Stepeni(Potencije) Operacije Sa Stepenima

5/26/2018 Stepeni(Potencije) Operacije Sa Stepenima

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M-1- rjeenja i kompletni postupakPOTENCIJE

Rjeenja svih zadataka s kompletnim postupkom i uputama

1. 2 3

put a

2 se ponavlja petput a p a u eksponentpi emo

Koristimo pravila:

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

, , ...

1) 2 2 2 2 2

n

n

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

= = =

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

5

5

5

3

6 3 2 6 3 2

3

4

grupiramo iste faktore...

itamo: 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 y

=

=

=

= = =

=

+ + + + = +

( ) ( ) ( ) ( ) ( )

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

4

3 2

3

7)

8)

x x

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

=

+ + =

= + + = +

=

2.

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

( ) ( ) ( ) ( ) ( )

( )

( )

na eksponent

na

2 3

put a

2 3

2

parni

3

neparni

4

Zakljuak:

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 15) 1 1 1 1 1

6) 1 1 1 1 1 1

n

n


=

= = =

= = = = = =

= =

= =

= =

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

eksponent

3

4

Imamo paran broj "minusa" paje produk t p ozi tivan bro j

Imamo neparan broj "minusa" pa e i umnoak biti negativno tj. imati e predznak minus

1

7) 2 2 2 2 8

8) x x x x x

=

= =

=

4x=

4

7


2/32

8

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

M.I.M-Sraga centar za poduku

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2 3

put a

2

3

5

2

3

Koristimo pravila: , , ...

1 1 1 19)

3 3 3 9

3 3 3 3 2710)

5 5 5 5 125

2 2 2 2 2 2 3211)

3 3 3 3 3 3 243

2 2 2 2 2 412)

3 3 3 3 3 9

2 213)

3

n

n


= = =

= =

= =

= =

= = =

=

4

2

3

4

2 2 2 2 2 8

3 3 3 3 3 3 27

3 3 3 3 3 8114)

4 4 4 4 4 256

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

4 4 4 4 4 4 4 6416)

5 5 5 5 5 5 5 125

4 417)

5 5

= =

= =

= = + =

= = =

=

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

2

2

3

3

2

4 4 4 4 4 4 4 256

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 = + =

2.


3/32


2.

( ) ( ) ( ) ( ) ( )

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

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

2 3

put a

3

4

2 3 4 5

Prebrojimo min

Koristimo pravila: , , ...

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

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

25) 1 1 1 1 1 1 1 1

n

n


= = =

= = =

= = + =

+ + + = + +

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

use -imaih paran broj pa e umnoakbit i poz iti van bro j

20 30 45

1 2 3 4

5 2

4 5

1 1

26) 1 1 1 1 1 1 1

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

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

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

= + =

+ + = + + =

+ + + = + + = + =

= = =

= = =

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

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

( )

3 5

2 3

2 22 3

2 2

49

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

1 1 1 1 1 1 1 1 1 1 2 131)

4 2 4 4 2 2 2 16 8 16 16

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

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

+ = + =

+ = + = = =

+ = + =

= = =

9


4/32

10



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3. ( ) ( )

( ) ( )

( )

( )

je potpuno isti izraz...

ili taj isti zadatak na malo dui ali sigurn

Koristimo pravila:

5

1) 2 3 2 3 5 5 il i

5

2) 2 4 1 2 4 7

n n n n n n

c a d a c d a c a d a c d a


x

x x x x x

x

x x x x x

+ = + =

+ = + =

+ = + = =

+ + = + + =

( )

iji nain :

Dakle:

Praksa je pokazala da velika veina aka radi istu greku: uzimate da je: to nije tono!!

dakle vi kada raunate u glavi greite na ovaj nain :

2 4 1 2 4 1 2 4 7 1

0

2

x x x x x x x x x x

x x

x x

+ + = + + = + + = =

=

+ ( )

( )

( )

( )

ili to nije tono!!!

Jednom zauvj ek treba zapamt it i pa to u zadat ku t reba izg ledat i ovako:

ili taj

4 6 2 4 0 2 4 6

1

2 4 1 2 4 1 2 4 7

3) 7 2 7 2 5 5

4) 9 2 3 9 2 3 1 9 9

x x x x x x x

x x

x x x x x x x x

a a a a a

y y y y y y y

+ = + + = + + ==

+ + = + + = + + =

= = =

+ = + = =

( )

( ) ( )

( )

( )

isti zadatak na malo dui ali sigurniji nain :

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 ab

+ = + = + =

+ + + =

= + + + = + + + = + = +

+ + = + + =

+ = + =

( )

( ) ( ) ( )

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

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

=

+ = + = =

+ + + =

= + + + =

= + + + =

= + + =

= + +


5/32

11


( ) ( )( ) ( )

( )

( ) ( ) ( )( ) ( )

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

3 3 3 3 3 3

2 2 2 2 2

ili taj isti zad

Koristimo pravila:

10) 7 2 4 7 2 4 9 9

11) 2 3 2 3 1

12) 3 4 3 4 1 6

n n n n n n



xy xy xy xy xy xy

x y x y x y x y x y

x y x y x y x y x y

+ = + =

+ = + =

+ = + = =

+ = + = + =

+ + + + = + + = +

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

( ) ( )

( )

( ) ( )

( ) ( )

2 3 2 3 2 3

2 3 2 3 2 3

2 3

2 3

2 3

2 2

atak na malo dui ali sigurniji nain :

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

3

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

x y

x y

x y z x y z x y 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 dui ali sigurniji nain :

4 3 5

1 3 5

3 5

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

3 1 6 1 1 3 1 6 1 1 1

3 6 1 1

2 1

x x

x x

x x

x y x y x y x y x y


x y

x y

+ =

= + =

= +

+ + + + = + + = +

+ + + + = + + + + =

= + + =

= +

3.


