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UNIVERZALNA ZBIRKA

POTPUNO RIJEŠENIH ZADATAKA PRIRUČNIK ZA SAMOSTALNO UČENJE

POTENCIJE

α

M-1- univerzalna zbirka potpuno riješenih zadataka

Autor: MLADEN SRAGA Grafički urednik: Mladen Sraga BESPLATNA - WEB-VARIJANTA Tisak: M.I.M.-SRAGA d.o.o. CIP-Katalogizacija u publikaciji Nacionalna i sveučilišna knjižnica, Zagreb © M.I.M-Sraga d.o.o. 1992./2012.

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M-1- rješenja i kompletni postupakPOTENCIJE

Samo zadaci su zadani od 3. do 12. strane …dio rješenja je

od 16. strane na dalje …

Sva rješenja ovih zadataka šaljemo besplatno

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3autor: Mladen Sraga


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

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( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 3

puta

Koristeći pravila: , , ...

Izračunaj:1) 2 2 2 2 2 2)3) 4)

5) 6)

7)

21. n

n

a a a a a a a a a a a a

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

x x x xx y x y x y x yy y y y

x y x y x y x y x

−

⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ ⋅ + ⋅ + ⋅ + ⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

− ⋅ + ⋅ − ⋅ − ⋅( )

( )( ) ( ) ( ) ( )

2 3

puta

22 3 4

3 4 3 4

2 3 5 2

8)

Koristeći pravila: , , ...

Izračunaj:

1) 5 2) 2 3) 3 4) 1

5) 1 6) 1 7) 2 8)

1 3 2 29) 10) 11) 12)3 5 3 3

213)3

22. n

n

a b a b a byc c c

a a a a a a a a a a a a

x

−

− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

−

− − − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛−

( )

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

( ) ( )

3 4 2 3

422 3

3 2 3 4

2 3 4 5 20 30 45

1 2 3 4 5 2 4 5

3 5

3 4 414) 15) 16)4 5 5

417) 18) 0,2 19) 0,2 20) 0,25

21) 0,2 22) 2,5 23) 2,5 24 2,5

25) 1 1 1 1 26) 1 1 1

27) 2 2 2 2 28) 2 3 29) 3 2

130) 2 2 31)4

⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞− −⎜ ⎟⎝ ⎠

− − − −

− + − + − + − − + − + −

− + − + − + − − −

⎛ ⎞− + − ⎜ ⎟⎝ ⎠

( ) ( )

( ) ( )

2 3 22 3

2 2 2 2 2

1 32) 0,2 0,22

Koristeći pravila:

1) 2 3 2) 2 4 3) 7 24) 9 2 3 5) 2 3 5 7 26) 2 3 7) 2 48) 4 2 9 9) 3 4 5 2 8

23. n n n n n nc a d a c d a c a d a c d a

x x x x x a ay y y y x a x a x axy xy xy ab ab abxy xy xy y x y z x y y z

⎛ ⎞ ⎡ ⎤+ − − + −⎜ ⎟ ⎣ ⎦⎝ ⎠

⋅ + ⋅ = + ⋅ ⋅ − ⋅ = − ⋅

+ + + −− + − + − + + −+ + − +

+ − + − − − + −

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

2

3 3 3 2 2

2 3 2 3 2 3

2 2

3 8

10) 7 2 4 11) 2 3

12) 3 4 13) 5 2 2 7 3 3

14) 3 3 5 4 5 1 15) 3 1 6 1 1

y x y

xy xy xy x y x y

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

x x x x x y x y x y

+

− + + − +

+ + + − + + − + + −

− + − − + + − − + − + + −


5

5 3 2 7 2 2 5 2

2 6 4 8 4 7 2 3 3 5

2 52 4 6 2 2 2 2 3 3 63 2

4 2

Koristeći pravila: : :

Izračunaj:1) 2 2 2) 5 5 3) 3 3 4) 7 75) 2 2 2 2 6) 3 2 7) 8)

