Algebarski Izrazi POTENCIJE 6

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**** MLADEN SRAGA ****2010.

UNIVERZALNA ZBIRKA

POTPUNO RIJEŠENIH ZADATAKAPRIRUČNIK ZA SAMOSTALNO UČENJE

POTENCIJE

α


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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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autor: Mladen Sraga

3

M-1- rješenja i kompletni postupakPOTENCIJE

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

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 x

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

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

y y y y

x y x y x y x y x

−

⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

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

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

− ⋅ + ⋅ − ⋅ − ⋅

( )

( )

( ) ( ) ( ) ( )

2 3

put a

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 b y

c c c


x

−

− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

−

− − − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ −

( )

( ) ( ) ( ) ) ( )

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

( ) ( ) ( ) ( )

( ) ( )

3 4 2 3

4

22 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 32

2 3

2 2 2 2 2

132) 0,2 0,2

2

Koristeći pravila:

1) 2 3 2) 2 4 3) 7 2

4) 9 2 3 5) 2 3 5 7 2

6) 2 3 7) 2 4

8) 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 a

y y y y x a x a x a

xy xy xy ab ab ab

xy 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

+

− + + − +

+ + + − + + − + + −− + − − + + − − + − + + −


4/41


5 3 2 7 2 2 5 2

2 6 4 8 4 7 2 3 3 5

2 5

2 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 7

5) 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 m

m

x x m m

a a

aa a a a a a a a a

a

a a x x x x x

x x x x x x x x x x a a

a a a

+ − −

− −

+ −

⋅ = = = =

⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ 2 5 7 3 3 2 2 2 3

3 4 2 5 2 3 6 3 2 2

2 3 4

2 1 3 2 2 4 3 2

3 12 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

mm 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 5

2 928) 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 x

a 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

ab

c

− − −

+ − − + + − + +

+ −

+ + −

− −

⋅ ⋅

⋅ ⋅ ⋅

⋅ − ⋅ − + ⋅ +

+ − ⋅ + − ⋅ + −

− ⋅ +

+ ⋅ − ⋅ + ⋅ −

⎛ ⎞⎜ ⎟⎝ ⎠

2 4 3 22 2 2

x y x y x y y x

ab ab ab

c c c

− − − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞

⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

NOVO:MALA ŠKOLA MATEMATIKE –1 na

DODATNE video UPUTEUZ OVE ZADATKE

-POTENCIJE

ALGEBARSKI IZRAZIALGEBARSKI RAZLOMCI

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


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25. P

5

( )

( )

( ) ( )

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 2

2 9 5 94)3 4 27 5

1 14 6) 9

2 3

7) 25 8) 2 35

19) 3 2 7 10) 2

2

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

yx x 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 7

5 8 2 4

2 3 1

3

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 x

a a a a a a a a a a a

x x x x

x x x x

x

− −

+

⋅

⋅

( ) ( ) ( ) ( )

( ) ( ) ( )

1 3 7 8 7 9 111 7 10

2 2 3 5 3 2 2 3

6 42 2

7 4 2 3 2 3

3 3

2 3 3 4 42 2 2

2 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 x

m m m

x x a x a

x x x a

a b a b

x y x y x y x y c c

a b a b a b x y x y x y

c c c

+ + +− +

+ + + −

+ +

− − −

− − −

⎛ ⎞ ⎛ ⎞

− − − − ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ ⋅ + + ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( ) ( ) ( ) ( )2 4 2 2 20) 1 1 31) 1 1 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 26. zadatak

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


autor: Mladen Sraga

http://www.mim-sraga.com/potencije.htmhttp://www.youtube.com/watch?v=lJssyca5z8g&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://www.mim-sraga.com/potencije.htmhttp://www.youtube.com/watch?v=-wWZviuFhRI&feature=plcphttp://www.youtube.com/watch?v=lJssyca5z8g&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://www.mim-sraga.com/potencije.htmhttp://www.youtube.com/watch?v=-wWZviuFhRI&feature=plcphttp://www.mim-sraga.com/potencije.htm


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( ) ( ) ( ) ( )( )

) ( ) ) ( ) ) )

) ) ( ) ) ( ) ) ( )

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

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

) ( ) )