6/32

12



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5 3 5 3 8

2 7 2 7 9

2 2 3

2 5 2 2 5 2 2 5 2 3 7

2 6 4 8 2 6 4 8 20

4 7 4 7 4 7 11

Koristimo pravila: : :

1) 2 2 2 2

2) 5 5 5 5

3) 3 3 3 3

4) 7 7 7 7 7

5) 2 2 2 2 2 2

6) 3 2 3 2 6 6

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

m

x x x x x

m m m m m m m

aa a a a a a a a a

a

a a a a a a

+

+

+

+

+ +

+ + +

+

= = = =

= =

= =

= =

= = =

= =

= = =

2 3 2 3 5

3 5 1 3 5 1 3 5 9 1

2 4 6 2 4 6 12

2 2 5 2 2 2 6 5 3 4 12 15 315223 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 esto radite ovakve greke:

to nije tono jer je:

14)

15)

16)

x x x x x x x

x y x x

a a a a a a a a

a a a a aa a

a a a a a

a a a a a a

a a a a a

a a a

+ +

+

+ +

+ + +

+ + + + +

+

= = = =

= = = = =

= =

= =

3 2 3 2 3 4 4

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

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

x x x x x x x x

+ + + + + + + + + +

+ +

= = =

= = =

4.


7/32


4.

( )

2 5 2 3 6 3 2 2

2 5 2 1 3 1 1 6 3 2 2

7 2 1 3 1 1 6 3 4

7 7 3 4 3 4


18) 2 3 4 2

2 3 4 22 3 4 2

2 4 3 2

2 4


m

aa a a a a a a a a

a

x x x y x y x x x y y

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

x 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... uobiajeni zapis je ovaj zadnji...

3 2

2 5

2 5

1 1 1 1 1

19) 2 2 2 2 2

20) m m m m m m

x x y

x x y

x 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 9

2 7 22 2 2 2 2

2 3 2 2 3 2 2 3 2 6 2

2 3 2 2

21)

22)

23)

24)

m

m m m m m m m

mm m m m m m mm

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

m n m n m

x

x x x x x

x x x x x x x

x x x x x x

x x x

+

+ + + + +

+ + ++ + + + + + + +

+ + + + + + + + + + +

= = =

= = = = =

= = =

2 3 2 2 2 3 2 2 7 4

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

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

2 2

25)

26) 2 3 2 3 6 6 6

27) 2 5 2 5 10

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

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

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

x y x y x y x y

x x x

a a a a a

a a a a a a

a a a

+ + + +

+ + + + +

+ + + + + + + +

+ + + + +

= = =

= = =

= = = =

= =

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

( ) ( ) ( ) ( )

2 3 2

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

2 2 1 2 1 3

3 4 3 4 7

10

3 32 9 2 9 2 3 328)

3 4 3 4 3 2 2 2 2

29)

30)

x x y y x y

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

a a

a a a a a a a


x y x y x y x y

+ + + +

+ + + + + +

+

+

=

= = = =

= = =

+ + = + = +

13


8/32


4.

( ) ( ) ( ) ( )

( ) ( )

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

( ) ( )

1 2 2 3 4 1 2 2 3 4

2 3 1 2 4 6 1

1 1 1 1 22 2 2 2 2 2

2 3


31) 1 1 1 1

1 1

32)

33)


m

m m m m m m

m m m m

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

aa a a a a a a a a

a

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 ab

c c c c c

ab

c

+ +

+ +

+

+ + +

+

= + + =

= + =

= + =

= +

= =

=

( ) ( )

4 2 2 3

62

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

56 4 6 4 5 6 5 4 1 11 5

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

1 1 135) 9 9 3 3 3 3

3 3 3

136) 25 5 5 5 5

5 5

37) 2 3 2 3 6

x x x y y y y

x y

ab

c

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

yxx y x y y x x y x y

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

+ +

+ +

+ +

+ +

=

=

= = = =

= = =

= =

( ) ( ) ( )

( ) ( )

4 4 2 5 7 6

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

2 4 2 3 2 2 2 1 2

6 5 4 3 2

2 2 3 3 2 2 2 3 2 3

1 2 2 1

6

38) 3 2 7 3 2 7

3 2 7

3 2 7

1 139) 2 2 2 2

2 2

2

z x y

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

x x x x x

x x x x x

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

x x y y

+

+ + + +

=

+ = + + + =

= + =

= +

+ = + =

= 1 1 2 3 1 3 2 1

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

3 3 2 5 4 3

12 2

2

2 2

2 2

x x y y x x y y

x y x y x y

x y x y x y

+ + + + + +

+ =

= + =

= +

z

14M.I.M-Sraga centar za poduku

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9/32

15


4

( )

9 5 9 5 4 4

7 5 7 5 2

5 3 5 3 2

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

U slijedeim zadacima koristimo pravilo:

Pazi: drugi eksponent se

. :

40) 2 : 2 2 2 2 2 2 2 2 16

41) 13 : 13 13 13 169

42) 3 : 3 3 3

43) 5 : 5 5 5 5 5

n m n m

x x x x x

m mm m m m m m m

a a a

+ +

=

= = = = =

= = =

= =

= = = =

3 2 2 3 2 2 3 2 2 2 4

mora staviti u zagradu...vrlo esto radite ovakvu greku:

to nije tono!

Ovdje je pogreka u predznaku kod zadnjeg lana, jer prvom promjenite predznak,

a drugom

5 : 5 5 5 5m m m m m m m = = =

4 2 4 2 2

6 2 6 2 4

7 2

ne, to se dogaa zbog toga to radite napamet... tj. preskaete korake...