9) 10) 11) 12)13) 1

24.n

n m n m n m n m n m n mm

x x m m

a a

aa a a a a a a a aa

a a x x x x x

x x x x x x x x x x a aa a a

+ − −

− −

+ −

⋅ = = = =

⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ 2 5 7 3 3 2 2 2 3

3 4 2 5 2 3 6 3 2 2

2 3 42 1 3 2 2 4 3 2

3 1 2 7 2 3 22

4) 15) 16)

17) 2 6 3 18) 2 3 4 2

1 1 119) 20) 21)2 2 2

22) 23) 24)

x x x x y x x y

m m m m

m m m n m n m n

a a a a a a a a a a

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

x x x x

x x x x x

+ + +

+ + − +

+ − + − +

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

− ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⋅ ⋅ ⋅

( ) ( ) ( ) ( )

( ) ( ) ( )

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

2 3 2 2

2 3 3 5 3 1 4 7 2

2 3 43 2 2 4

1 2 2 3 4

2 2

2 3 2 1 3

32

25) 26) 2 3 27) 2 52 928) 29) 30)3 4

31) 1 1 1

32)

33)

34)

m n m n m n

m n m n m n m n x y x y

m n m n

m m m

m n m n

m m

x x xa a a a a a

a a x 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

abc

− − −

+ − − + + − + +

+ −

+ + −

− −

⋅ ⋅

⋅ ⋅ ⋅

⋅ − ⋅ − + ⋅ +

+ − ⋅ + − ⋅ + −

− ⋅ +

+ ⋅ − ⋅ + ⋅ −

⎛ ⎞⎜ ⎟⎝ ⎠

2 4 3 22 2 2x y x y x y y xab ab abc c c

− − − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

NOVO: MALA ŠKOLA MATEMATIKE –1 na

DODATNE video UPUTE

UZ OVE ZADATKE -

POTENCIJEALGEBARSKI IZRAZI

ALGEBARSKI RAZLOMCI

link: http://www.mim-sraga.com/Mala-skola-matematike--video.htm

autor: Mladen Sraga

http://www.mim-sraga.com/Mala-skola-matematike--video.htm






25.

P

( )

( )

( ) ( )

2 2 3 3 2 5 3

2 3 4 6 2 2

2 3 4 3 2 5 4 2 2

56 4 2 3 4 3 4 2

2 4 3 2 2 2 3 3

omnoži:2 3 2) 5 22 9 5 94)3 4 27 51 14 6) 92 3

7) 25 8) 2 35

19) 3 2 7 10) 22

x y x y x y x y

a b ab a b a b

a b c a b c x y x y

yxx y x y z x y z

1)

x x x x x xy x y xy x y

⋅ ⋅

⎛ ⎞⋅ − ⋅ −⎜ ⎟⎝ ⎠

⋅ − ⋅

⋅ ⋅ −

⎛ ⎞⋅ − − + − − ⋅ − +⎜ ⎟⎝ ⎠

3)

5)

9 5 7 5 5 3 3 2 2

4 2 6 2 7 2 3 7 3 4

7 1 21 12 2 29 3 52 2

4 2 8 3 2 2 3 7 2 3

3 75 8 2 4

2 3 13

Izračunaj:

1) 2 : 2 2) 13 : 13 3) 3 : 3 4) 5 : 5

5) : 6) : 7) : : 8) :

9) : 10) : 11) : 12) : :13) : 14) : : 15) : : 16) :

17) 18) 19) 20)

21)

26.x x m m

x x

m

x x x x x x x x x x

x x x x x x x x xa a a a a a a a a a a

x x x xx x x

xx

− −

+

⋅

⋅

( ) ( ) ( ) ( )

( ) ( ) ( )

1 3 7 8 7 9 111 7 10

2 2 3 5 3 2 2 3

6 42 27 4 2 3 2 3

3 3

2 3 3 4 42 2 22 4 4 2 4

3 3 3

22) 23) : 24) :

