POTENCIJE
potencije - MIM-Sraga
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**** MLADEN SRAGA ****<br />
2010.<br />
UNIVERZALNA ZBIRKA<br />
POTPUNO RIJEŠENIH ZADATAKA<br />
PRIRUČNIK ZA SAMOSTALNO UČENJE<br />
<strong>POTENCIJE</strong><br />
α
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
Autor:<br />
MLADEN SRAGA<br />
Grafički urednik:<br />
Mladen Sraga<br />
BESPLATNA - WEB-VARIJANTA<br />
Tisak:<br />
M.I.M.-SRAGA d.o.o.<br />
CIP-Katalogizacija u publikaciji Nacionalna i sveučilišna knjižnica, Zagreb<br />
© M.I.M-Sraga d.o.o. 1992./2012.<br />
Potpunu garanciju na kompletnu zbirku daje: centar za dopisnu poduku M.I.M.-SRAGA -<br />
dakle sve što vam se čini nejasno krivo ili sumnjivo - zovite 01-4578-431 ili 01-4579-130<br />
i tražite dodatne upute i objašnjenja ...<br />
Dodatne upute i objašnjenja možete zatražiti i na mail: mim-sraga@zg.htnet.hr<br />
Ovo je jako skraćena varijanta naše zbirke …<br />
M.I.M.-SRAGA d.o.o. zadržava sva prava na reproduciranje , umnažanje , prodaju ove<br />
zbirke potpuno riješenih zadataka isključivo u okviru svog programa poduke i dopisne<br />
poduke.<br />
Nikakva komercijalna upotreba ove zbirke nije dozvoljena bez pismene dozvole nakladnika !<br />
www.mim-sraga.com 2 © M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
Samo zadaci su zadani od 3. do 12. strane …dio rješenja je<br />
od 16. strane na dalje …<br />
Sva rješenja ovih zadataka šaljemo besplatno<br />
u obliku PDF dokumenta mailom<br />
dovoljno je da nam pošaljete mail na<br />
našu adresu: mim-sraga@zg.htnet.hr s porukom da trebate<br />
kompletna rješenja zadataka iz potencija …<br />
autor: Mladen Sraga<br />
3
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
2 3<br />
n<br />
21. Koristeći pravila: aa ⋅ = a , aaa ⋅ ⋅ = a , aaa ⋅ ⋅ ⋅...<br />
⋅ a=<br />
a<br />
<br />
n−puta<br />
Izračunaj:<br />
1) 22222 ⋅ ⋅ ⋅ ⋅<br />
2) xxx ⋅ ⋅<br />
( ) ( ) ( )<br />
3) xyxyxzxyxxz ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 4) xy⋅ xy⋅<br />
xy<br />
5) ( x+ y) ⋅ ( x+ y) ⋅ ( x+ y) ⋅ ( x+ y)<br />
6)<br />
⎛ x⎞ ⎛ x⎞ ⎛ x⎞ ⎛ x⎞<br />