2 32 2 2 3 2

4 2 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 4

4 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 3

2 2

4 23 2 3 2

2 3 2

4 5 43 4

2 3 2 3

2 2 36 5 4 3 5 7

2 3 3 5 2 4

19 20

21 2 22 23 2 24 3

2 225 26 27 28

3 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

x x x x m n

y x 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 4

3 2

3 3 32 3 3

2 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 : 4

3 2 2

48 3 49 50

51 : 52 2 3 53 3 4

x m m m x 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 zadatakhttp://www.mim-sraga.com/potencije.htm

ili



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7

( ) ( )

3 4 4 5

5 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) 4

3 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 b

b 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 z

z yz z yz x y z x y

x y z x y x

x z x y x xy y

++

⎞ ⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎠ ⎝ ⎠ ⎝ ⎠

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

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎛ ⎞⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟

+ + +⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

5 5

x y

x y

⎛ ⎞+⋅⎜ ⎟

−⎝ ⎠


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

ili


Novo: MALA ŠKOLA MATEMATIKE –1 na

DODATNE video UPUTEUZ OVE ZADATKE

-POTENCIJE



autor: Mladen Sraga

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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) 1

3 7 2

29.n n nn n

n n

a a a b b

b 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) 3

2 3 3 5

21) 22) 23) 24)

25)

x x x x

y y y y

x x x x y

y y y z

x y

z

− −

−

⎞ ⎛ ⎞⎟ ⎜ ⎟ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

− −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

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 y

y z z z

x y x y x x y

y y x z z

x a b a y x x y c y a

− −

− −

− − −− −

−−−

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

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

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

22 2

3 4

2 2

3 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

− −

− −

−− − − − −−

−

⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⋅ + ⋅⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − ⋅ ⋅⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦



ili


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9/41


( )

( ) ( ) ( ) (

( ) ( ) ( )

( )

0 1 1

02 4

0 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

9

30.

)0 0

n

n

m

a a a a aa a

x y x x y

z

x

x y x y x y z x y z

x x x x x

x x y

− −

−− − −

= = = =

⎛ ⎞+⎜ ⎟

⎝ ⎠

− − −

+ − − + − +

⋅

−

( )

( )

( )

( ) ( )

52 3 2

1

1

1 1

1 1 2 4

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

1

2 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

a

b

x y x y x y

y

−− − −

−

−

− −

− − − −

− − − + − − − −

−

− −

−

−

⎛ ⎞⎜ ⎟⎝ ⎠−

−

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⎛ ⎞+ ⋅ + ⋅ ⋅ ⋅ ⋅⎜ ⎟⎝ ⎠

⋅

21 1 2 2 21 3

3 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 12 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 n

a b a b a b

c 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 b

b b c

a b x bc x b c

c d y c d y c d

− − −

− −−

− −

− − −

− − −

− − − −

− − − − −

+ −

⋅ ⋅

⋅

autor: Mladen Sraga


10/41


11/41


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 x

y y y y

x x x x

y y y y

a b a x y x y

c 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

+

− − +

+ +

⋅ +

− − ⋅ +

+

11autor: Mladen Sraga


12/41


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 nm

n m n m n n m

n

n zm

n m n m n n m z n

m n

c a d a c d a

a aa a a abc a b c a a

b b

a b ba a a a a a

b a aa

aa a a a a a a

a 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 3

5 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 4

47 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 x x x x x

x x 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 y

x y y xy

2 0 1

0

3

x x x x

x x x x

− −

− − −

+ − + ⋅ +

− ⋅ − + −

+

−

⋅ ⋅ x x x− −

2


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14/41


Novo: MALA ŠKOLA MATEMATIKE –1 na

BESPLATNA video poduka i instrukcijeUČIMO ZAJEDNO

POTENCIJEALGEBARSKI IZRAZIALGEBARSKI RAZLOMCI



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15/41


Rješenja svih zadataka s kompletnim postupkom i uputama

2 3

put a

2 se ponavlja pet put a p a u eksponent piš em

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


−

−

⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =

⋅ ⋅ ⋅ ⋅

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

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 2

2)

3)

4)

5)

6)

x x x x

x 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 x

y y

= →

⋅ ⋅ ⋅ ⋅ =

⋅ ⋅ =

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

⋅ ⋅ =

+ ⋅ + ⋅ + ⋅ + = +

⎛ ⎞ ⎛ ⎞⋅ ⋅⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

( ) ( ) ( ) ( ) ( )

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

4

3 2

3

7)

8)

x x

y 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 b

c c c c

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

− ⋅ + ⋅ − ⋅ − ⋅ + =

= − ⋅ − ⋅ − ⋅ + ⋅ + = − ⋅ +

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

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

15

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

( ) ( ) ( ) ( ) ( )

