kompletni postupak i rješenja - m.i.m.-sragam-1- univerzalna zbirka potpuno riješenih zadataka...

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

UNIVERZALNA ZBIRKA

POTPUNO RIJEŠENIH ZADATAKA PRIRUČNIK ZA SAMOSTALNO UČENJE

POTENCIJE

α

M-1- univerzalna zbirka potpuno riješenih zadataka

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

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠− ⋅ + ⋅ − ⋅ − ⋅ + =

= − ⋅ − ⋅ − ⋅ + ⋅ + = − ⋅ +

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

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

( )( )

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=

www.mim-sraga.com 2 © M.I.M-Sraga centar za [email protected]

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autor: Mladen Sraga

M-1- rješenja i kompletni postupakPOTENCIJE

2 3

puta

2

3

5

2

3

Koristimo pravila: , , ...

1 1 1 19)3 3 3 9

3

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⋅ − = + ⋅ =


( ) ( ) ( ) ( ) ( )

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

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

2 3

puta

3

4

2 3 4 5

Prebrojimo mi

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


= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

− = − ⋅ − ⋅ − = − ⋅ ⋅ =

− = − ⋅ − ⋅ − ⋅ − = + ⋅ ⋅ ⋅ =

− + − + − + − = + ⋅ − ⋅ + ⋅ −

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

nuse -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 0,00102

=

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

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

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

= − = = 4


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( ) ( )( ) ( )

( )

( )

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

= −

+ − = + − ⋅ = − ⋅ = −

+ − − − + − + =

= − − + − + − + =

= − − ⋅ + − ⋅ + − + ⋅ =

⎫= − ⋅ + ⋅ + ⋅ = ⎪⎬

= − + + ⎪⎭

5autor: Mladen Sraga

www.mim-sraga.com 6


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

tel-01-4578-431

( ) ( )( ) ( )

( )

( ) ( ) ( ) ( ) ( )

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

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

− ⋅ − + =

= − ⋅ − + =

= − + −

+ − − + − + + − = − + ⋅ + − = − + −

+ − − + − + + − = + − − + − + + − =

= − + ⋅ + − =

= − + −

autor: Mladen Sraga

7


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

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

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

+ + + + + + + + + +

+ +

= = =

− ⋅ ⋅ ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ ⋅ ⋅ = − ⋅ = −


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

24.n


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...

4 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

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 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 1

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

0 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

+ + + +

+ − + − + + − + + − −

+

+

⋅ =

⋅⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ = ⋅ =

⋅

− ⋅ − = − ⋅ − = − = −

+ ⋅ + = + = +


tel-01-4578-431


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

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

( ) ( )

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)

24.n


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

x x x y y y y

x yabc

+ + − − − − +

−

=

⎛ ⎞= ⎜ ⎟⎝ ⎠

Novo MALA ŠKOLA MATEMATIKE –1 na

BESPLATNA video poduka i instrukcije

UČIMO ZAJEDNO POTENCIJE

ALGEBARSKI IZRAZIALGEBARSKI RAZLOMCI

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


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





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 2 366

2) 5 2 5 21010

2 9 2 93)3 4 3 4

2

25.n


aa a a a a a a a aa

x y x y x x y yx y

x y

x y x y x x y yx y

x y

a b ab a a b b

+ −

+ +

+ +

⋅ = = = =

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ =

=

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ =

=

⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ =

=3

3⋅

32⋅ 2 1 3 4

2 1 7

3 7

6 2 2 6 2 2

23232

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

13

1 15) 4 42 2

422

1 16) 9 93 3

13 33

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



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

25.n


aa a a a a a a a aa

+ −⋅ = = = = −

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 USA OBJAŠNJENJIMA I PO

NALAZE SE Nlink: http://www.mim-sraga.

MALA Š

DODATNE U

ALGEALGEB

autor: Mladen Sraga

Z OVE ZADATKE iz POTENCIJA STUPCIMA RJEŠEVANJA ZADATAKA A NAŠOJ WEB-STRANICI com/Mala-skola-matematike--video.htm

KOLA MATEMATIKE

PUTE UZ OVE ZADATK POTENCIJEBARSKI IZRAZI

ARSKI RAZLOMCI

11

mailto:[email protected]





www.mim-sraga.com 12



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 2 162) 13 : 13 13 13 1693) 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 xx x x x

−

−

−

= =

= =

= 3 2

7 3 4 7 3 4 6

2 1 2 1 2 1

1 1 2 2 1 4 1 322 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

8

8) :19) : :

10) :

11) :

12) : :13) :14) :

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 aa a

−

+ −

− −

⋅ − −−

− ⋅ −−

⋅ − ⋅ − ⋅ − −− − −

−

=

⋅ = =

= = = =

= = = =

= = = =

= = = =

= =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 3 3 2 12 112 2 2 2

ili

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

a a a aa

x x x x xxx x x x xx

x x x x x xx

− −

− − −

− +

−

−

−

−−

= =

= = = =

⋅ = =

= =

= = =

= = =

= = = =

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

13

20) :x x x x x x xx

⋅ − ⋅ −−

= = = = =


U slijedećim zadacima koristimo pravilo: :

Rješenja ZADATAKA od 21) do 31) šaljemo u besplatnom PDF dokumentu samona zahtjev upućem mailom !