⎜ ⎟⋅⎜ ⎟⋅⎜ ⎟⋅⎜ ⎟<br />
⎝ y⎠ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠<br />
7) ( x− y) ⋅ ( x+ y) ⋅( x− y) ⋅( x− y)<br />
⋅( x+ y)<br />
8)<br />
⎛a−b⎞ ⎛a−b⎞ ⎛a−b⎞<br />
⎜ ⎟⋅⎜ ⎟⋅⎜ ⎟<br />
⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠<br />
22.<br />
n<br />
2 3<br />
Koristeći pravila: , , ...<br />
Izračunaj:<br />
n−puta<br />
( − )<br />
2 3 4<br />
2<br />
1) 5 2) 2 3) 3 4) 1<br />
( − ) ( − ) ( − ) ( −x)<br />
3 4 3 4<br />
5) 1 6) 1 7) 2 8)<br />
2 3 5 2<br />
⎛1⎞ ⎛3⎞ ⎛2⎞ ⎛ 2⎞<br />
9) ⎜ ⎟ 10) ⎜ ⎟ 11) ⎜ ⎟ 12) ⎜−<br />
⎟<br />
⎝3⎠ ⎝5⎠ ⎝3⎠ ⎝ 3⎠<br />
13)<br />
a = a⋅ a a = a⋅a⋅ a a = a⋅a⋅a⋅ ⋅a<br />
<br />
3 4 2 3<br />
⎛ 2 ⎞ ⎛3⎞ ⎛ 4⎞ ⎛ 4⎞<br />
⎜ − ⎟ 14) ⎜ ⎟ 15) ⎜−<br />
⎟ 16) ⎜−<br />
⎟<br />
⎝ 3 ⎠ ⎝4⎠ ⎝ 5⎠ ⎝ 5⎠<br />
4<br />
⎛ 4 ⎞<br />
17) ⎜−<br />
⎟ 18) 0,2 19) ( −0,2)<br />
20) 0,2<br />
⎝ 5 ⎠<br />
( − ) ( − ) ( − ) ) ( − )<br />
( − ) + ( − ) + ( − ) + ( − ) ( − ) + ( − ) + ( − )<br />
( ) ( ) ( ) ( )<br />
2 2<br />
3<br />
3 2 3 4<br />
21) 0,2 22) 2,5 23) 2,5 24 2,5<br />
2 3 4 5 20 30 45<br />
25) 1 1 1 1 26) 1 1 1<br />
1 2 3 4 5 2 4 5<br />
27) − 2 + − 2 + − 2 + −2 28) 2 −3 29) 3 −2<br />
3 5 ⎛1<br />
⎞<br />
30) ( − 2) + ( −2)<br />
31) ⎜ ⎟<br />
⎝4<br />
⎠<br />
2 3<br />
1<br />
2 3<br />
2<br />
⎛ ⎞ ⎡ ⎤<br />
( ) ( )<br />
+ ⎜− ⎟ 32) − 0,2 + −0,2<br />
⎝ 2 ⎠ ⎣ ⎦<br />
n n n n n n<br />
23. Koristeći pravila: ca ⋅ + da ⋅ = c+ d ⋅a ca ⋅ −da ⋅ = c−d ⋅a<br />
( ) ( )<br />
1) 2x+ 3x 2) x+ 2x+ 4x 3) 7a−2a<br />
4) 9y− 2y+ 3y− y 5) 2x+ 3a− x+ 5a+ 7x−2a<br />
6) 2xy + 3xy + xy 7) 2ab − 4ab + ab<br />
2 2 2 2 2<br />
8) 4 2 9 9) 3 4 5 2 8<br />
2<br />
xy + xy − xy y + x y − z − x y − y + z −3y+<br />
8x y<br />
( ) ( ) ( )<br />
( ) ( )<br />
10) 7 2 4 11) 2 3<br />
3 3 3 2 2<br />
xy − xy + xy x + y − x + y<br />
2 3 2 3 2 3<br />
12) 3 + + 4 + − + 13) 5 + 2 − 2 + 7 + 3 −3<br />
x y x y x y x y z x y z x y z<br />
2 2<br />
( x − x+ ) − ( x − x+ ) ( x+ y− ) − ( x+ y− ) + ( x+ y−<br />
)<br />
14) 3 3 5 4 5 1 15) 3 1 6 1 1<br />
www.mim-sraga.com 4<br />
© M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
Koristeći pravila: a ⋅ a = a a : a = a<br />