( )

( )

na eksponent

na

2 3

put a

2 3 4

2

parni3

neparni

4

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 produk t p ozi tivan bro j

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

↓

= −

− = − ⋅ − ⋅ − = −

− = − ⋅ − ⋅ − ⋅ −

4 x=




autor: Mladen Sraga

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16/41

www.mim-sraga.com 16



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 8

3 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 4 256

5 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⋅ − = + ⋅ =




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17/41


( ) ( ) ( ) ( ) ( )

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

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

2 3

put a

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 1

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

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

⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

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

= − = =




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18/41


( ) ( )( ) ( )

( )

( )

je potpuno isti izraz...

ili taj isti zadatak na malo duži ali sigur

Koristimo pravila:

5

1) 2 3 2 3 5 5 il i

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 a


x

x 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 zauvj ek t reba zapamt it i pa t o u zadat ku t reba i zgledat i 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 is ti zada tak na malo duži ali si gurnij i naćin :

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

5) 2 3 5 7 2

2 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 a

x 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

2 Ovo je potpuno isti izraz

8) 4 2 9 4 2 9 3 3

9) 3 4 5 2 8 3 8

2 3 8 4 3 5 8

1 2 3 8 4 3 5 8

4 4 6

4 4 6

b ab

xy xy xy xy xy xy

y x y z x y y z y x y

y y y z z x y x y x y

y z x y

y z x y

y z x y

= −

+ − = + − ⋅ = − ⋅ = −

+ − − − + − + =

= − − + − + − + =

= − − ⋅ + − ⋅ + − + ⋅ =

⎫= − ⋅ + ⋅ + ⋅ = ⎪

⎬= − + + ⎪⎭


tel-01-4578-431


19/41

autor: Mladen Sraga

19


( ) ( )( ) ( )

( )

( ) ( ) ( ) ( ) ( )

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

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



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 3

5 2 3 2 7

5 2 3 2 7

6 9

6 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 z

x y x y x y z z

x y z

x y z

x 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

− ⋅ − + =

= − ⋅ − + =

= − + −

+ − − + − + + − = − + ⋅ + − = − + −

+ − − + − + + − = + − − + − + + − =

= − + ⋅ + − =

= − + −


20/41




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


m

x x x x x

m m m m m m m

aa a a a a a a a a

a

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 6 1

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 a a aa 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

+ + + + + + + + + +

+ +

= = =

− ⋅ ⋅ ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ ⋅ ⋅ = − ⋅ = −




http://www.mim-sraga.com/potencije.htmhttp://www.youtube.com/watch?v=wtiaJyXemMQ&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://www.youtube.com/watch?v=wtiaJyXemMQ&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://www.mim-sraga.com/potencije.htm


21/41


22/41


2 2 3 2 2 3

2 2 1 3

4 4

3 2 5 3 3 5 2 3

3 5 2 3

8 5

2 3 4 2 3 4


1) 2 3 2 36

6

2) 5 2 5 2

10

10

2 9 2 93)3 4 3 4

2

25.n


m

aa a a a a a a a a

a

x y x y x x y y x y

x y

x y x y x x y y

x y

x y

a b ab a a b b

+ −

+ +

+ +

⋅ = = = =

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ == ⋅ ⋅ =

=

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ =

=

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ =

=3

3⋅

3

2

⋅ 2 1 3 4

2 1 7

3 7

6 2 2 6 2 2

2

3

2

3

2

5 9 5 94)

27 5 27 5

5

a a b

a b

a b

a b a b a a b b

+

+

⋅ ⋅ ⋅ =⋅

= ⋅ ⋅

=

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

= +3 9⋅

9⋅

5

( ) ( )

6 2 2 1

8 3

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

2 3 3 2 4 5

5 5 9

4 2 2 4 2 2 1

4 2 2 1

6 3 6 3

1

3

1 15) 4 4

2 24

2

2

1 16) 9 9

3 3

13 3

3

3 3

a b

a b

a b c a b c a a b b c c

a b c

a b c

x y x y x x y y

x y

x y x y

+ +

+ + +

+ +

⋅ ⋅ =

=

⋅ − = ⋅ − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =

= − ⋅ ⋅ ⋅ =

= −

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ =



−


tel-01-4578-431

http://www.mim-sraga.com/potencije.htmhttp://www.youtube.com/watch?v=-wWZviuFhRI&feature=plcphttp://www.youtube.com/watch?v=-wWZviuFhRI&feature=plcphttp://www.mim-sraga.com/potencije.htm