25) : 26) 2 3 : 2 3 27) :

28) 2 2 : 2 29) :

3

m m mm m

m m m m

x y x y

x x xm m m

x x ax ax x x a

a b a bx y x y x y x yc c

a b a b a bx y x y x yc c c

+ + +− +

+ + + −

+ +

− − −− − −

⎛ ⎞ ⎛ ⎞− − − − ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ ⋅ + + ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( ) ( ) ( ) ( )2 4 2 2 20) 1 1 31) 1 1x x x x x x− ⋅ + + + ⋅ − +

Pogledaj dodatnu VIDEO uputu uz ovaj zadatak http://www.mim-sraga.com/potencije.htm ili direktni link na : video uputu …uz 26. zadatak



tel-01-4578-431

http://www.mim-sraga.com/potencije.htm

http://www.youtube.com/watch?v=-wWZviuFhRI&feature=plcp


http://www.youtube.com/watch?v=lJssyca5z8g&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcp


( ) ( ) ( ) ( )( )

) ( ) ) ( ) ) )

) ) ( ) ) ( ) ) ( )

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

) ( ) )

2 32 2 2 3 2

42 2 32 3 2 2 2

5 2 3 42 3 4 3 4 3 4

3 3 2 32 2 3 4

25

Koristeći pravila:

Izračunaj:

3 21 2 2 3 3 44 3

15 6 7 82

9 10 2 11 2 12 2

13 14 15 16

17 18

27. mn nn n n n n n n m n m z n m zab a b abc a b c a a a a

x x x y xy

x y x x x

x x y x y x y

y y y y

x x

⋅ ⋅ ⋅= = = =

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ −⎜ ⎟⎝ ⎠

⎡ ⎤− − − − −⎣ ⎦

− − − −( ) ) ( ) ) ( )

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

) ) ) ( ) ) ( )

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

22 2 25 5 5

3 4 2 32 2 3 3 2

2 32 32 2

4 23 2 3 2

2 3 2

4 5 43 42 3 2 3

2 2 36 5 4 3 5 72 3 3 5 2 4

19 20

21 2 22 23 2 24 3

2 225 26 27 283 3

29 30 31 32

33 2 34 3 35 2 36 2 3

37 38 39 40

41 42 43 :

m n

m xm n m n x y x y

xx x x m n

yx x

x x

ab a b y y

x x x y

x y x y a b a b

x y x y

x x y y x x

⎡ ⎤⎡ ⎤ − − −⎣ ⎦ ⎢ ⎥⎣ ⎦

− −

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ ⋅ ) ( )( ) ( )( )) ( ) ) ( ) ) ( )

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

8 56 43 2

3 3 32 3 32 3 2 2 3 3 2 2 3 3 2

2 1 2 1 12 3 3 3 2

2 1 1 3 2 4 6 35 3 2 3 3 2 4

44 :

2 1 145 : 2 46 4 46 : 43 2 2

48 3 49 50

51 : 52 2 3 53 3 4

x m m mx x

m m

a a

a b ab x y x y x y x y

a b a b x x x

x x x x a a a

+ − −

− +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⋅ ⋅

+ − +

Pogledaj dodatnu VIDEO uputu uz ovaj zadatak http://www.mim-sraga.com/potencije.htm

ili direktni link na : video uputu …uz 27. zadatak



https://www.youtube.com/watch?v=s18SssjoGhk&feature=plcp


Samo zadaci su zadani od 3. do 12. strane …dio rješenja je

od 15. strane na dalje …

Sva rješenja ovih zadataka šaljemo besplatno




tel-01-4578-431



9

( ) ( )