n<br />
a<br />
m<br />
a<br />
= a : a = a<br />
Izračunaj:<br />
24.<br />
1) 2 ⋅2 2) 5 ⋅5 3) 3 ⋅3 4) 7 ⋅7<br />
5 3 2 7 x 2x 2m−5 m−2<br />
5) 2 ⋅2 ⋅2 ⋅2 6) 3⋅a ⋅2⋅a 7) x ⋅x 8) x⋅x ⋅x<br />
2 6 4 8 4 7 2 3 3 5<br />
2 5<br />
2 4 6 3 2 2<br />
2 2+ a a−2 3 3 6<br />
9) x ⋅x ⋅x 10) x ⋅x ⋅x 11) x ⋅x ⋅x ⋅x 12) a ⋅a<br />
4 2<br />
13) 1<br />
a ⋅a ⋅a<br />
n m n+ m n m n−m n m n−m<br />
4) a ⋅a ⋅a ⋅a 15) a ⋅a ⋅a 16) a ⋅a ⋅a<br />
2 5 7 3 3x x+ 2 2x x+ y 2x x+<br />
3y<br />
17) −2⋅x⋅6⋅x ⋅3⋅x 18) 2⋅x ⋅ x + 3⋅x ⋅y ⋅x⋅y−4⋅x⋅ x + 2⋅x ⋅y ⋅y<br />
3 4 2 5 2 3 6 3 2 2<br />
2 3 4<br />
⎛1⎞ ⎛1⎞ ⎛1⎞<br />
19) ⎜ ⎟ ⋅⎜ ⎟ ⋅⎜ ⎟<br />
20) x ⋅x 21) x ⋅x<br />
⎝2⎠ ⎝2⎠ ⎝2⎠<br />
2m+ 1 3m+ 2 2m− 4 3m+<br />
2<br />
3<br />
m+<br />
1<br />
2 2m− 7 m+ n 2m− n 3m+<br />
2n<br />
22) x ⋅x 23) x ⋅x ⋅x<br />
24)<br />
25) a ⋅a 26) 2a ⋅3a 27) 2a ⋅5a<br />
( x+ y− ) ⋅ ( x+ y− ) ⋅ ( x+ y−<br />
)<br />
2m n m 2n<br />
( x −y ) ⋅ ( x + y )<br />
m−<br />
( x+ y) ⋅( x− y) ⋅ ( x+ y) ⋅( x−<br />
y)<br />
2m−n 3m−n 2m−2n<br />
2m+ 3n 3m−5n m− 3n+ 1 4m+ n− 7 x+ y 2x+<br />
y<br />
2 9<br />
3 4<br />
( ) ( ) ( ) ( )<br />
3m+ 2n 2m−4n<br />
2 3 4<br />
28) a ⋅ a 29) x− y ⋅ x− y 30) x+ y ⋅ x+<br />
y<br />
2<br />
m+ 1 2m+ 2 3m−4<br />
31) 1 1 1<br />
32)<br />
33)<br />
34)<br />
⎛<br />
⎜<br />
⎝<br />
ab<br />
c<br />
⎞<br />
⎟<br />
⎠<br />
2 3 2 1 m−3<br />
3x−2y 2<br />
4x−y 2<br />
x−y 2<br />
3y−2x<br />
⎛ab ⎞ ⎛ab ⎞ ⎛ab<br />
⎞<br />
⋅⎜ ⎟ ⋅⎜ ⎟ ⋅⎜ ⎟<br />
⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠<br />
x ⋅x ⋅x<br />
NOVO:<br />
MALA ŠKOLA MATEMATIKE –1 na<br />
DODATNE video UPUTE<br />
UZ OVE ZADATKE<br />
-<br />
<strong>POTENCIJE</strong><br />
ALGEBARSKI IZRAZI<br />
ALGEBARSKI RAZLOMCI<br />
link: http://www.mim-sraga.com/Mala-skola-matematike--video.htm<br />
autor: Mladen Sraga<br />
5
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
25.<br />
Pomnoži:<br />
1)<br />
2xy⋅3xy 2) 5xy⋅2xy<br />
2 2 3 3 2 5 3<br />
3)<br />
2 2 3 9 4 5 6 2 ⎛ 9 2 ⎞<br />
ab⋅ ab 4) − ab ⋅ ⎜ − ab ⎟<br />
3 4 27 ⎝ 5 ⎠<br />
5)<br />
1 2 3 4 3 2 5 4 2 1 2<br />
abc⋅−<br />
( 4abc)<br />
6) 9xy⋅<br />