23/41



Rješenja ZADATAKA 7), 8) , 9) , 10) - šaljemo mailom

25.n


m

aa a a a a a a a a

a

+ −⋅ = = = = −



ili


Dakle ako trebate preostala rješenja ovog zadatka pošaljite nam poruku na:Mail: [email protected] sa tekstom : Trebam preostala rješenja zadataka iz zbirke POTENCIJE

DODATNE UPUTE UZ OVE ZADATKE iz POTENCIJASA OBJAŠNJENJIMA I POSTUPCIMA RJEŠEVANJA ZADATAKA

NALAZE SE NA NAŠOJ WEB-STRANICIlink: http://www.mim-sraga.com/Mala-skola-matematike--video.htm

MALA ŠKOLA MATEMATIKE

DODATNE UPUTE UZ OVE ZADATK POTENCIJEALGEBARSKI IZRAZI

ALGEBARSKI RAZLOMCI


http://www.mim-sraga.com/potencije.htmhttp://www.youtube.com/watch?v=-wWZviuFhRI&feature=plcpmailto:[email protected]://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.youtube.com/watch?v=-wWZviuFhRI&feature=plcphttp://www.mim-sraga.com/potencije.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmmailto:[email protected]://www.mim-sraga.com/Mala-skola-matematike--video.htm


24/41




tel-01-4578-431

( )

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 mo

:

1) 2 : 2 2 2 2 2 2 2 2 16

2) 13 :13 13 13 169

3) 3 : 3 3 3

4) 5 : 5 5 5 5 5

26. 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

ra 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 ne,

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

↓

4 2 4 2 2

6 2 6 2 4

7 2 3 7 2

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

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

5) :

6) :

7) : :

x x x x

x x x x

x x x x

−

−

−

= =

= =

= 3 2

7 3 4 7 3 4 6

2 1 2 1 2 1

1 1 2 2 1 4 1 32

2 2 2 2 2 2

7 1 7 1 7 1 3 7 3 4

9 3 9 3 9 9 9

2 2 1 2 2 2 10 1 5 4 20 5 2112

25 5 2 10 10 102

4 2 4 2 2

8

8) :

19) : :

10) :

11) :

12) : :

13) :

14) :

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 x x x x

a a a a

a a

−

+ −

− −

⋅ − −−

− ⋅ −−

⋅ − ⋅ − ⋅ − −− − −

−

=

⋅ = =

= = = =

= = = =

= = = =

= = = =

= =3 2 8 3 2 3

2 3 1 2 3 1 2 3 4

4

7 2 3 7 2 3 8

55 2 5 2 3

2

88 3 8 3 5

3

33 3 3 2 12 1

12 2 2 2

il i

U slijedećim zadacima koristimo pravilo:

:

115) : : : :

16) :

:

17) :

18) :

19) :

nn m n m

m

a a a

a a a a a a a aa

a a a a a

aa a a

a

x x x x x

x

x x x x x

x

x x x x x x

x

− −

− − −

− +

−

−

−

−−

= =

= = = =

⋅ = =

= =

= = =

= = =

= = = =

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

1

3

20) : x

x x x x x x

x

⋅ − ⋅ −−

= = = = =



http://www.mim-sraga.com/potencije.htmhttp://www.youtube.com/watch?v=lJssyca5z8g&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://www.youtube.com/watch?v=lJssyca5z8g&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://www.mim-sraga.com/potencije.htm


25/41


U slijedećim zadacima koristimo pravilo: :

Rješenja ZADATAKA od 21) do 31) šaljemo u besplatnom PDF dokumentu samo

na zahtjev upućem mailom !

25

26.n

n m n m

m

aa a a

a

−= =


ili


Ako trebate preostala rješenja ovog zadatka pošaljite nam poruku na:Mail: [email protected] sa tekstom : Trebam preostala rješenja zadataka iz zbirke POTENCIJE

DODATNE UPUTE UZ OVE ZADATKE iz POTENCIJASA OBJAŠNJENJIMA I POSTUPCIMA RJEŠEVANJA ZADATAKA

NALAZE SE NA NAŠOJ WEB-STRANICIlink: http://www.mim-sraga.com/Mala-skola-matematike--video.htm