3 4 4 55 5 3 5

5 5 2 2 4

Koristeći pravilo:Izračunaj:1) 2 5 2) 3 4 3) 4 6 4) 2 4

1 3 2 15) 3 2 6) 9 7) 8) 43 2 3 2

2 99) 10) 11)3 2

28. n nn n n n n

x x x x a a y y

a b a b ili a b c abc

ac b a bb c b a

⋅ = ⋅ =

⋅ ⋅ ⋅ ⋅

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝

4 3 3

2 22 3 12 3 1 2 3 6

2 4 3 2

52 3 6 2

4 3 2 2 2

3 812)4 3

413) 14) 15) 16)2

117) 18)1 2

mm

m m

xy xy x xy x x y zz yz z yzx y z x y

x y z x y xxz x y x xy y

++

⎞ ⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎛ ⎞⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟+ + +⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

5 5x yx y

⎛ ⎞+⋅⎜ ⎟−⎝ ⎠



Novo: MALA ŠKOLA MATEMATIKE –1 na

DODATNE video UPUTE

UZ OVE ZADATKE -


ALGEBARSKI RAZLOMCI


autor: Mladen Sraga






www.mim-sraga.com 10



tel-01-4578-431

2 2 3 4

1 1 2 3

2 2

Izračunaj:

Koristeći pravila:

2 2 1 21) 2) 3) 4)3 3 2 3

5 2 4 25) 6) 1 7) 8)7 7 5 3

1 2 19) 2 10) 1 11) 13 7 2

29.n n nn n

n n

a a a b bb b ab a

−

−

− − − −

− −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛⎜ ⎟ ⎜ ⎟ ⎜⎝ ⎠ ⎝ ⎠ ⎝

3 2

2 2 3 4

2 2 3 2

4 3 2 22 3 3 2 3

3 4 4 4

22 5

6

312)4

13) 14) 15) 16)

1 2 2 417) 1 18) 2 19) 2 20) 32 3 3 5

21) 22) 23) 24)

25)

x x x xy y y y

x x x x yy y y z

x yz

− −

−

⎞ ⎛ ⎞⎟ ⎜ ⎟⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

3 3 24 2 3 4 2

2 3 4 3

2 3 1 2 34 2 4 2

3 3

3 323 2 5 2

3 3

2 226) 27) 28)3 2 5

2 229) 30) 31) 32)5 5

2 3 133) 34) 35) 36)2 3

x x y x yy z z z

x y x y x x yy y xz z

x a b a yx x y c y a

− −

− −

− − −− −

−−−

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞−⎛ ⎞ ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟+ −⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

22 2

3 4

2 23 2

21 2 3 2 32

3

:

3 1 337) 2 2 38)2 8 4

2 1 1 3 2 339) 40)3 3 2 4 23

ya

− −− −

−− − − − −−

−

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⋅ + ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − ⋅ ⋅⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦





( )

( ) ( ) ( ) (

( ) ( ) ( )

( )

0 1 1

02 40 0 0 0

00 0 0

0 0 0 0 0 0

2 0 00 0 0

11 1 1

Izračunaj:

1 1Koristeći pravila: 1 , , ,

1) 2 2) 3) 4)

5) 3 6) 3 7) 3 8) 3

9) 10) 2 7 11) 2 7 12) 2 7

13) 14) 15) 16) 4

17) 5 18) 19) 0,2 20)

2

11

30.

)0 0

nn

m

a a a a aa a

x yx x yz

x

x y x y x y z x y z

x x x x x

x x y

− −

−− − −

= = = =

⎛ ⎞+⎜ ⎟

⎝ ⎠

− − −

+ − − + − +

⋅

−

( )

( )

( )

( ) ( )