xy<br />
2 3<br />
7)<br />
5<br />
6 4 yx<br />
2 3 4 3 4 2<br />
25xy⋅<br />
8) 2xyz⋅( −3xyz)<br />
5<br />
9) ⋅( − − 3 + 2 −7) 10) ( −2<br />
⎛ 1<br />
) ⋅⎜<br />
−<br />
⎝ 2<br />
+<br />
2 4 3 2 2 2 3 3<br />
x x x x x xy x y xy x y<br />
⎞<br />
⎟<br />
⎠<br />
Pogledaj dodatnu VIDEO uputu uz ovaj zadatak<br />
http://www.mim-sraga.com/potencije.htm<br />
ili<br />
direktni link na : video uputu …uz 25. zadatak<br />
26.<br />
Izračunaj:<br />
1) 2 : 2 2) 13 : 13 3) 3 : 3 4) 5 : 5<br />
9 5 7 5 5x 3x 3m−2 m−2<br />
5) x : x 6) x : x 7) x : x : x 8) x ⋅ x : x<br />
4 2 6 2 7 2 3 7 3 4<br />
1 7 1 2 1<br />
2 2 2 9 3 5 2 2<br />
9) x: x 10) x : x 11) x : x 12) x : x : x<br />
13) a : a 14) a : a : a 15) a: a : a 16) a : a ⋅ a<br />
4x<br />
2x<br />
8 3 2 2 3 7 2 3<br />
3 7<br />
5 8 2 4<br />
17)<br />
x x x x<br />
18) 19) 20)<br />
2 3 1<br />
x x x<br />
3<br />
x<br />
21)<br />
m+<br />
1 3m+ 7 8m+ 7 9m+<br />
11<br />
x<br />
x x m<br />
a<br />
22) 23) : x 24) : a<br />
2m+ 2 3m+ 5 3m+ 2 2m−3<br />
x x x a<br />
− 1 7m+<br />
10<br />
2<br />
x−2 2<br />
3x−3 2<br />
4x−4<br />
2<br />
6<br />
2<br />
4<br />
7 4 2x+ 3y 2x+<br />
3y<br />
⎛ab⎞ ⎛ab⎞<br />
25) ( x− y) :( x− y) 26) ( 2x−3 y) :( 2x−3y)<br />
27) ⎜ :<br />
3 ⎟ ⎜ 3 ⎟<br />
⎝ c ⎠ ⎝ c ⎠<br />
2m−4 4m−2 m−4<br />
⎛ab⎞ ⎛ab⎞ ⎛ab⎞<br />
28) ( 2x+ y) ⋅ ( 2 x+ y) :( 2x+ y)<br />
29) ⎜ :<br />
3 ⎟ ⋅⎜ 3 ⎟ ⎜ 3 ⎟<br />
⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠<br />
( ) ( ) ( ) ( )<br />
30) x 2 −1 ⋅ x 4 + x 2 + 1 31) x 2 + 1 ⋅ x− x<br />
2 + 1<br />
Pogledaj dodatnu VIDEO uputu uz ovaj zadatak<br />
http://www.mim-sraga.com/potencije.htm<br />
ili<br />
direktni link na : video uputu …uz 26. zadatak<br />
www.mim-sraga.com 6 © M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
27.<br />
Koristeći pravila:<br />
Izračunaj:<br />
( )<br />
n<br />
n<br />
m<br />
( ) ( ) ( ) ( )<br />
2 3<br />
2 2 ⎛3 2 3⎞ ⎛2<br />
2⎞<br />
) ( ) ) ( ) ) )<br />
1 2x 2 3x 3 ⎜ x y ⎟ 4 ⎜ xy ⎟<br />
⎝4 ⎠ ⎝3<br />
⎠<br />
4<br />
1<br />
5 ⎜ ⎟ 6 7 8<br />
⎝2<br />
⎠<br />
⎛ 2 3⎞<br />
2 2 2<br />
) x y ) ( x ) ) ( −x ) ) ( x )<br />