DODATNE UPUTE UZ OVE ZADATK POTENCIJE


autor: Mladen Sraga

http://www.mim-sraga.com/potencije.htmhttp://www.youtube.com/watch?v=lJssyca5z8g&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcpmailto:[email protected]://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.youtube.com/watch?v=lJssyca5z8g&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://www.mim-sraga.com/potencije.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmhttp://www.mim-sraga.com/Mala-skola-matematike--video.htmmailto:[email protected]


26/41




tel-01-4578-431

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

Koristimo pravila:

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

Pojavio nam se jedan problem a taj je:

22.ogledajmo opet u kako smozadata k

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

parni

3

neparni

3

Imamo neparan broj "minusa" pa će i umnožak

1 1Zaključak:

to tamo rješili:

4) 1 1 1 1

5) 1 1 1 1 1

7) 2 2 2 2 8

↓

− =

−

⎫− = − ⋅ − = ⎪⎪

− = − ⋅ − ⋅ − = − ⎬

⎭

− = − ⋅ − ⋅ − = −

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

4 4

Imamo paran broj "minusa" pa je produkt poz iti van bro j

53

53 puta bi to trebali

biti negativno tj. imati će predznak minus

8)

a se javlja ovakav problem: ...

x x x x x x

x x x x

− = − ⋅ − ⋅ − ⋅ − =

− = − ⋅ − ⋅ ⋅ −

( )

( )

napisatia to je previše pisanja pa ćemo rađe gledati eksponent

53 53

54alje logički je pitanje koliko je eksponent je 54, 54 je paran b

u eksponentu je 53 ,

an broj pa je:

?

x x

x

=

− = −

−

( )

( )

( )

( )

( )

( )

na eksponent

na eksponent

54 54

55 55

56 56

2 par ni 2 parn u

2neparni neparnu

roj pa je:

Zaključak:

jer je eksponent 55, a 55 je neparan broj

jer je eksponent 56, a 56 je paran broj

ili

n n x x

x x

x x

x x

x x

x x

x

− =

− = −

− =

− = − =

− = =

− =

−

27.

P

22.

( ) ( ) ( ) ( ) ( )( )4 1 1

6) 1 1 1 1 1 1− =⎪

− = − ⋅ − ⋅ − ⋅ − = ⎪

Sad

53 je nepar

d

( )

( )

( )

1 2 1

4 4

3 3

2 je oznaka za parn i b ro j

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 br

22

n n

n

n

x

x x

+ +

+

= −

− = =

− = − = − =

( ) ( ) ( )

( ) ( )

( )( ) ( )

( )

( ) ( ) ( )

3 3 3 3

3 3 3 3

Svaki negativan brojda se zapisati u obliku:1

4 4 4 4 4 4I

oj

Postoji 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

−− = − ⋅

− = − ⋅ = − ⋅ = − ⋅ = −

− = − ⋅ = − ⋅ = − ⋅ = −

− = − ⋅ = − ⋅ = ⋅ =

I način


27/41

autor: Mladen Sraga

27


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

) ( )) ( )

) ( ) ( )

2 2 2 2 2

2 2 2 2 2

2 2 22 2

2 3 2 3 2 2 3 2 4 6

2

Koristimo pravila:

Postupili smo prema prvom pravilu...

1 2 2 4 4

2 3 3 9 9

3 3 3 93

4 4 164

27. z

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

n n n

ab a b abc a b c a a a a

ab a b

x x x x

x x x x

x y x y x y x y

a

⋅ ⋅ ⋅

⋅ ⋅

= = = =

=

= ⋅ = ⋅ == ⋅ = ⋅ =

⎛ ⎞ ⎛ ⎞= ⋅ ⋅ = ⋅ ⋅ =⎜ ⎟ ⎜ ⎟

⎝ ⎠

( ) ( )

)


ilidirektni link na : video uputu …uz 27. zadatak

⎝ ⎠

( )

) ( ) ( )

) ( )

) ( ) ( )

3 3 3

2 3 2 3 2 3 3 63

4 4 44 4

2 3 2 3 2 4 3 4 8 12

4

22 2 2 4

2 22 2 2 2 4

Postupili smo prema drugom i trećem pravilu...