52 3 2

11

1 1

1 1 2 4

5 4 4 3 7 9 2 1 3 2 3 6 2 3 2 2

12 2 0 0 0 1

0

1) 3 22) 23) 0,3 24) 2 5

1 1 225) 4 26) 27) 28)34 4

2 2 3 229) 30) 31) 32)5 25

33) 3 3 3 34) 2 2 2 2 35) 5 5

136) 37) 27 23

38) 2

x x x x x x x x

m n n m

x x

ab

x y x y x y

y

−− − −

−−

− −

− − − −

− − − + − − − −

−− −

−

−

⎛ ⎞⎜ ⎟⎝ ⎠−

−

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⎛ ⎞+ ⋅ + ⋅ ⋅ ⋅ ⋅⎜ ⎟⎝ ⎠

⋅21 1 2 2 2

1 33 3 3

22 22 13 1 2 1

2 4 1 2 1 2

0 0 1 0 0 1 0 1 2

2 3 3

1 1 2 39) :2 5

2 32 2 2 340) 41) 42)2 2 2 3 2 3

43) 3 2 3 44) 3 2 3 45) 2 2 2

46) 2 2 47) 2 2

m n m n na b a b a bc c c

+ +−−

−− −− −− − − −

− − − − − −

−

− − − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞⋅− −⎜ ⎟⎜ ⎟ ⎜ ⎟+ ⋅ +⎝ ⎠ ⎝ ⎠⎝ ⎠

⋅ − + − + −

− − 2 1 2 3

3 2 2 12

3 2 2 1

1 2 1

1 2 1

2 3 1 2 1 2 2

1 2 2 3 1 2 3 1

2 48) 3 2

2 3 2 4249) 50) 51)3 4 3 6

52) 53) 54)

55) 56) 57)

a a a bb b ca b x bc x b cc d y c d y c d

− − −

− −−

− −

− − −

− − −

− − − −

− − − − −

+ −

⋅ ⋅⋅

autor: Mladen Sraga




tel-01-4578-431

5 3 2 3

( ) ( )

( ) ( ) ( )

32 3 2

4 2

3 3 5

Zapiši u obliku potencija s bazom 2:

Pojednostavni i rezultati zapiši kao potenciju:

)

1) 4 32 16 2) 4 8 16 3) 2 4 8

) Zapiši u obliku potencija s bazom 3:

1) 3 9 27 2) 9 81 3 3) 3 9 27

)1) 2 2 2) 3 2

31.a

b

c

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

+ ⋅ 5 4

2 2 3 3

6 5 4 3 7 3 5 2

8 3 3 3 9 6 5 6 3 4

1 1 2

1 1

1

2 3) 6 2

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

9 3 4 21) : 2) :2 2 3 15

5 1 25 53) : 4) :7 7 49 7

16 2 4 161) 2)8 32

9 273)81

32.

33.x x x

x x

x x

x y x y x y x y

x y x y a b c a b c

+ − +

− −

−

+ ⋅

⋅ − ⋅ ⋅ + ⋅

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⋅

⋅ 1 5

1 3

25 54)125

x x

x x

+ +

+ −

⋅

42 4− ⋅


2 2 2 22 4 2 4

3 2 3 2

3 2 3 23 2 3 2

4 3 4 3

2 2 47 2 2 2 3 6 2

3 3 2 4 2

1) 2)

3 4 3 33) 4)2 3 2 4

5 5 3 35) 6)6 3 5 5

34.x x x xy y y y

x x x xy y y y

a b a x y x yc b c z z

−− −

− −

−

− −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛− ⋅ − ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝

2⎞⎟⎠

)a b

a b

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

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

2 3 4 2 2 5 2

32 2 2 3 2 3 2

2 3 4 2 2 3 2 2 3 2

3 2 3 2 3 2 2

2 3 4 2 2 3 2

3 5 2 2 3 3 5 3 4 2

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

3 2 2 4 3

1) 2) 2 3)

4) 4 5) 2 6) 3

7) 4 8)

9) 2 10) 2 3

35. )

1) 2) 3)

4) 5) 6)

7) 8)

9)

35.x x

ab x y x y

a b c a b c a b c

a b x y xy

b

x y

a b a b a b x y

a b a b a b a b a b a b

x y x y

+

− − +

+ +

⋅ +

− − ⋅ +

+



36 . I sada primjenom svih pravila za potencije riješite dopunske zadatke

( )