26.n

n m n mm

a a a aa

−= =

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



MALA Š

DODATNE U

ALGEALGEB

autor: Mladen Sraga


KOLA MATEMATIKE


ARSKI RAZLOMCI

13









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

parni3

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 pozitivan broj

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 parni 2parnu

2 neparni neparnu

roj pa je:

Zaključak:

jer je eksponent 55, a 55 je neparan broj

jer je eksponent 56, a 56 je paran broj

ilin nx 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 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 br

22

n 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 I

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

autor: Mladen Sraga


( ) ( ) ( ) ( )( )

( )

) ( )) ( )

) ( ) ( )

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

27.zm 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

⋅ ⋅ ⋅

⋅ ⋅

= = = =

=

= ⋅ = ⋅ =

= ⋅ = ⋅ =

⎛ ⎞ ⎛ ⎞= ⋅ ⋅ = ⋅ ⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠

( ) ( )

)

⎝ ⎠

15

( )

) ( ) ( )

) ( )

) ( ) ( )

3 3 32 3 2 3 2 3 3 6

3

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...

2 2 2 843 3 273

1 1 1 152 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

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

43 4 4 3

I 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 n ma 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

⋅⋅

⋅

⋅

⋅ ⋅

⋅ ⋅

⎫⎪⎪⎪⎪⎪⎪⎪ =⎬

− = − ⋅ = − ⋅ = ⋅ = ⎪⎪⎪= =⎪⎪⎪= = ⎪⎭

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ( )

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

4 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

⋅ ⋅

⋅

⋅

⋅ = ⋅ ⋅ =

⎡ ⎤− = = =⎣ ⎦

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




tel-01-4578-431

27.

( )( )

( ) ( )

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

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

14 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

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:


17 1 1 1

18

18 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 naj

1 1 1 1

18

x x x x

x x x x x

⋅

⋅

⋅ − ⋅ = − ⋅ = ⋅ =

⎡ ⎤ ⎡ ⎤− − = + = = =⎣ ⎦⎣ ⎦

bolji....

↓

autor: Mladen Sraga

17


1 1

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 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 neparnoristimo 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

− =

− = −

− ⋅ = − ⋅ = ⋅ =

− = − − = − ⋅ = − ⋅ = − ⋅ = −

− = − ⋅ = − ⋅ = ⋅ =

− = − ⋅ − =

− = − ⋅ − ⋅ − = −

27.

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

) ( ) ( )( ) ( )

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:

19

19 20

21

22 23

( )( )

( )

) ( ) ( ) ( )

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

2 2

3u 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 27

y y y y⋅

− = =

− = − = −

− = − = −

↑

− = − ⋅ = − ⋅ = −

↓

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


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

na zahtjev upućem mailom !


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 098-237-534 4. pošaljite nam pismo na . M.I.M.-Sraga centar za poduku i dopisnu

poduku , Kraljevečki ogranak 24 , Zagreb


tel-01-4578-431

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




( ) ( )

( ) ( )( ) ( )( )55 5 5

3 3 33

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 93 3 3

28. n nn n n n n

x xx 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

4 44

1

5 5 55 5

5 5 5

1 1 11

1 1 48) 4 4 2 322 2 2

2 9 2 9 29)3 2 3 2

⎛ ⎞ ⎛ ⎞⎜ ⎟ = = =⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

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

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ = ⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 3

3⋅

32⋅

5

5

22 22

4 4 44

33 3 33

2 22

2 2

3 243

10)

11) 1 1

4 23 8 3 8 312) 2 84 3 4 3 4 3

2 24 413)2 2 2

a cac b b ab c b c

a b a bb a b a

x yxy xyx 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

xx xxy x xy x x x xz yz z yz z z z z zz

xy xz yz

⋅

⋅

++

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⋅ ⋅ ⎝ ⎠⎝ ⎠

= =

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

⎛ ⎞⎛ ⎞ ⋅⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

( )( )

( )

( )

m+1

111 2 2 12 2 2

2 1 2 22 12

nakon kraćenja

mmm m m

m mm

xy xz yz

xx x x x xz z z zzz

+++ ⋅ + +

+ +⋅ +

⎛ ⎞= ⋅ = =⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞⋅= = = = =⎜ ⎟⎜ ⎟⋅⎝ ⎠ ⎝ ⎠





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( ) ( )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⋅ = ⋅ =






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

−

−

⎛ ⎞ =⎜ ⎟⎝ ⎠

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

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛⎝

29. 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 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

mješoviti broj treba prvo pretvoriti u razlomak...

primjenili smo pravilo: 75

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

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 7 410) 17 7 7 7 9 9

−− − −

−− − −

→

=

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

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

981




tel-01-4578-431

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

29.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 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

xy

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

⎛ ⎞ ⎛ ⎞− ⋅ = − ⋅ = ⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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


23

Koristimo pravila:

ješenja ovog ZADATKA od 18) do 27) šaljemo u besplatnom PDF dokumentu samona zahtjev upućem mailom !

ješenja ovog ZADATKA od 28) pa do 4

n n nn n

n n

a a a b bb b ab a

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

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

29.