2 2 2 843 3 273

1 1 1 15

2 2 162

6

7

mn n n n n n mbc 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

⋅

⋅

⋅ ⋅

⋅

⋅

= =

⎛ ⎞ ⎛ ⎞= ⋅ ⋅ = ⋅ ⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= ⋅ ⋅ = ⋅ ⋅ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

= =

− = = =

) ( ) ( ) ( ) ( )

) ( )

) ( )

( )

3

) ( ) ( ) ( )

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

I način

II način :

Prema pravilu:7 1 1 1

8

9

10 2 2 4 4

mn n ma a

x x x x x

x x x

x x x

x y x y x y x y

⋅

⋅

⋅

⋅

⋅ ⋅

⎫⎪⎪⎪⎪⎪⎪⎪

=⎬− = − ⋅ = − ⋅ = ⋅ = ⎪

⎪⎪= =⎪⎪⎪= =⎪⎭

= ⋅ ⋅ = ⋅ ⋅ =


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27. Rješenja ovog ZADATKA od 25) do 53) šaljemo u besplatnom PDF dokumentu samo






KAKO NARUČITI KOMPLETNU ŠTAMPANU VARIJANTU OVE ZBIRKE ?

1. javite nam se mailom na: [email protected] 2. nazovite nas na telefon 01-4578-431

3. nazovite nas na mobitem 098-237-5344. pošaljite nam pismo na . M.I.M.-Sraga centar za poduku i dopisnu

poduku , Kraljevečki ogranak 24 , Zagreb


tel-01-4578-431

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( ) ( )

( ) ( )

( ) ( )

( )5

5 5 5

3 3 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) 3 2 3 5 15

3 31 1 96) 9 9

3 3 3

28. n nn n n n n

x x x x x x x x

a ya a a y y y

a b a b a b c abc⋅ = ⋅ =

⋅ = ⋅ = ⋅ = ⋅ =

⋅ = ⋅ = ⋅ = ⋅ =

⋅ = ⋅ =

⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ = ⋅ = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 3

3

3

14 4 4

3 27

3 2 3 2 37)

2 3 2 3

⎛ ⎞= =⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ = ⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 2

2⋅

1

3

44

4

1

5 5 5

5 5

5 5 5

11 1

1

1 1 48) 4 4 2 32

2 2 2

2 9 2 9 29)

3 2 3 2

⎛ ⎞ ⎛ ⎞⎜ ⎟ = = =⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ = ⋅ = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ = ⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 3

3⋅

3

2

⋅5

5

22 2

2

4 4 4

4

33 3 3

3

2 22

2 2

3 243

10)

11) 1 1

4 23 8 3 8 312) 2 8

4 3 4 3 4 3

2 24 413)

2 2 2

a cac b ba

b c b c

a b a b

b a b a

x y xy xy

x y x y

⎛ ⎞= =⎜ ⎟

⎝ ⎠

⋅⎛ ⎞⎛ ⎞ ⎛ ⎞⋅ = ⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ = ⋅ = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

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

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⋅ ⋅⎛ ⎞⋅ = ⋅ = ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( )

( )

2 2

2

2 2

333 23 33 2 2 3 6

2 3 2 3 62

11

možemo ostaviti u ovom obliku

ili dalje:

nakon kraćenja

2

2 4

14)

15)

mm

x x y x

x x

x x x xy x xy x x x x

z yz z yz z z z z z z

xy x

z yz

⋅

⋅

++

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⋅ ⋅ ⎝ ⎠⎝ ⎠

= =

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

⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞⋅⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

( )

( )

( )

( )

m+1

111 2 2 12 2 2

2 1 2 22 12

nakon kraćenja

mmm m m

m mm

xy x

z yz

x x x x x x

z z z z z z

+++ ⋅ + +

+ +⋅ +

⎛ ⎞= ⋅ = =⎜ ⎟

⎝ ⎠

⎛ ⎞⎛ ⎞⋅= = = = =⎜ ⎟⎜ ⎟

⋅⎝ ⎠ ⎝ ⎠



30/41




tel-01-4578-431

( ) ( )Koristimo pravilo:

Rješenja ovog ZADATKA od 16) do 18) šaljemo u besplatnom PDF dokumentu samona zahtjev upućem mailom !

28. n nn n n n na b a b a b c abc⋅ = ⋅ =


Novo MALA ŠKOLA MATEMATIKE 1 na


POTENCIJEALGEBARSKI IZRAZI

ALGEBARSKI RAZLOMCI


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31/41



2 2

2

2 2 2

2

po pravilu :

pr imjeni li 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 a

b b

a a a b b

b b ab a

a a

b b

−

−

⎛ ⎞=⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛

⎝

29.