( )

( )

( )( )

1

1

0

POTENCIJE

1:

1: 1

n n n

n nnn m n m n n n

n

n n nmn m n m n n mn

n zmn m n m n n m z nm n

c a d a c d a

a aa a a abc a b c a ab b

a b ba a a a a ab a aa

a a a a a a a aa a

+

−− ⋅ −

− ⋅ ⋅ −

⋅ ± ⋅ = ± ⋅

⎛ ⎞⋅ = = = =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= = = = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= = = = =

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

2 3 4 7 2 17 8 3 4 2

2 4 5 3 8 4 2 7 5 6

3 2 2 42 2 7 3 9 4 7

2 35 6 2 3 4 4 3 4 2

4 3 2 5 6 5 4 7 2

Izračunaj:1 21) 2 5 2) : 3)3 5

4) 5) : 6) :

7) : 8) : 9) 25 :5

1 110) 7 : 11) 2 3 447 7

12) 3 4 13) 2 5

36.

m

x x x x x x x x a a

a a a a a a a

a a x x x y x y

x x x a b c a b c ab c

a a a a a a x x x x

+

⋅ + ⋅ ⋅ − ⋅

⋅ ⋅

⎛ ⎞⋅ ⋅ ⋅⎜ ⎟⎝ ⎠⋅ + ⋅ + ⋅ ⋅ + ⋅ −

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

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

3 6

5 3 10 2 3 5 4 2 2 3 2

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

4 6 3 4 6 33 2 4 3 2 4

4 6 3 43 2 4 7 2 3 3

2 3 4 37 2 3 3 2 3 2 4

3

14) 7 2 : 3 15) 3 2

16) : 17) :

18) 19) :

20) 3 2 21) :

22) : 23)

24) 1

b m m b

m n m n m n m n m n

x xx x x x

xx x

x y

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

x y x y x x x

a a a a a a

a a a x y x y

x y x y a b a b

x

−

+ − + − + − − + + −

⋅

⋅ + − ⋅ ⋅

⋅

⋅ ⋅ ⋅

+ −

⋅

+ ( ) ( )

( ) ( ) ( )

( )

0 00 0 0

0 00 0 2 3 0 0

0 0

0 0

5 2 4 2 3 2

8 3 9 2

3 4 25) 2 2

26) 27) 2 3 5

3 328) 29) 30)2 2

31) 32) : 33) :

34) : 35) 12 :3

x x x

x y xy xy x y x y xy

xy x x yx y y xy

2 0 1

0

3

x x x x

x x x x

− −

− − −

+ − + ⋅ +

− ⋅ − + −

+−

⋅ ⋅x x x− −

2


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Rješenja su od 16.strane na dalje… Iz naše ponude izdvajamo Zbirke potpuno riješenih zadataka priručnici za samostalno učenje: Matematika-1-prvo polugodište: Sve dodatne informacije o ovim zbirkama zatražite na mail: [email protected] ili na naše telefone 01-4578-431 , 4579-130 Dodatne informacije i PDF ogledne primjerke potražite na našoj web-stranici : www.mim-sraga.com


http://www.mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-2.htm

http://www.mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-ALG-RAZL.htm

http://www.mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-ALGEBARSKI-RAZLOMCI-2011.htm

http://www.mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htm


http://www.mim-sraga.com/


Novo: MALA ŠKOLA MATEMATIKE –1 na

BESPLATNA video poduka i instrukcije

UČIMO ZAJEDNO POTENCIJE

ALGEBARSKI IZRAZIALGEBARSKI RAZLOMCI



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Rješenja svih zadataka s kompletnim postupkom i uputama 2 3

puta

2 se ponavlja petputa pa u eksponent pišem

Koristimo pravila:

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

, , ...