Preporuka: im imamo vielane eksponente koristite zagrade i ne preskaite korake...

44) :

45) :

46) : :

x x x x

x x x x

x x x

= =

= =3 7 2 3 2

7 3 4 7 3 4 6

2 1 2 1 2 1

1 1 2 2 1 4 1 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

47) :

148) : :

49) :

50) :

51) : :

52) :x x x x

x x

x 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 x

a a a a

+

= =

= =

= = = =

= = = =

= = = =

= = = =

= =8 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 32 1

12 2

il i

U slijedeim zadacima koristimo pravilo:

53) : :

154) : : : :

55) :

:

56) :

57) :

58) :

x

nn m n m

m

a a a a a

a a a a a a a aa

a a a a a

aa a a

a

xx x x x

x

xx x x x

x

xx x x x

x

+

= =

= = = =

= =

= =

= = =

= = =

= = =3 2 1

2 2

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

1

3

59) :

x

xx x x x x x

x

=

= = = = =


10/32


4.

( )

( )

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

2 2

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

3 5

8 71 8

3 2

U slijedeim zadacima koristimo pravilo: :

60) :

61) :

62) :

nn m n m

m

mm mm m m m m m m

m

mm mm m m m m m

m

mm m

m

aa a a

a

xx x x x x x

x

xx x x x x xx

xx x

x

++ ++ + + +

+

++ ++ + + +

+

+

+

= =

= = = = =

= = = = =

= ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( )

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

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

2 3

9 2 7 11 3 10 4

7 4 7 4 3

2 3

: :

63) : : :

64) :

65) 2 3

m m mm m m m m m m m m

mm m mm m m m m m m

m

m m m

x y

x x x x x x

aa a a a a a

a

a a

x y x y x y x y

x y

+ + + + + + + + +

++ ++ + + + +

+ +

+

= = = =

= = = =

= =

= =

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( )

( )

( ) ( )

2 3 2 3 2 3 2 3 2 3 0

6 4 6 4 22 2 2 2

3 3 3 3

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

2 4 4 4 2 5

: 2 3 2 3 2 3 2 3 1

66) :

67) 2 2 : 2 2 2

2 2

x y x y x y x y x y

m m m m m m m m m

m m m m

x y x y x y x y

a b a b a b a b

c c c c

x y x y x y x y x y

x y x y

+ + + +

+ + +

+ +

= = = =

= =

+ + + = + = + =

= + = +

( )

2

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

3 3 3 3 3

3 4 4 2 3 12 2 3

3 3 2

U ovom sluaju moete ostaviti i ljevo rjee

il i

68) :

x x x x x x x x x

x x x

a b a b a b a b a b

c c c c c

a b a b c

c c a b

+ + +

+ +

= = =

= = =

( ) ( )

nje...jer jo nismo obradi li pra vilo :"na minus prvu "

2 4 2 2 4 2 2 2 4 2

2 4 2 2 2 4 2

Mnoimo svaki lan prve zagrade sa svakim lanom druge zagrade...

69) 1 1 1 1 1 1 1

1

x x x x x x x x x x

x x x x x+ +

+ + = + + =

= + + =

( ) ( ) ( ) ( )

6 4 2 4 2

6 4 4 2 2

6

2 2 2 1 2 2 2 2

2 1 2 2 2 2

3 4 2 2

4 3

1

1

1

70) 1 1 1 1 1 1 1

1

1

1

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

+ +

= + + =

= + + =

=

+ + = + + + + + =

= + + + =

= + + + =

= + + +


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11/32

17


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

Koristimo pravila:

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

vio nam se jedan problem a taj je

2.

:

ogledajmo opet u zada kako smok tt oa

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

n

ab a b abc a b c a a a a

x

= = = =

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

( ) ( ) ( ) ( ) ( )

( )

( )

( ) ( ) ( ) ( )

na eksponent

na eksponent

2

parni

3

neparni

4

3

Imamo neparan broj "minusa" pa e i umnoak bi

1 1Zakljuak:

1 1

tamo rjeili:

. 4) 1 1 1 1

5) 1 1 1 1 1

6) 1 1 1 1 1 1

7) 2 2 2 2 8

2

=

=

= =

= =

= =

= =

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

4 4

Imamo paran broj "minusa" paje produkt p ozi tivan bro j

53

53 puta bi to trebali na

ti negativno tj. imati e predznak minus

8)

vlja ovakav problem: ...

x x x x x x

x x x x

= =

=

( )

( )

pisatia to je previe pisanjapa emo rae gledati eksponent

53 53

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

u eksponentu je 53 ,

an broj pa je:

?

x x

x

=

=

( )

( )

( )

( )

( )

( )

na eksponent

na eksponent

54 54

55 55

56 56

2parni 2parn u

2 1neparni neparnu

pa je:

Zakljuak:

jer je eksponent 55, a 55 je neparan broj

jer je eksponent 56, a 56 je paran broj

ili

n n

n

x x

x x

x x

x x

x x

x x

x +

=

=

=

= =

= =

=

( )

( )

2 1

4 4

3 3

2 je oznaka za parn i b ro j

2 1 je oznaka za neparan broj

Sada to primjenimo u zadatcima:

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

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

Po

2

n

n

n

x

x x

+

+

=

= =

= = =

( ) ( ) ( )

( ) ( )

( )

( ) ( )

( )

( ) ( ) ( )

3 3 3 3

3 3 3 3

Svaki negativan brojda se zapisati u obliku:

1

4 4 4 4 4 4II na

stoji i drugi nain rjeavanja ovakvih zadataka:

II nain

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

=

= = = =

= = = =

= = = =

in

5.