R

Rm


autor: Mladen Sraga




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 du

Koristimo pravila:1 11 , , ,

1) 2 1 1

2) 1

3) 1

3)

30. n

na a a a aa a

a

x

x yz

− −= = = =

= =

=

⎛ ⎞ ⎧= ⎨⎜ ⎟

⎩⎝ ⎠

( ) ( )

( )

( )

0 00 2 42 4

0

0 0

0

0 0

0 0 0

0

0

žim postupkom:

po pravilu:

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

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

je 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

autor: Mladen Sraga


( )

( )

( )

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:

Koristimo pravila:1 11 , , ,

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

− −

− −

−

−

= = = =

⋅ = ⋅ =

= =

=

=

= =

=

30.

( )( )

1 1

1

1

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 93

122)

3 10 10 10023) 0,310 3 93

24) 2

nn

nn

x y ax y ax y

aa

xx

x

− −

− −

− −

−

−−

⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− = = =−−

= = =

=

⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

−( )( )

5

5

1

152 5

125) 44

x−

−

=−

=

25


0 1 1Koristimo pravila:1 11 , , ,

Rješenja ovog ZADATKA od 26) do 41) šaljemo u besplatnom PDF dokumentu samona zahtjev upućem mailom !

ešenja ovog ZADATKA od 42) pa do 57) ogu se nab

nna a a a a

a a− −= = = =

aviti samo u štampanoj ( prodajnoj ) varijante ove zbirke !

30.


Rjm


tel-01-4578-431


autor: Mladen Sraga

27


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 !

99 3 9 3 9 221) : : : :

32 2 2 2 2 32

3 39 22 3

32.x y

x y x y x x y y x yx y

x y

− −

− −

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

⋅= ⋅ ⋅ ⋅ =

22

⋅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 ZADATKA mogu 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 xxx

x xx x x x

x y x y

+ + + + +

− −−−

+ − −+ − + − +

⋅ ⋅ =

⋅ ⋅= = = =

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


x x x x− + + − + −= =

www.mim-sraga.com


( )( )

( )( )

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 xx x y x x x x x x x

y y x y y y y y y yy y

− −− ⋅ +

− ⋅ +

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

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Preostala rješenja ovog ZADATKA 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 69 6

35.

ab a b a b a b

x y x x y y

x x y y

x x y yx x y y

⋅

⋅

⋅

= ⋅ ⋅ = ⋅ ⋅ =

+ = + ⋅ ⋅ + =

= ⋅ + + =

= ⋅ + + =

= + +

e ove zbirke !




MALA Š

DODATNE U

ALGEALGEB


KOLA MATEMATIKE


ARSKI RAZLOMCI

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

tel-01-4578-431






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

Preostala rješenja ovog ZADATKA 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 !

=






KAKO NARUČITI KOMPLETNU ŠTAMPANU VARIJANTU OVE ZBIRKE ?

5. javite nam se mailom na: [email protected] 6. nazovite nas na telefon 01-4578-431 7. nazovite nas na mobitem 098-237-534 8. pošaljite nam pismo na . M.I.M.-Sraga centar za poduku i dopisnu

poduku , Kraljevečki ogranak 24 , 10 000 Zagreb









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

UNIVERZALNA ZBIRKA

POTPUNO RIJEŠENIH ZADATAKA PRIRUČNIK ZA SAMOSTALNO UČENJE

POTENCIJE

SAMO ZADACI

α


tel-01-4578-431

autor: Mladen Sraga

31


( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

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

n

n

a b a b a byc c c

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

x

−

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

= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

−

− − − −

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

⎛−

22.

( )

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

( ) ( )

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

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

+

− + + − +

+ + + − + + − + + −

− + − − + + − − + − + + −

23.


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


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

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


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25.

P

33

( )

( )

( ) ( )

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− ⋅ + + + ⋅ − +

autor: Mladen Sraga


( ) ( ) ( ) ( )( )

) ( ) ) ( ) ) )

) ) ( ) ) ( ) ) ( )

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

) ( ) )

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

+ − −

− +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⋅ ⋅

+ − +


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35

( ) ( )

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

⎛ ⎞+⋅⎜ ⎟−⎝ ⎠

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) 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

37

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




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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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