1) 2 2 2 2 2

21. n

n

a a a a a a a a a a a a−

−

⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =

⋅ ⋅ ⋅ ⋅

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

5

o 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

= →

⋅ ⋅ ⋅ ⋅ =

⋅ ⋅ =

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

⋅ ⋅ =

+ ⋅ + ⋅ + ⋅ + = +

⎛ ⎞ ⎛ ⎞⋅ ⋅⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠( ) ( ) ( ) ( ) ( )

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

4

3 2

3

7)

8)

x xy y 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

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠− ⋅ + ⋅ − ⋅ − ⋅ + =

= − ⋅ − ⋅ − ⋅ + ⋅ + = − ⋅ +

− − − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ ⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

17

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

( )( )

na eksponent

na

2 3

puta

2 3 4

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

22. 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=


autor: Mladen Sraga


http://www.youtube.com/watch?v=8eURnnGczoE&list=UU_OImhcINU0KEQhZ-64S-eQ&index=2&feature=plcp




tel-01-4578-431

2 3

puta

2

3

5

2

3

Koristimo pravila: , , ...

1 1 1 19)3 3 3 9

3 310) ⎛ ⎞ = ⋅⎜ ⎟3 3 27

5 5 5 5 125

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

2 22 2 2 412)3 3 3 3 3 9

213)3

n

n


= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

⋅ =⎝ ⎠

⎛ ⎞ = ⋅ ⋅ ⋅ ⋅ =⎜ ⎟⎝ ⎠

⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − ⋅ − = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞− = −⎜ ⎟⎝ ⎠

22.

4

2

3

4

2 2 22 2 2 83 3 3 3 3 3 27

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

4 44 4 4 1615)5 5 5 5 5 25

4 4 44 4 4 4 6416)5 5 5 5 5 5 5 125

4 417)5 5

⋅ ⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ − ⋅ − = − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⋅ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ = ⋅ ⋅ ⋅ =⎜ ⎟⎝ ⎠

⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − ⋅ − = + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⋅ ⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − ⋅ − ⋅ − = − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⋅ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞− = −⎜ ⎟⎝ ⎠

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )

2

2

3

3

2

4 4 4 44 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⋅ − = + ⋅ =

Pogledaj dodatnu VIDEO uputu uz ovaj zadatak

http://www.mim-sraga.com/potencije.htm ili direktni link na : video uputu …uz 22. zadatak


http://www.youtube.com/watch?v=1ZQAhjN5U04&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcp


( ) ( ) ( ) ( ) ( )

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

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

2 3

puta

3

4

2 3 4 5

0 0

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

22. n

n


= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

− = − ⋅ − ⋅ − = − ⋅ ⋅ =

− = − ⋅ − ⋅ − ⋅ − = + ⋅ ⋅ ⋅ =

− + − + − + − = + + − + + + − =

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

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

20 30 45

1 2 3 4

5 2

4 5

3 5

2

0 0 0

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 49

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

1 131)4 2

+ =

− + − + − = + ⋅ + ⋅ − = −

− + − + − + − = − + − + = + − − =

− = ⋅ ⋅ ⋅ ⋅ − ⋅ = − =

− = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ = − =

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

⎛ ⎞ ⎛+ −⎜ ⎟⎝ ⎠ ⎝

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

( )

3

2 22 3

2 2

1 1 1 1 1 1 1 1 2 14 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

−⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⋅ + − ⋅ − ⋅ − = − = = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎡ ⎤ ⎡ ⎤− + − = − ⋅ − + − ⋅ − ⋅ − =⎣ ⎦⎣ ⎦

= − = =




http://www.youtube.com/watch?v=1ZQAhjN5U04&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcp


( ) ( )( ) ( )

( )

( )

je potpuno isti izraz...

ili taj isti zadatak na malo duži ali sigur

Koristimo pravila:

51) 2 3 2 3 5 5 ili

5

2) 2 4 1 2 4 7

23. 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

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

⋅ ⎫⎪+ = + ⋅ = ⋅ = ⎬⎪⎭

+ + = + + ⋅ =

( )niji 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

+ + = + + = + + ⋅ = =

=

+ ( )

( )

( )

( )

ili što nije točno!!!