2 3 4 3 4 3 4<br />
) ( x ) ) ( x y ) ) ( x y ) ) ( x y )<br />
2<br />
) ⎡<br />
2 3 4<br />
( −y) ⎤<br />
) ( −y ) ) ( −y ) ) −( −y<br />
)<br />
2<br />
5<br />
) −( −x<br />
) )<br />
2 2 3<br />
5 2 3 4<br />
9 10 2 11 2 12 2<br />
3 3 2 3<br />
13 14 15 16<br />
⎣ ⎦<br />
17 18<br />
ab = a b abc = a b c a = a a = a<br />
n n n n n n n⋅m n m z n⋅m⋅z<br />
2<br />
5 5 5<br />
( ) ) ( ) ) ( x )<br />
⎡− −x<br />
⎤ 19 ⎡ x ⎤<br />
⎣ ⎦<br />
− − 20 −<br />
⎢⎣<br />
⎥⎦<br />
2 2 3 3 2<br />
) ( ab ) ) ( a b ) ) ( − y ) ) ( − y )<br />
⎛ 2⎞ ⎛ 2⎞<br />
m<br />
n<br />
) − x ) − x ) ( x ) ) ( y )<br />
4 m<br />
2<br />
m n m n<br />
) ( ) ) ( ) ) ( 3 x 2 y) ) ( 3 x 2 y<br />
x y x y a b a b )<br />
2 3 x<br />
2<br />
x x x m n<br />
) ( ) ) ( ) ) ( ) ) ( )<br />
3<br />
4<br />
4<br />
5<br />
x<br />
4<br />
x<br />
y<br />
2 3 2 3<br />
( x ) ) (( y ) ) ) (( x ) ) ) (( y ) )<br />
6<br />
2<br />
5 4 3<br />
2<br />
5<br />
3<br />
7<br />
2 3 3 5 2 4<br />
3 2<br />
( x ) ⋅( x ) ) ( y ) ⋅ (( y ) ) ) (( x ) ) ( x ) 44 ) (( a ) ) :(( a ) )<br />
) ( )<br />
) ( )<br />
2 2 2<br />
3 4 2 3<br />
21 2 22 23 2 24 3<br />
2 3<br />
2 2<br />
25 ⎜ ⎟ 26 ⎜ ⎟ 27 28<br />
⎝ 3 ⎠ ⎝ 3 ⎠<br />
2 3<br />
29 30 31 32<br />
33 2 34 3 35 2 36 2 3<br />
37 38 39 40<br />
41 42 43 :<br />
⎛ 2 3⎞ 2 ⎛ 2 3⎞ 3 2 ⎛ 2 3⎞<br />
3 2<br />
) ab ( ab) ) xy ⋅( xy) ) xy ( xy)<br />
x 2 m+ 1 2m−1 m−1<br />
2 x 3x<br />
3 3 2<br />
) ( ab) ⋅( ab ) ) ( x) ) ( x) ⋅( x )<br />
m− m+<br />
5 3 2 3 3 2 4<br />
) ( x ) ( x ) ) ( x ) + ( x ) ) ( a ) − ( a ) + ( a )<br />
x<br />
8 5<br />
6 4<br />
3 3 3<br />
2 3 3<br />
2 1 1<br />
45 ⎜ ⎟ : 2 46 ⎜ ⎟ 4 46 ⎜ ⎟ : 4<br />
⎝3 ⎠ ⎝2 ⎠ ⎝2<br />
⎠<br />
48 3 49 50<br />
2 1 1 3 2 4 6 3<br />
51 : 52 2 3 53 3 4<br />
Pogledaj dodatnu VIDEO uputu uz ovaj zadatak<br />
http://www.mim-sraga.com/potencije.htm<br />
ili<br />
direktni link na : video uputu …uz 27. zadatak<br />
autor: Mladen Sraga<br />
7
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
Samo zadaci su zadani od 3. do 12. strane …dio rješenja je<br />
od 15. strane na dalje …<br />