Poja

P

Sada se ja

53 je nepar

dal

( )


12/32

18



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

) ( )) ( )

) ( ) ( )

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


n n n


ab a b

x x x x

x x x x

x y x y x y x y

ab

= = = =

=

= = =

= = =

= = =

( ) ( )

) ( )

) ( ) ( )

) ( )

) ( ) ( )

3 3 3

32 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 treem pravilu...

I

2 2 2 843 3 273

1 1 1 15

2 2 162

6

7

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

xy x y x y x y

x y x y x y x y

x x x

x x x x

= =

= = =

= = =

= =

= = =

) ( ) ( ) ( ) ( )

) ( )

) ( )

( )

) ( ) ( ) ( )

) ( ) ( ) ( )

) ( ) ( )

2 2 222 2 2 2 2 4

32 2 3 6

52 2 5 10

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

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

4 43 4 4 3

nain

II nain :

Prema pravilu:7 1 1 1

8

9

10 2 2 4 4

11 2 2 8 8

12 2 2

mn na a

x x x x x

x x x

x x x

x y x y x y x y

x y x y x y x y

x y x

= = = = =

= == =

= = =

= = =

= ( )

) ( ) ( )

) ( ) ( ) ( ) ( )

44 3 4 4 4 12 16

3 32 2 2 3 6

3 3 3 332 2 2 2 2 2 2 3 6II nain :

16 16

13

13 1 1 1

y x y x y

y y y y

y y y y y y y

= =

= = =

= = = = = =

5.

m


13/32

19


5.

( )

( )

( ) ( )

na eksponent

na eksponent

parni par nu

neparni neparnu

3 32 2 2 3 6

Koristimo pravilo: prilikom raunanja nainom u

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

to je bio I nain rjeavanja

I

14)

II nain

1

x x

x x

y y y y

=

=

= = =

) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

) ( ) ( ) ( ) ( )

( ) ( )

3 3 332 2 2 2 3 6 6

3

2

2 2 223 3 3 3 2 6 6

2 23 3 3 2 6

Prema pravilu:

Prema pravilu:

to je bio II nain rjeavanja

to je bio I

4 1 1 1 1

1 1 1 1

1 1 1 1 1

15 1 1 1 1

15)

neparnu

par nu

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

nain rjeavanja


II nain


16

16 1 1 1

17

1

par nu par nu

y 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 nain

U 18) pokazat emo tri naina rjeavanja:


7 1 1 1

18

18 1

1 1

x x x x x x

x x x x x x

x x

x

= = = = =

= + = = = =

= =

= = = ( ) ( ) ( )

) ( ) ( ) [ ]

2 225 5 5 2 10

2 225 5 5 5 2 10

II nain

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

U ovom 18) zadatku najbri ne III nain rjeavanja... dok je u 19) definitivno II nain najb

1 1 1

18

x x x x

x x x x x

= = =

= + = = =

olji....


14/32

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5.

) ( ) ( ) ( ) ( ) ( )

( )

) ( ) ( )( ) ( )

2 22 2 2 2 4 4

5 5 5 5 5 5 4 20

2 22 2 25 5

I nain

I nain u ovom zadatku pomalo zbunju je pa je bolje to rjeavati na II nain izluivan jem 1 :

II nain :

1 1

x x x x x x x

x x x

= + = = = = =

= =

( )( )( )

( ) ( ) ( )

) ( ) ( ) ( )

) ( ) ( )

) ( ) ( ) ( )

) ( ) ( ) ( )

( ) ( )

22

5

25 2

210

22 210 10 10 2 20

25 5 2 10 10

3 32 3 3 2 3 2 3 3 6

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

2 223 3 3 2 6

2 2

1

1

1 1 1

2 2 8 4

2 2 4 4

2 4 jer je: 2

x

x

x x x x

x x x x

ab a b a b a b

a b a b a b a b

y y y y

=

= =

= =

= = = =

= = =

= = =

= = =

= = =

= = ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )

( )

na eksponent

na eksponent

2 2 2

3 3 3 3 3

4 4 4 4

2

3

parni parnu

neparni neparnuoristimo pravilo:

1 2 1 2 1 4 4

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

2 1 2 1 2 1 16 16 itd.

ili ovako: 2 2 2 4

2 2 2 2 8

ili kx x

x x

=

=

= = =

= = = = =

= = = =

= =

= =

( )

( )

( )

) ( )

( ) ( )

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

2 2

3 3

3 3

3 332 2 3 2 3 6

3 3 3 33

pa je :

il i

2 2 4

2 2 8

3 3 27

24 3 3 3 27

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

y y y y

= =

= =

= =

= = =

= = = = = =

19

19

20

21

22

23

27


15/32


5.

25

25

25

26

26

27

) ( )

) ( )

) ( ) ( )

2 2 2 22

2 2 2 2 2 4

2

2 2 22

2 2 2 2 4

2

2 2 2 2222 2 2 2 2 4

2

25)zadatak na tri nai

2 2 2 2 4

3 3 3 93

2 2 2 4

3 3 932 2 2 2 4

1 1 13 3 3 93

x x x x x

x x x x

x x x x x

= = = =

= = + =

= = = =

) ( )

) ( )

) ( )

) ( )

) ( ) ( ) ( )

) ( ) ( ) ( )

3 3 3 33

2 2 2 2 3 6

3

3 3 33

2 2 2 3 6

3

22 2

33 3

4 4 44 4 4 4

na

I nain

II nain

2 2 2 2 8

3 3 3 273

2 2 2 8

3 3 273

28

29

30

m m m

n n n

m n m n m n m n

m m mm n m n

x x x x x

x x x x

x x x

y y y

x y x y x y x y

x y x y

= = = =

= = =

= =

= =

= = =

= =

) ( ) ( ) ( )

) ( ) ( ) ( )

) ( ) ( )

) ( ) ( )

) ( )

) ( ) ( ) ( ) ( ) ( )

2

2

2

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

3 2 3 2 3 2 3 2

22 2 2

33 3 3

2 2 22 2 2 2 2 2

31

32

33 2 2 2 2 4

34 3 3 3 3 27

35 2 2 2

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

m m n m m n m

x y x y x y x y

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

xx x x x

xx x x x

xx x x x

m nm n m n m n m n m

x y x y

a b a b a b a b

a b a b a b a b

=

= = =

= = =

= = = =

= = = =

= =

= = = = = 9 4 9n m n =

21


16/32


5.

