)2 3 2 3 2 3

134)1

35)

y yy x xyx x x x y

xx xx x x

x y x yx y

ca b c c cc a b a b a ba b

−

−

− ⋅

⋅ ⋅

⋅= ⋅ = =

⋅

+− + = = + − −

− = = − −

= = = = ⋅ ⋅

6 9z


tel-01-4578-431


7.

( )( ) ( )

( )( )

3 23 2 2 222 2 2

3 3 4 3 2 23 3 4

2 3 2 2 2

3 3 3 2 4 2

6 2 4

9 6 8

6 2 4

7 2 6 8

4

7 8

7

kratimo

Koristimo pravila:

36) : :

:

:

:

1 :

1

n n nn n

n n

a a a b bb b ab a

a ya y y yy a a y a a

a y yy a aa y yy a aa y yy y a a

yy a

y

−

⋅ ⋅

⋅ ⋅ ⋅

= = =

⋅ = ⋅ =

= ⋅ =

= ⋅ =

= ⋅ =⋅

= =

= ⋅8

4

8 8 8

7 4 7 4 11

2 2 23 2

3 2 2

2 2 2

2

1 2

3 2 1 1 2 1 1 4 1 137) 2 22 3 8 4 9 4 2 42 2 3

1 2 1 91 1 1 1 119 2 4 18 4 36 36

8 21 3 1 4 1 4 1 16 1 238)8 4 8 3 8 8 9 8 9 93

2 1 139)3 3 2

ay

a a ay y y y+

−− −

−

− −

=

= = =⋅

⋅ + = ⋅ + = ⋅ + = ⋅ + = ⋅ ⋅ + ⋅

= + = + = =⋅

⋅ ⋅ = ⋅ = ⋅ = ⋅ = ⋅ =

+ −

( )

3 1 2 3

2 2 22 3 2 32 3 2 3

3 2 2 3

22 2

3 3 2 3 3 3 2 59 8 12 1 1 2 2 2 2 2

3 2 3 4 3 2 4 27 240)4 2 3 3 43 2 3 3

9 316 8 8 8 14 3 129 4 27 27 27

−

− − −− −−

−

−− −

= + − = + − = + = + =

⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ =

⋅ = ⋅ ⋅ = ⋅ ⋅ = ⋅ = 2

8 1 8 127 144 27 48612⋅ = ⋅ =

31


8. 0 1 1

0 0

0

02 4

po pravilu:

bez obzira što je u zagradi ako je eksponent

te zagrade nula sve je jednako jedan !

ili taj isti zadatak duž

Koristimo pravila:1 11 , , ,

1) 2 1 1

2) 1

3) 1

3)

nna a a a a

a a

a

x

x yz

− −= = = =

= =

=

=

( ) ( )

( )

( )

0 00 2 42 4

0

0 0

0

0 0

0 0 0

0

0

im postupkom:

po pravilu:

bez obzira što je u zagradi ako je to na nultu sve j

1 1 1 11 1

4) 1 1 2

3 1

6) 3 1 1

7) 3 1 zato što je: 3 1 3 1 1 1

8) 3 3 1 3

9) 1

x yx yz z

x y

a

x

x y

⋅ ⋅= = = =

+ = + =

=

− = =

− = − − = − ⋅ = − ⋅ = −

− = − ⋅ = −

+ =

( )

( ) ( )

( )

0

0 0 0

0 00

0 0

20 2

e jednako jedan !

bez obzira što je u zagradi ako je eksponent

te zagrade nula sve je jednako jedan !10) 2 7 1

11) 2 7 1 2 1 7 1 1 2 7 6

12) 2 7 1 1 1 1

13) 1 1 1

14) 1 1

x y

x y z

x y z

x x

x

− =

− + = − ⋅ + ⋅ = − + =

− + = − + =

⋅ = ⋅ =

= =

5)


tel-01-4578-431

33


8.

( )

( )

( )

0 1 1

0 0

20 2

0

0

1 1

1

1

bez obzira što je u zagradi ako je to na nuultu sve je jednako jedan !

po pravilu:


13) 1 1 1

14) 1 1

15) 1

16) 4 1

1 117) 5 5

118)

19) 0, 2

nn

m

a a a a aa a

x x

x

x

x

aa

xx

− −

− −

−

−

= = = =

⋅ = ⋅ =

= =

=

=

= =

=

=

( )( )

( )

1 1

11

22

33

2 2 22

2

prvo decimalni broj pretvorimo u razlomak...

po pravilu:

po pravilu:

2 1 5 510 5 1

1 1 120)

1 1 121) 3 3 9

122)

3 10 10 10023) 0,310 3 3 9

24) 2 5

nn

nn

x y ax y ax y

aa

xx

x

− −

− −

− −

−

−−

= = =

− = = =−−

= = =

=

= = = =

−( )

55

1

12 5

125) 44

x−

−

=−

=


8.