Sva rješenja ovih zadataka šaljemo besplatno<br />
u obliku PDF dokumenta mailom<br />
dovoljno je da nam pošaljete mail na<br />
našu adresu: mim-sraga@zg.htnet.hr s porukom da trebate<br />
kompletna rješenja zadataka iz potencija …<br />
www.mim-sraga.com 8 © M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
28.<br />
Koristeći pravilo:<br />
Izračunaj:<br />
n<br />
( ) ( )<br />
n n n n n<br />
a ⋅ b = a⋅ b ili a b c = abc<br />
1) 2 ⋅5 2) 3 ⋅4 3) 4 ⋅6 4) 2 ⋅4<br />
x x x x a a y y<br />
n<br />
3 4 4 5<br />
5 5 ⎛1⎞ 3 ⎛3⎞ ⎛2⎞ ⎛1⎞<br />
5<br />
5) 3 ⋅2 6) ⎜ ⎟ ⋅9 7) ⎜ ⎟ ⋅⎜ ⎟ 8) ⎜ ⎟ ⋅4<br />
⎝3⎠ ⎝2⎠ ⎝3⎠ ⎝2⎠<br />
5 5 2 2 4<br />
4 3 3<br />
⎛2⎞ ⎛9⎞ ⎛ac ⎞ ⎛b ⎞ ⎛a ⎞ ⎛b<br />
⎞ ⎛3⎞ ⎛8⎞<br />
9) ⎜ ⎟ ⋅⎜ ⎟ 10) ⎜ ⎟ ⋅⎜ ⎟ 11) ⎜ ⎟ ⋅⎜ ⎟ 12) ⎜ ⎟ ⋅⎜ ⎟<br />
⎝3⎠ ⎝2⎠ ⎝ b ⎠ ⎝c⎠ ⎝b⎠ ⎝a⎠ ⎝4⎠ ⎝3⎠<br />
2 2 3 3 m+<br />
1 m+<br />
1<br />
2 3<br />
2<br />
6<br />
2<br />
⎛ xy ⎞ ⎛ 4 ⎞ ⎛ xy ⎞ ⎛ x ⎞ ⎛ xy ⎞ ⎛ x ⎞ ⎛ x y ⎞ ⎛ z ⎞<br />
13) ⎜ ⎟ ⋅⎜ 14) 15) 16)<br />
2 ⎟ ⎜ ⎟ ⋅⎜ ⎟ ⎜ ⎟ ⋅⎜ ⎟ ⎜ ⋅<br />
4 ⎟ ⎜ 3 2 ⎟<br />
⎝ 2 ⎠ ⎝ x y⎠ ⎝ z ⎠ ⎝ yz⎠ ⎝ z ⎠ ⎝ yz⎠<br />
⎝ z ⎠ ⎝ x y ⎠<br />
2 3<br />
m<br />
6<br />
m<br />
5<br />
2<br />
⎛ x y ⎞ ⎛ z ⎞ ⎛ x−<br />
y⎞<br />
⎛ x −1<br />
⎞<br />
17) ⎜ 18)<br />
4 ⎟ ⋅⎜ 3 2 ⎟ ⎜ ⎟ ⋅⎜ 2 2 ⎟<br />
⎝ z ⎠ ⎝ x y ⎠ ⎝ x + 1 ⎠ ⎝ x + 2xy+<br />
y ⎠<br />
5 5<br />
⎛ x+<br />
y⎞<br />
⋅ ⎜ ⎟<br />
⎝ x−<br />
y⎠<br />
Pogledaj dodatnu VIDEO uputu uz ovaj zadatak<br />
http://www.mim-sraga.com/potencije.htm<br />
ili<br />
direktni link na : video uputu …uz 28. zadatak<br />
Novo: MALA ŠKOLA MATEMATIKE –1 na<br />
DODATNE video UPUTE<br />
UZ OVE ZADATKE<br />
-<br />
<strong>POTENCIJE</strong><br />
ALGEBARSKI IZRAZI<br />
ALGEBARSKI RAZLOMCI<br />
link: http://www.mim-sraga.com/Mala-skola-matematike--video.htm<br />
autor: Mladen Sraga<br />
9
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
29.<br />
Koristeći pravila:<br />
Izračunaj:<br />
2 −2 3 4<br />
⎛2⎞ ⎛2⎞ ⎛1⎞ ⎛2⎞<br />
1) ⎜ ⎟ 2) ⎜ ⎟ 3) ⎜ ⎟ 4) ⎜ ⎟<br />
⎝3⎠ ⎝3⎠ ⎝2⎠ ⎝3⎠<br />
−1 −1 −2 −3<br />
⎛5⎞ ⎛ 2⎞ ⎛4⎞ ⎛2⎞<br />
5) ⎜ ⎟ 6) ⎜1 ⎟ 7) ⎜ ⎟ 8) ⎜ ⎟<br />
⎝7⎠ ⎝ 7⎠ ⎝5⎠ ⎝3⎠<br />
−2 −2<br />
n n<br />
−n n<br />
⎛a⎞ a ⎛a⎞ ⎛b⎞<br />