37

38

39

) ( )( ) ( )( )

) ( )( )

) ( )( )

) ( )( )

) ( )( ) ( )

) ( ) ( )( )

) ( )( ) ( )

43

2 2 3 4 24

54

3 3 4 5 60

42 2 4 8

3 3 3

26 5

2 3 2 6 2 3 5 24 15 24 15 39

24 3

3 5 3 4 5 3 2 12 30 12 30 42

35 7

2 4 2 5 3 4 7

Prema pravilu:

40

41

42

43 : :

zm

n n m z

xx x

yx

x y xy

x x x a a

y y y

x x x

y y y

x x x x x x x x

y y y y y y y y

x x x x

+

+

= = =

= =

= =

= =

= = = =

= = = =

=

) ( )( ) ( )( )

) ( ) ( ) ( ) ( )

( )

( )

30 28 30 28 2

8 56 4

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

3 32 3 3 2

2 3 2 2 3 2 2 2

32 3 3 3 2 2 2

3

6 9 2 4

6 2 9 4

6 2 9 4

:

44 : : :

2 245 : 2 : 2

3 3

2: 4

3

8: 427

8:4 :

27

2

27

x x x x

a a a a a a a a

a b ab a b a b

a b a b

a b a b

a a b b

a b

= = =

= = = =

= =

= =

= =

= =

= =

4 52

27a b=


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17/32


23

( ) ( ) ( ) ( )( )

) ( ) ( ) ( ) ( ) ( )

( )

( )

3 33 3 3 3 3

2 3 3 2 2 3 3 3 2

32 3 3 3 3 3 2 3

3

6 9 9 6

6 9 9 6

6 9 9 6

Koristimo pravila:

1 14 4

2 2

164

2

164

8

164

8

64

8

zm mn nn n n n n n n m n n m z ab a b abc a b c a a a a

x y x y x y x y

x y x y

x y x y

x x y y

x y

+ +

= = =

= =

= =

= =

= =

= =

=

) ( ) ( ) ( ) ( ) ( )

( )

( )

) ( )

15 15

15 15

3 33 3 3 3 3

2 3 3 2 2 3 3 3 2

32 3 3 3 3 3 2 3

3

6 9 9 6

6 9 9 6

6 9 9 6

3 3

3

3

3

3

2

8

8

1 1: 4 : 4

2 2

1: 64

2

1: 64

8

1:64 : :8

1

8 64

1

512

1 1

512

512

48 3

x y

x y

x y x y x y x y

x y x y

x y x y

x x y y

x y

x y

yx

y

x

a b

=

=

= =

= =

= =

= =

= =

= =

= =

=

( ) ( ) ( ) ( )

( )

( )

) ( ) ( )

2 2 23 2 3

2 2 3 2

2 2 6

2 2 6

2 2 6

4 7

1 3 13 3 3

3

3

3

3

3

3

49

x xx x x x x x

x x x x x

x x x x x

x x x x x

x x x x x

x x x

m m m

a b a b a b

a b a b

a b a b

a a b b

a b

a b

x x x

+ +

+ + +

= =

= =

= =

= =

= =

=

= =

5. =

46

47


18/32


5.

49

50

51

52

53

( ) ( ) ( ) ( )( )

) ( ) ( )

) ( ) ( ) ( ) ( )

) ( ) ( ) ( ) ( )

1 3 13 3 3

2 1 1 3 2 1 2 13 2

6 3 2 2

6 3 2 2

6 2 3 2

8 5

2 1 1 3 2 1 2 15 3

6 3 2 2

6

Koristimo pravila:

: :

:


m m m

m m m m

m m

m m

m m

m

m m m m

m m

m


x x x

x x x x

x x

x

x

x

x x x x

x x

x

+ + +

+

+

+

= = =

= =

= =

= =

= =

= =

=

= =

= =

= ( )

) ( ) ( )

( )

) ( ) ( ) ( )

( )

3 2 2

6 3 2 2

6 2 3 2

4 1

3 22 3 2 3 3 2

6 6

6

6

4 6 33 2 4 3 4 2 6 4 3

12 12 12

12

12

12

2 3 2 3

2 3

2 3

5

3 4 3 4

3 4

1 3 4

2

2

m

m m

m m

m

x

x

x

x x x x

x x

x

x

a a a a a a

a a a

a

a

a

+

+

=

= =

= =

=

+ = + =

= + =

= + =

=

+ = + =

= + =

= + =

= =

=

=


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19/32


6. ( )

( ) ( )

( ) ( )22 2 4 4 4

2 4

3 3

Koristimo pravilo:

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

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

5) 6) 1 1

3 8 3 87)

4 3 4 3

nn n n

x xx x x x x x

a ya a a y y y

a b c abc

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

=

= = = =

= = = =

= = = = =

=

33

3

2 2 22 2

2 2

2

2 2

3 33

moemo ostaviti u ovom obliku

ili dalje:

nakon kraenja

4 232 8

4 3

2 24 4 28)