( ) ( )( ) ( )

0 1 1

1

1 1

1

1 1

1 1

ili na drugi način:

ili na drugi način:


11 1 1 4 1 1 4126) 4 41 14 1 1 4 1

4 4

11 4 1 41 1 1 4 1127) 4 41 1 1 1 1 14 4

4 4

2 3 328) pr3 2 2

nna a a a a

a a− −

− −

− −

−

= = = =

⋅ ⋅= = = = = =

⋅

⋅ − ⋅ −⋅= = = − = − = = = −

⋅− −−−

= =

( )( )

11

11

1 1

1

1

Isti taj zadatak po pravilu ispada: vidimo da je rezultat isti!

ili kraće:

ema pravilu:

11 2 1 3 1 31: 2 23 2 1 2

3 3

2 2 5 229) 2 5 10 2 5 2 5 105 1 5

2 5230) 215

n n n

na b bb a a

aa

−

−−

− −

−

= =

⋅ = = = = = ⋅

⋅= = ⋅ = = ⋅ = ⋅ =

⋅ −= =

−( )

2

2

44

4

5 4 4 3 7 9 5 4 4 3 7 9 5 4 9 4 3 7 0

5 10

3 3 231) 3 4 122 1

2 232) 21

33) 3 3 3 3 3 3 1x x x x x x x x x

a a b abb

−

−

− − − − + − + − + − − − +

⋅ − = −

⋅= = ⋅ =

⋅= =

⋅ ⋅ = = = =


tel-01-4578-431

35


8.

( ) ( ) ( ) ( ) ( )

0 1 1

2 1 3 2 3 6 2 1 3 2 3 6 2 3 1 3 2 6 0

2 3 2 2 2 3 2 2 2 2 3 2 1

2 2 2 2 2 2 0


34) 2 2 2 2 2 2 2 1

135) 5 5 5 5 55

36) 1

3

nn

x x x x x x x x x

x x x x x x

m n n m m n n m m m n n

a a a a aa a

x y x y x y x y x y

− −

+ − − + + + − + − − + + + + −

− − − + − − − + −

− − − + − − + −

= = = =

⋅ ⋅ ⋅ = = = =

⋅ = = = =

+ ⋅ + = + = + = + =

( )

1 10 0 0 1

1 1 10 1 3

3

2 22 2 2 2 2

3 3 3 3 3

1 37) 27 2 1 1 1 2 1 2 3 63 1

1 1 1 1 5 1 1 1 1 538) 2 2 1 1 52 5 2 2 1 2 2 8 322

39) :m n m n n m n m n n

x y

y

a b a b a b a b a bc c c c c

−

−− −

+ + + − + +

⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ =

⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =

⋅ = =

2 02

3

22 2

23 1 3 1

2 4

2 4

2 2

2

1

1 2 41 1 1 12 2 8 2 82 240)

1 1 1 1 1 4 12 24 16 162 2

1 8 77 168 8

4 1 5 8 516 16

m n m n na bc

+ − − +

−− −

−− −

− −

− −

−

= =

− ⋅ − − −= = = = ⋅ ++ + +

− − ⋅= = = − + ⋅

( )

2

2 2 2 22

2

2 2 2

2 22 1 2

1 2

2

7 25

14 5 5 5 25 251 1 15 14 14 196 19614

1 1 1 1 12 3 183 4 32 1241)

1 1 1 1 1 122 32 2 9 183

6 36 2

−

−

− − −

− −− −

− −

⋅ = − =

= − = − = − ⋅ = − ⋅ = ⋅ =

⋅ ⋅ ⋅ = = = = = ⋅ ⋅ ⋅

⋅= ⋅

2 2 2 2

2

3 2 2 42 3 93

− − = = = =


8. 0 1 1

22 2 2

2 22 1 2

1 2

2


3 41 1 1 1 14 3 18 12 3 3 4 32 1242)