2 2 2

2

po pravilu

3 3

3

4 4

4

1 1



3 3 9

2 42

1 1 13)

2 82

2 2 164)

3 813

5 75)

7 5

n na b

b a

n n

n

n n

n

a a

b b

a a

b b

−

−

⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

−

⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟

⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

11 1

2 2 2

2

3 3

mješoviti broj treba prvo pretvoriti u r azlomak...

pr imjeni li smo pravilo:7

5

1 7 22 9 76) 1

7 7 7 9

4 5 5 257)

5 4 164

2 3 38)

3 2

n n n

n

a b b

b a a

−

−− −

−

−

→

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⋅ +⎛ ⎞⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

3

3

22 2 2 2 2

2

22 2 2 2 2

2

mješoviti broj treba prvo pretvoriti u razlomak...

27

82

2 3 11 6 1 7 3 3 99) 2

3 3 3 3 7 497

1 7 22 7 2 9 7 7 410) 1

7 7 7 7 9 9

−− − −

−− − −

→

=

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

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

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

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

9

81


32/41




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33 3 3 3 3

3

2

mješoviti broj treba prvo pretvoriti u r azlomak...

Koristimo pravila:

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

3 412)

4 3

29.n n nn n

n n

a a a b b

b 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 16

93

13)

14) I k

14) jer je:

par nu par nu

x x

x x

x x

y y

x x x x x x

y y y y y y

− =

− = −

⎞= =⎜ ⎟

⎠

⎛ ⎞=⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = = − = − = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( )

2

2 2 2 2 22

2 2II način

način rastavili smo na:

.. .

14) 1 1 1

II 1 ...

x

y

x x x x x

y y y y y

x x

y y

⎛ ⎞=⎜ ⎟

⎝ ⎠

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

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− − ⋅⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Koristimo pravila:

Rješenja ovog ZADATKA od 15) do 27) šaljemo u besplatnom PDF dokumentu samo


Rješenja ovog ZADATKA od 28) pa do 4

29.n n nn n

n n

a a a b b

b b ab a

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

0)

mogu se nabaviti samo u štampanoj ( prodajnoj ) varijante ove zbirke !


33/41




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Dodatne informacije i PDF ogledne primjerkepotražite na našoj web-stranici : www.mim-sraga.com


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34/41


0 1 1

0 0

0

02 4

po pravilu:

bez obzi ra što je u zagrad i ako je eksponent

te zagrade nula sve je jednako jedan !

ili taj isti zadatak du

Koristimo pravila:1 1

1 , , ,

1) 2 1 1

2) 1

3) 1

3)

30. n na a a a aa a

a

x

x y

z

− −= = = =

= =

=

⎛ ⎞ ⎧= ⎨⎜ ⎟

⎩⎝ ⎠

( ) ( )

( )

( )

0 00 2 42 4

0

0 0

0

0 0

0 0 0

0

0

žim postupkom:

po pravilu:

bez obzi ra što je u zagradi ako je to na nu ltu sve

1 1 1 1

1 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 y x y

z z

x y

a

x

x y

⋅⎛ ⎞ ⋅= = = =⎜ ⎟

⎝ ⎠

+ = + =

=

− = =

− = − − = − ⋅ = − ⋅ = −

− = − ⋅ = −

+ =

( )

( ) ( )

( )

0

0 0 0

0 00

0 0

20 2

je jednako jedan !

bez obzi ra 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


35/41

autor: Mladen Sraga


( )

( )

( )

0 1 1

0 0

20 2

0

0

1 1

1

1

bez obzi ra št o je u zagradi ako je to na nuul tu sve je jednako jedan !

po pravilu:


1 , , ,

13) 1 1 1

14) 1 1

15) 1

16) 4 1

1 117) 5

5

118)

19) 0,2

n

n

m

a a a a aa a

x x

x

x

x

aa

x x

− −

− −

−

−

= = = =

⋅ = ⋅ =

= =

=

=

= =

=

30.

( )( )

1 1

1

1

2

2

3

3

2 2 22

2

prvo decimalni broj pretvorimo u razlomak.. .

po pravilu:

po pravilu:

2 1 55

10 5 1

1 1 120)

1 1 121) 3

93

122)

3 10 10 10023) 0,3

10 3 93

24) 2

n

n

n

n

x y a x y a x y

aa

x x

x

− −

− −

− −

−

−

−

⎛ ⎞ ⎛ ⎞= = = =

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− = = =−−

= = =

=

⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

−( )( )

5

5

1

15

2 5

125) 4

4

x

−

−

=−

=

35


36/41


0 1 1


1 , , ,

Rješenja ovog ZADATKA od 26) do 41) šaljemo u besplatnom PDF dokumentu samo


ešenja ovog ZADATKA od 42) pa do 57)

ogu se nab

n

na a a a a

a a

− −= = = =

aviti samo u štampanoj ( prodajnoj ) varijante ove zbirke !