2 2 2

2 4

9)

x yxy xy

x x y xx y x y

x x

xy x xy x

z yz z yz

= = =

= = = =

= =

= =

( )( )

( )

( )

( )

( )

333 22 2 3 6

2 3 2 3 62

1 m+11

111 2 2 12 2 2

2 1 2 22 12

22 3 6

4 3 2

nakon kraenja10)

11)

mm

mmm m m

m mm

xx x x x x

z z z z zz

xy x xy x

z yz z yz

xx x x x x

z z z zzz

x y zz x y

++

+++ + +

+ + +

= = = = =

= = =

= = = = =

( )

( )

22 22 2 1 4 22 3 6

4 3 2 4 2 1 2

22 22 2 2 4

1 2 1 2 21

2 2 1 4 22 3 6 2 3 6

4 3 2 4 3 2 4 2 1 2

nakon kraenja

nakon krae12)

mm m m

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

zz z z

x x xx

x y y z zx y z x y z

z x y z x y z x x y

= = = =

= = = =

= = =

( ) ( )

( )

( ) ( )

( )

( )

( )

( ) ( )

( ) ( )

1 21 2 1 2 2

1 11

5 555 2 2 2

2 2 2 2

5

2

nja

.

1 113)

1 12 2

1 1

1

1 1

1

m mmm m m m

m m m

y zy z y z y z

x xx

x y x x y x y x x y

x x y x x yx xy y x xy y

x xx y x y

x x yx y

x y x x x

x x y x y

=

= = = =

+ + = =

+ + + + + +

+ + = = + +

+=

+ + +

( )

( )

( )

( )

5

55

5

pa krat imo

11

y

x y

xx

x y x y

+= =

= =

+ +

x

25


20/32


7.

2 2

2

2 2 2

2

po pravilu :

pr imjeni li smo pravilo:

Koristimo pravila:

Izraunaj:

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

=

= = =

= = =

= = =

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 7 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 3

mjeoviti broj treba prvo pretvoriti u razlomak...

pr imjeni li smo pravilo: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

22 2 2 2 2

2

22 2 2 2 2

2


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 4910) 1

7 7 7 7 9 9

=

+ + = = = = = =

+ + = = = = = =

81


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21/32

27


33 3 3 3 3

3

2


Koristimo pravila:

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

3 412)

4 3

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

nain 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 x

y y y y yy

=

=

= =

=

= = = = =

( )

( )

2

2 2 2 2 22

2 2

3 3 3

3

3 3 3

3

II nain

nain rastavili smo na:

I nain

.. .

14) 1 1 1

II 1 ...

15)

15) 1 1

y

x x x x x

y y y y y

x x

y y

x x x

y y y

x x x

y y y

=

= = = =

= =

= = =

( )

( )

( )

na eksponent

na eksponent

3 3

3 3

parni parnu

neparni neparnu

4 4 4na parni eksponent parnu

4

4

II nain

nain oristimo pravilo: pa je:

II nain :

1

16) I k

16) jer je:

16)

x x

x x

x x

y y

x x xx x

y y y

x

y

=

=

=

= = =

=

( )

4 4 4 44

4 4

22 2 2 2

2

1 1 1

1 2 11 2 1 3 317) 1

2 2 2 2 2

x x x x

y y y y

= = =

+ + = = = =

7.


22/32


7.

22 2 2 2 2

2

22 2

I nain

II nain :

Koristimo pravila:

2 3 22 6 2 8 8 8 6418) 2

3 3 3 3 3 93

2 3 22 6 2 818) 2

3 3 3 3

n n nn n

n n

a a a b b

b b ab a

= = =

+ + = = = = = = =

+ + = = = =

( )

2 2 2 22

2

33 3 3 3 3

3

33 3 3

I nain

II nain :

8 8 8 61 1 1

3 3 93

2 3 22 6 2 8 8 8 51219) 2

3 3 3 3 3 273

2 3 22 6 2 8 819) 2 1

3 3 3 3 3

= = = =

+ + = = = = = = =

+ + = = = = =

( )

( )( )

( )

( )

3 3 33

3

22 2 2 2

2

4

4 22 2 4 8

3 4 3 4 123

33 3 3

3 3 3 3 9

4 4 3 4 3 124

3

4

I nain

8 8 51 1

3 23

3 5 44 15 4 19 19 36120) 3

5 5 5 5 255

21)

22)

22)

xx x x

y y yy

xx x x x

y y y yy

x

y

= = =

+ + = = = = =

= = =

= = = =

( )

( )

( )

( )

( )

( ) ( )

( )

33 3 3

3 3 3 93

4 3 4 3 124

22 2 33 3 3 2 6

4 4 2 4 2 84

22 2 33 3 3 2 6

2

4 4 2 4 2 84

II nain

I nain

II nain

1 1 1

23)

23) 1 1 1

xx x x

y y yy

xx x x x

y y y yy

xx x x x

y y y yy

= = = =

= = = =

= = = =

12

7

4


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23/32


7.

( ) ( )

( )

( )

( ) ( ) ( )

2 22 2 3 2 2 3 22 3 4 6

4 2 4 2 84

2 22 5 6

6 2 5

26

2 22 5

Koristimo pravila:

po pravilu:

po pravilu:

24)

25)

n n nn n

n n

n n n

n

n

a a a b b

b b ab a

x y x yx y x y

z z zz

x y z a b b

b az x y a

za

x y

= = =

= = =

= = = =

= =

( )

( )

6 2

2 2 5 2

12

4 10

33 3 4 4 34 12

2 3 2 3 3 6 33 2 3

3 32 3 3 3 4

3 4 2

Najk rae objanjenje:

Sve to ima minus u eksponentu

"skae" iz brojnik

2 82 826)

3 27 273

227)

2

mn ma

z

x y

z

x y

x xx x

y z y z y zy z

x y y z

z x

=

= =

=

= = =

= =

( ) ( )

( )

33 4

2

3 33 3 4

32

3 3 4 3 9 12

2 3 6

2 3

3 4

2 3 3 3 4

3 4

a u nazivnik i obrnuto...