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

ili ta

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 x

a a a a a

y y y y y y y

+ = + + = + + ⋅ =

=

+ + = + + = + + ⋅ =

− = − ⋅ = ⋅ =

− + − = − + − ⋅ = ⋅ =

( )

( ) ( )

( )

( )

j 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 a

− + − = − + − = − + − ⋅ =

+ − + + − =

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

+ + = + + ⋅ =

− + = − + ⋅ = − ⋅

( )

( ) ( ) ( )

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

b 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

= −

+ − = + − ⋅ = − ⋅ = −

+ − − − + − + =

= − − + − + − + =

= − − ⋅ + − ⋅ + − + ⋅ =

⎫= − ⋅ + ⋅ + ⋅ = ⎪⎬

= − + + ⎪⎭


tel-01-4578-431

autor: Mladen Sraga

21


( ) ( )( ) ( )

( )

( ) ( ) ( ) ( ) ( )

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

3 3 3 3 3 3

2 2 2 2 2

ili taj isti za

Koristimo pravila:

10) 7 2 4 7 2 4 9 9

11) 2 3 2 3 1

12) 3 4 3 4 1 6

23.

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

datak 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

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:

3 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 x y x y x y x y

x y

x y

− ⋅ − + =

= − ⋅ − + =

= − + −

+ − − + − + + − = − + ⋅ + − = − + −

+ − − + − + + − = + − − + − + + − =

= − + ⋅ + − =

= − + −




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 1

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

24.n


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

+ − −

+

+

+

− − − + − + − − −

+ + +

+

⋅ = = = =

⋅ = =

⋅ = =

⋅ = =

⋅ = = =

⋅ ⋅ ⋅ = =

⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ = 1

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

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

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

x x x x x x x x

+ + + + + + + + + +

+ +

= = =

− ⋅ ⋅ ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ ⋅ ⋅ = − ⋅ = −

Pogledaj dodatnu VIDEO uputu uz ovaj zadatak

http://www.mim-sraga.com/potencije.htm ili direktni link na : video uputu …uz 24. zadatak


http://www.youtube.com/watch?v=wtiaJyXemMQ&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcp


Rješenja svih zadataka šaljemo besplatno



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

2 3 2 1 3 2 2 1 3 3

2 2 1 3 3

2 2 1

24.rješenj od 18) do 32) zadatka narućite preko maila !

Koristimo pravila: : :

33)

24.n


m m m m

m m

m m

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 y x y

x y x y

+ − −

− − − −

+ − + −

+ −

⋅ = = = =

+ ⋅ − ⋅ + ⋅ − = + ⋅ + ⋅ − ⋅ − =

= + ⋅ − =

= + ⋅ −

( ) ( )

3 3

2 1

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

3 4 2 2 32

62

34)

m m

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

x x x x y y y y

x y

x y x y

ab ab ab ab abc c c c c

abc

abc

+ −

+

− − − − − + − + − + −

+ + − − − − +

−

=

= + ⋅ −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅ = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞= =⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠




Iz naše ponude izdvajamo Zbirke potpuno riješenih zadataka priručnici za samostalno učenje: Matematika-1-prvo polugodište: Matematika-1-prvo polugodište: Matematika –2- univerzalna zbirka potpuno riješenih zadataka Sve dodatne informacije o ovim zbirkama zatražite na mail: [email protected] ili na naše telefone 01-4578-431 , 4579-130 Dodatne informacije i PDF ogledne primjerke potražite na našoj web-stranici : www.mim-sraga.com


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http://www.mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-ALGEBARSKI-RAZLOMCI-2011.htm

http://www.mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htm





http://www.mim-sraga.com/Matematika--1--prvo-polug--spec-ponuda.htm

http://www.mim-sraga.com/Matematika--1--prvo-polug--spec-ponuda.htm



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