30.


Rj

m


tel-01-4578-431

mailto:[email protected]:[email protected]


37/41

autor: Mladen Sraga

37


4 32 16 2 2 2 2 2

2) 4 8 16 2 2 2 2 2 2 2

Preostala rješenja ovog ZADATKA

mogu se nabaviti samo u štampanoj ( prodajnoj ) varijant

+ +

+ + ⋅

⋅ ⋅ = ⋅ ⋅ = =

⋅ ⋅ = ⋅ ⋅ = = = =

31.

a

1)

( ) ( ) ( ) ( )

2 5 4 2 5 4 11

2 2 3 4 2 2 3 4 2 9 2 9 2 18

) Zapiši u obliku potencija s bazom 2:

6 5

6 5 4 3 6 4 5 3 6 4 5 3

4 3

6 4 5 3

e ove zbirke !

9

9 3 9 3 9 221) : : : :32 2 2 2 2 3

2

3 39 2

2 3

32.

x y

x y x y x x y y x y

x y

x y

− −

− −

⎛ ⎞ ⎛ ⎞= = ⋅ ⋅ = ⋅ ⋅ ⋅ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⋅= ⋅ ⋅ ⋅ =

2

2⋅

3

( ) ( )

( )

2 2 2 2

1 4 1 4 1 5

1 3 33 13 1

5 3 35 3 3 5 3 3

3

Preostala rješenja ovog ZADATKAmogu se nabaviti samo u štampanoj ( prodajnoj ) varijante ove zbirke !

16 2 2 2 2 21)

8 22 2

2 :2 2 2

33. x x x x

x x x x

x x x x x x

x y x y

+ + + + +

− −−−

+ − −+ − + − +

⋅ ⋅ =

⋅ ⋅= = = =

= = = = 3 5 3 2 8 8 22 2 2



x x x x− + + − + −= =


38/41




tel-01-4578-431

( )( )

( )( )

2 2 2 24 2 2 22 4 3 2 4 2 8 4 8 4 8 12

3 2 2 3 2 2 4 6 4 6 4 102 2 3 21)

34.

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

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

− −− ⋅ +

− ⋅ +

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

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠


mogu se nabaviti samo u štampanoj ( prodajnoj ) varijant

( ) ( )

( ) ( ) ( )

( )

2 2 2 2 2 2 2 2 2 2 4

2 3 2 2 2 2 3 3 2

2 2 2 2 3 3 2

2 2 2 3 6

4 2 3 6

4) 4 4 16 16

6) 3 3 2 3

3 6

9 6

9 6

35.

ab a b a b a b

x y x x y y

x x y y

x x y y

x x y y

⋅

⋅

⋅

= ⋅ ⋅ = ⋅ ⋅ =

+ = + ⋅ ⋅ + =

= ⋅ + + =

= ⋅ + + =

= + +

e ove zbirke !



DODATNE UPUTE UZ OVE ZADATKE iz POTENCIJA

SA OBJAŠNJENJIMA I POSTUPCIMA RJEŠEVANJA ZADATAKANALAZE SE NA NAŠOJ WEB-STRANICI



DODATNE UPUTE UZ OVE ZADATK POTENCIJE


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( ) ( ) ( ) ( ) ( ) ( )2 3 4 2 4 3 2 4 3 4 2 3 4 3 2 4 3 4 2 3 4 3 8 12 6 12

8 6 12 12 8 6 12 12 14 24

23)


mogu se nabaviti samo u štampanoj ( prodajnoj ) varijante ove zbi

36.

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

a a b b a b a b

⋅ ⋅ ⋅ ⋅

+ +

⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ = ⋅ =

rke !

=

Novo MALA ŠKOLA MATEMATIKE –1 na

BESPLATNA video poduka i instrukcije

UČIMO ZAJEDNOPOTENCIJEALGEBARSKI IZRAZI

ALGEBARSKI RAZLOMCI


KAKO NARUČITI KOMPLETNU ŠTAMPANU VARIJANTU OVE ZBIRKE ?

5. javite nam se mailom na: [email protected] 6. nazovite nas na telefon 01-4578-4317. nazovite nas na mobitem 098-237-534

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