Sada dodatna uputa: Razlomak

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

8

8

512 512

2

2

2

y z

x

y z

x

y z y z

x x

x y

z

x y y z

z

= =

= =

= =

= 2 22

3 4 3 4

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

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

: 2 2

x xx

i z i z

29


24/32


7.

( )

( ) ( )

2 24 2 4

3 2 3

22 4 4 2 8

2 2 2 2 3 2 4 62 2 3

2 24 2 4

3 2 3

22 3

4

2

Koristimo pravila:

2 228)

5 5

2 4 4

25 255

2 229)

5 5

5

2

5

n n nn n

n n

a a a b b

b b ab a

x y x

z y z

x x x

y z y zy z

x y x

z y z

y z

x

= = =

= =

= = =

= =

= =

= ( ) ( )

( )

( )

( ) ( )

2 22 3

22 4

2 2 3 2

4 2

4 6

8

33 3 3 4 4 34 2 4 12

3 2 3 3 3 2 3 3 33 2 3

1 1

2 3 2 3 2 3

3 2

2

25

4

25

4

2 82 2 830)

5 5 125 1255

31)

32)

y z

x

y z

x

y z

x

x xx y x x

z y z y z yy z

x y y

y x x

x y y y y y

y x x x x

=

= =

=

= = = =

= =

= =

( )

( )

( )

( )

( ) ( )

2 12 1

3 2 2 1 1

33 3

3

2 22

33 3 3

2 5 3 3 3 9

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

2 32 3 2 333)

2 3 2 3 2 3

134)

1

35)

y yy x x

yx x x x y

xx x

x x x

x yx y

x y

ca b c c c

c a b a b a ba b

= = =

+ + = =

+

= =

= = = =

6 9z


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25/32


7.

( )

( ) ( )

( )

( )

3 23 2 2 222 2 2

3 3 4 3 2 2

3 3 4

2 3 2 2 2

3 3 3 2 4 2

6 2 4

9 6 8

6 2 4

7 2 6 8

4

7 8

7

kratimo

Koristimo pravila:

36) : :

:

:

:

1:

1

n n nn n

n n

a a a b b

b b ab a

a ya y y y

y a a y a a

a y y

y a a

a y y

y a a

a y y

y y a a

y

y a

y

= = =

= =

= =

= =

= =

= =

= 8

4

8 8 8

7 4 7 4 11

2 2 23 2

3 2 2

2 2 2

2

1 2

3 2 1 1 2 1 1 4 1 137) 2 2

2 3 8 4 9 4 2 42 2 3

1 2 1 91 1 1 1 11

9 2 4 18 4 36 36

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

8 4 8 3 8 8 9 8 9 93

2 1 139)

3 3 2

a

y

a a a

y y y y+

=

= = =

+ = + = + = + =

+ = + = + = =

= = = = =

+

( )

3 1 2 3

2 2 22 3 2 32 3 2 3

3 2 2 3

22 2

3 3 2 3 3 3 2 59 8 1

2 1 1 2 2 2 2 2

3 2 3 4 3 2 4 27 240)

4 2 3 3 43 2 3 3

9 316 8 8 8 14 3 12

9 4 27 27 27

= + = + = + = + =

= = =

= = = =

2

8 1 8 1

27 144 27 48612 = =

31


26/32


8. 0 1 1

0 0

0

02 4

po pravilu:

bez obzi ra to je u zagradi 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)

n

na a a a a

a a

a

x

x y

z

= = = =

= =

=

=

( ) ( )

( )

( )

0 00 2 42 4

0

0 0

0

0 0

0 0 0

0

0

im postupkom:

po pravilu:

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

1 1 1 1

1 1

4) 1 1 2

3 1

6) 3 1 1

7) 3 1 zato to je: 3 1 3 1 1 1

8) 3 3 1 3

9) 1

x yx y

z z

x y

a

x

x y

= = = =

+ = + =

=

= =

= = = =

= =

+ =

( )

( ) ( )

( )

0

0 0 0

0 00

0 0

20 2

e jednako jedan !

bez obzi ra to 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)


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27/32

33


8.

( )

( )

( )

0 1 1

0 0

20 2

0

0

1 1

1

1

bez obzi ra t o je u zagradi ako je to na nuultu 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

xx

= = = =

= =

= =

=

=

= =

=

=

( )( )

( )

1 1

1

1

2

2

3

3

2 2 22

2

prvo decimalni broj pretvorimo u razlomak .. .

po pravilu:

po pravilu:

2 1 5

510 5 1

1 1 120)

1 1 121) 3

3 9

122)

3 10 10 10023) 0,3

10 3 3 9

24) 2 5

n

n

n

n

x y ax y ax y

aa

xx

x

= = =

= = =

= = =

=

= = = =

( )

5

5

1

1

2 5

125) 4

4

x

=

=


28/32


8.