b<br />
⎜ ⎟ =<br />
n ⎜ ⎟ = ⎜ ⎟ =<br />
⎝b⎠ b ⎝b⎠ ⎝a⎠<br />
a<br />
⎛ 1⎞ ⎛ 2⎞ ⎛ 1⎞ ⎛3<br />
⎞<br />
9) ⎜2 ⎟ 10) ⎜1 ⎟ 11) ⎜1<br />
⎟ 12) ⎜ ⎟<br />
⎝ 3⎠ ⎝ 7⎠ ⎝ 2⎠ ⎝4<br />
⎠<br />
n<br />
n<br />
−3 −2<br />
2 2 3 4<br />
⎛ x⎞ ⎛ x⎞ ⎛ x⎞ ⎛ x⎞<br />
13) ⎜ ⎟ 14) ⎜− ⎟ 15) ⎜− ⎟ 16) ⎜−<br />
⎟<br />
⎝ y⎠ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠<br />
2 2 3 2<br />
⎛ 1⎞ ⎛ 2⎞ ⎛ 2⎞ ⎛ 4⎞<br />
17) ⎜1 ⎟ 18) ⎜−2 ⎟ 19) ⎜−2 ⎟ 20) ⎜3<br />
⎟<br />
⎝ 2⎠ ⎝ 3⎠ ⎝ 3⎠ ⎝ 5⎠<br />
2<br />
4<br />
3<br />
3<br />
3<br />
2<br />
2 3<br />
2<br />
⎛ x ⎞ ⎛ x ⎞ ⎛ x ⎞ ⎛ x y ⎞<br />
21) ⎜ 22) 23) 24)<br />
3 ⎟ ⎜−<br />
4 ⎟ ⎜−<br />
4 ⎟ ⎜ 4 ⎟<br />
⎝ y ⎠ ⎝ y ⎠ ⎝ y ⎠ ⎝ z ⎠<br />
25)<br />
⎛ x y<br />
⎜ 6<br />
⎝ z<br />
2 5<br />
⎞<br />
⎟<br />
⎠<br />
−2<br />
4<br />
3<br />
−2 3<br />
3<br />
4 −2<br />
2<br />
⎛ 2x ⎞ ⎛ x y ⎞ ⎛2x y ⎞<br />
26) ⎜ 27) 28)<br />
2 ⎟ ⎜ −3 −4 ⎟ ⎜ 3 ⎟<br />
⎝3yz⎠ ⎝2 z ⎠ ⎝ 5z<br />
⎠<br />
4 −2 −2 4 −2<br />
3 −1 −2 3<br />
⎛2x y ⎞ ⎛2x y ⎞ ⎛ x⎞ ⎛ x⎞ ⎛ y⎞<br />
⎜ ⋅<br />
3 ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟<br />
29) 30) 31) 32)<br />
⎝ 5z<br />
⎠ ⎝ 5z<br />
⎠<br />
⎝ y⎠ ⎝ y⎠<br />
⎝ x⎠<br />
−3 −2<br />
2 5<br />
−3 2<br />
3<br />
2 2<br />
⎛2x−<br />
3⎞<br />
⎛ 1 ⎞<br />
⎛a b ⎞ ⎛a ⎞ ⎛ y ⎞ ⎛ y ⎞<br />
33) ⎜ ⎟ 34) ⎜ ⎟ 35) ⎜ 36) ⋅<br />
3 ⎟ ⎜ 3 ⎟ ⎜ 3 ⎟ : ⎜ 4 ⎟<br />
⎝2x+ 3⎠ ⎝ x−<br />
y⎠ ⎝ c ⎠ ⎝ y ⎠ ⎝a<br />
⎠ ⎝a<br />
⎠<br />
2<br />
−2 −2<br />
−3 −2<br />
⎛3⎞ 1 ⎛3⎞<br />
37) ⎜ ⎟ ⋅ 2 + 2 38) ⋅⎜ ⎟<br />
⎝2⎠ 8 ⎝4⎠<br />
−1 −2 −3 −2 −2<br />
−2<br />
−3<br />
⎛2⎞ ⎛1⎞ ⎛1⎞ ⎡⎛3⎞ 2 ⎤ ⎛3⎞<br />
39) ⎜ ⎟ + ⎜ ⎟ −⎜ ⎟ 40) ⎢⎜ ⎟ ⋅ ⋅<br />
−3<br />
⎥ ⎜ ⎟<br />
⎝3⎠ ⎝3⎠ ⎝2⎠ ⎢⎣⎝4⎠ 3 ⎥⎦<br />
⎝2⎠<br />
Pogledaj dodatnu VIDEO uputu uz ovaj zadatak<br />
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ili<br />
direktni link na : video uputu …uz 29. zadatak<br />
www.mim-sraga.com 10<br />
© M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
30.<br />
0 1 −1<br />
Koristeći pravila: 1 , , ,<br />
Izračunaj:<br />
1 −n<br />
1<br />
a = a = a a = a =<br />
n<br />
a<br />
a<br />
2 4<br />
0<br />
0 0<br />
⎛ x y ⎞<br />
0 0<br />
1) 2 2) x 3) ⎜ ⎟ 4) x + y<br />