1 1 1 1 9 2 11 11 122 32 2 9 2 9 183

3 611 2

nna a a a a

a a− −

−− − −

− −− −

− −

= = = =

− − − − ⋅ ⋅− = = = = = − = + ⋅+ + + ⋅

⋅= −

⋅( )

2 2 2 2 22

2

0 0 1

0 0 1

0 1 22

2 3

3 22 22 22 484 4841 1 16 22 3 3 93

1 3 11 1 3 1 243) 3 2 3 1 1 13 3 3 3 3

2 3 11 1 6 1 544) 3 2 3 1 1 23 3 3 3 3

3 4 11 1 12 1 1145) 2 2 2 1 2 34 4 4 42

46) 2 2

− −

−

− −

= − = − = − ⋅ = − ⋅ = ⋅ = ⋅

⋅ − −⋅ − = ⋅ − = − = = =

⋅ − −+ − = + − = − = = =

⋅ − −+ − = + − = − = = =

− = 2 3

3 2 13 2 1

2 32 3

2 2

3

3

2

3ili isti zadatak na drugi način:

1 2 11 1 1 1 2 1 14 8 8 8 82 2

1 1 2 1 41 1 1 1 1 1 1 2 4 347) 2 2 28 4 2 8 8 82 2 2

1 1 1 1 8 9 1 148) 3 29 8 9 8 72 723 2

1 11 272 272 449)

1 1 4 1 43273

2 3 3

− − −

− −

−

−

−

−

⋅ − −− = − = = =

− ⋅ + ⋅ − +− + = − + = − + = = =

− −− = − = − = = = −

⋅

⋅= = = =

⋅

=3

2

3 2 2

2

3 2 3

2 2 2

vidi u 7. zadatku 27), 28), 29)

ili isti zadatak na drugi način:

2742

1 882 3 8 1 1 13 950)1616 9 16 9 2 1841

2 3 2 8 1 116 9 2 9 184 4 3

−

−

=

⋅⋅ ⋅= = = = =

⋅ ⋅

⋅= = = =

⋅ ⋅⋅

9


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37


8. 0 1 1

2 1

2 1

2 1 1

2 1 1

1

1

ili isti zadatak na drugi način:

ili isti zadatak na drugi nač


1 4 142 4 1 2 24 4 151)1 9 3 1 3 33 6 96 6 2

2 4 4 6 6 29 33 6 9 4

1152)

1 1

nna a a a a

a a

ba baa ab

b

− −

−

−

−

−

−

−

= = =

⋅⋅ ⋅= = = = =

⋅⋅ ⋅

⋅ ⋅= = =

⋅ ⋅

⋅= = =

⋅

1

1

22 2 2 22

2 2 2 2 2

2

11 11

1 1 1

1

22

22 3 23 3

1 2 2 2 2 2 22

1

in:

ili drugi način:

ili drugi način:

ili drugi

1153)

1 1

1

54)1 1 1

1

55)1

a bab

ba b a bab a a b a

b

bb b c b ca b bc a b bca aa a ac c

cc

aa a ca b a cb bc d d b d b dd

c c

−

−

− −

− −

− −

− −

−

−

=

⋅= = = =

⋅

⋅ ⋅ ⋅= = = = = =

⋅

⋅ ⋅= = = =

⋅⋅

2 12 3 2

1 2 3 2 3 2

22

2 2 2 21 2 21

2 3 1 3 3 2 13

2 1 2 1

2 2 1 2 21 2 2

2 3 1 3 1 2 1

1 2

način:

ili drugi način:

1

56)1 1

57)

a ca b a cc d b d b d

b cb c b c y d b c y dx bc bdyx xcxy c d c x c x c cc

y d y db c y d b c y dx bc bdy

cxy c d c x c c x

x b

−

−

−

− −

−

− −

− −

⋅= =

⋅

⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

= = = = =⋅ ⋅ ⋅⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅= = =

⋅ ⋅ ⋅

22

2 2 3 2 32 51 2

2 3 1 2 2 2 2 2 2 22

3 1 3

3 1 2 3 21 2 2 5

2 3 1 1 2 2 2 2 2 2ili drugi način:

1 1

1 1

cc x b c c d c dc cx b

y c d y x b y b xy b xyyc d c d

c d c c dx b c c dy c d x y b b xy b xy

+

− −

+− −

− −

⋅ ⋅ ⋅ ⋅ ⋅ ⋅= = = = =

⋅ ⋅⋅ ⋅⋅

⋅ ⋅ ⋅= = =

⋅ ⋅

d

a

=


9.


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