( ) ( )

( ) ( )

0 1 1

1

1 1

1

1 1

1 1

ili na drugi nain :

ili na drugi nain :


1 , , ,

11 1 1 4 1 1 4126) 4 4

1 14 1 1 4 14 4

11 4 1 41 1 1 4 1127) 4 4

1 1 1 1 1 14 44 4

2 3 328) pr

3 2 2

n

na a a a a

a a

= = = =

= = = = = =

= = = = = = =

= =

( )

( )

1

1

11

1 1

1

1

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

ili krae:

ema pravilu:

11 2 1 3 1 31:

2 23 2 1 2

3 3

2 2 5 229) 2 5 10 2 5 2 5 10

5 1 5

2 5230) 2

15

n n n

n

a b b

b a a

aa

= =

= = = = =

= = = = = =

= =

( )

2

2

44

4

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

5 10

3 3 231) 3 4 12

2 1

2 232) 2

1

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

a a bab

b

+ + + +

=

= = =

= =

= = = =


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35


8.

( ) ( ) ( ) ( ) ( )

0 1 1

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

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

2 2 2 2 2 2 0


1 , , ,

34) 2 2 2 2 2 2 2 1

135) 5 5 5 5 5

5

36) 1

3

n

n

x x x x x x x x x

x x x x x x

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

a a a a aa a

x y x y x y x y x y

+ + + + + + + + +

+ +

+ +

= = = =

= = = =

= = = =

+ + = + = + = + =

( )

1 1

0 0 0 1

1 1 1

0 1 3

3

2 22 2 2 2 2

3 3 3 3 3

1 37) 27 2 1 1 1 2 1 2 3 6

3 1

1 1 1 1 5 1 1 1 1 538) 2 2 1 1 5

2 5 2 2 1 2 2 8 322

39) :

m n m n n m n m n n

x y

y

a b a b a b a b a b

c c c c c

+ + + + +

= = =

= = =

= =

2 02

3

22 2

23 1 3 1

2 4

2 4

2 2

2

1

1 2 41 1 1 1

2 2 8 2 82 240)1 1 1 1 1 4 12 2

4 16 162 2

1 8 7

7 168 84 1 5 8 5

16 16

m n m n n

a b

c

+ +

= =

= = = = ++ + +

= = = +

( )

2

2 2 2 22

2

2 2 2

2 22 1 2

1 2

2

7 2

5

14 5 5 5 25 251 1 1

5 14 14 196 19614

1 1 1 1 1

2 3 183 4 32 1241)1 1 1 1 1 122 3

2 2 9 183

6 3

6 2

= =

= = = = = =

= = = = =

=

2 2 2 2

2

3 2 2 4

2 3 93

= = = =


30/32


8. 0 1 1

22 2 2

2 22 1 2

1 2

2


1 , , ,

3 41 1 1 1 1

4 3 18 12 3 3 4 32 12

42) 1 1 1 1 9 2 11 11 122 3

2 2 9 2 9 183

3 6

11 2

n

na a a a a

a a

= = = =

= = = = = = + + + +

=

( )

2 2 2 2 22

2

0 0 1

0 0 1

0 1 2

2

2 3

3 22 22 22 484 4841 1 1

6 22 3 3 93

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

3 3 3 3 3

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

3 3 3 3 3

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

4 4 4 42

46) 2 2

= = = = = =

= = = = =

+ = + = = = =

+ = + = = = =

= 2 3

3 2 1

3 2 1

2 3

2 3

2 2

3

3

2

3ili isti zadatak na drugi nain :

1 2 11 1 1 1 2 1 1

4 8 8 8 82 2

1 1 2 1 41 1 1 1 1 1 1 2 4 347) 2 2 2

8 4 2 8 8 82 2 2

1 1 1 1 8 9 1 148) 3 2

9 8 9 8 72 723 2

1 1

1 272 272 449)1 1 4 1 43

273

2 3

3

= = = =

+ + + = + = + = = =

= = = = =

= = = =

=3

2

3 2 2

2

3 2 3

2 2 2

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

ili isti zadatak na drugi nain :

27

42

1 88

2 3 8 1 1 13 950)1616 9 16 9 2 184

1

2 3 2 8 1 1

16 9 2 9 184 4 3

=

= = = = =

= = = =

9


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37


8. 0 1 1

2 1

2 1

2 1 1

2 1 1

1

1

ili isti zadatak na drugi nain :

ili isti zadatak na drugi na


1 , , ,

1 4 14

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

9

6 6 22 4 4 6 6 2

9 33 6 9 4

1

152)

1 1

n

na a a a a

a a

ba ba

a ab

b

= = =

= = = = =

= = =

= = =

1

1

22 2 2 22

2 2 2 2 2

2

11 11

1 1 1

1

2

222 3 23 3

1 2 2 2 2 2 22

1

in :

ili drugi nain :

ili drugi nain :

ili drugi

11

53)1 1

1

54)1 1 1

155)

1

a b

ab

ba b a ba

b a a b a

b

bb

b c b ca b bc a b bca a

a a ac c

cc

aaa ca b a cb b

c d d b d b d d

c c

=

= = = =

= = = = = =

= = = =

2 12 3 2

1 2 3 2 3 2

22

2 2 2 21 2 21

2 3 1 3 3 2 13

2 1 2 1

2 2 1 2 21 2 2

2 3 1 3 1 2 1

1 2

nain :

ili drugi nain :

1

56)1 1

57)

a ca b a c

c d b d b d

b cb c

b c y d b c y d x bc bd yx x

cxy c d c x c x c ccy d y d

b c y d b c y d x bc bd y

cxy c d c x c c x

x b

= =

= = = = =

= = =

2

22 2 3 2 32 51 2

2 3 1 2 2 2 2 2 2 22

3 1 3

3 1 2 3 21 2 2 5

2 3 1 1 2 2 2 2 2 2ili drugi nain :

1 1

1 1

cc

x b c c d c dc cx b

y c d y x b y b xy b xyy

c d c dc d c c d x b c c d

y c d x y b b xy b xy

+

+

= = = = =

= = =

d

a

=


32/32


9.


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