⎝ z ⎠<br />
( )<br />
0 0<br />
0 0<br />
5) 3 6) −3 7) −3 8) −3x<br />
0 0 0 0 0 0<br />
( + ) ( − ) − + − ( ) + (<br />
9) x y 10) 2x 7y 11) x 2y 7z 12) x 2y 7z<br />
m<br />
( ) ( ) ( )<br />
0 0 0<br />
2 0 0<br />
13) 14) 15) 16) 4<br />
x ⋅ x x x x<br />
−1 −1 −1<br />
17) 5 18) 19) 0,2 20)<br />
( − )<br />
x x y<br />
−2 −3 −2<br />
21) 3 22) x<br />
23) 0, 3 24) 2x−<br />
5<br />
−1<br />
25) 4 26) 27) 28)<br />
−1 −1<br />
( −5)<br />
( − )<br />
−1 −1 −2 −4<br />
−1<br />
( )<br />
1 1 ⎛2⎞<br />
⎜ ⎟<br />
4 4<br />
⎝3<br />
⎠<br />
2 2 3 2a<br />
29) 30) 31) 32)<br />
5 2<br />
b<br />
33) 3 ⋅3 ⋅3 34) 2 ⋅2 ⋅2 ⋅2 35) 5 ⋅5<br />
5x−4 4x−3 7− 9x 2x+ 1 3 2−3x x−6 2x−3 2−2x<br />
−1<br />
−5<br />
)<br />
0 0<br />
2m−n n−2m<br />
0 0 0 1 ⎛1<br />
⎞<br />
36) ( x+ y) ⋅ ( x+ y)<br />
37) 27 ⋅x ⋅y<br />
⋅2<br />
⋅ ⎜ ⎟<br />
⎝3<br />
⎠<br />
1 −1 2<br />
m+ n<br />
2<br />
m+<br />
2n 2<br />
n<br />
0 −1 ⎛1⎞ ⎛1⎞<br />
−3<br />
⎛ab⎞ ⎛ab⎞ ⎛ab⎞<br />
38) y ⋅2<br />
⋅ ⋅ ⋅2 39) ⎜ :<br />
3 ⎟ ⎜ ⋅<br />
3 ⎟ ⎜ 3 ⎟<br />
⎜ ⎟ ⎜ ⎟<br />
⎝2⎠ ⎝5⎠ ⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠<br />
2<br />
2 1<br />
−2<br />
−3 −1 −<br />
− −<br />
−2 −1<br />
−2<br />
⎛ 2 −2 ⎞ ⎛2 ⋅3<br />
⎞<br />
⎛2 −3<br />
⎞<br />
40) ⎜ 41) 42)<br />
−2 −4 ⎟ ⎜ −1 −2 ⎟<br />
⎜ −1 −2<br />
⎟<br />
⎝2 + 2 ⎠ ⎝2 ⋅ 3 ⎠<br />
⎝2 + 3 ⎠<br />
43) 3 ⋅2 − 3 44) 3 + 2 − 3 45) 2 + 2 −2<br />
0 0 1 0 0 1 0 1 −2<br />
−2 −3 −3<br />
−<br />
46) 2 2 47) 2 2<br />
− − 2 − 1 −<br />
2 48) 3 2 −<br />
+ −2<br />
3<br />
−2<br />
2<br />
2 ⋅3 2 ⋅4<br />
49) 50) 51)<br />
3 4 3 ⋅6<br />
−1<br />
3 −2 2 −1<br />
−3 2 2 −1<br />
−1 −2 −1<br />
a a a b<br />
52) 53) 54)<br />
−1 −2 −1<br />
b b c<br />
ab x bc x b c<br />
55) 56) 57)<br />
c d y c d y c d<br />
2 −3 −1 2 −1 −2 2<br />
−1 2 −2 3 −1 2 −3 −1<br />
autor: Mladen Sraga<br />
11
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
31.<br />
a)<br />
Zapiši u obliku potencija s bazom 2:<br />
2 3 2<br />
( ) ( )<br />
1) 4⋅32⋅16 2) 4⋅8⋅16 3) 2 ⋅4 ⋅8<br />
3<br />
b) Zapiši u obliku potencija s bazom 3:<br />
4 2<br />
( ⋅ ⋅ ) ( ⋅ ⋅ ) ( ⋅ ⋅ )<br />
1) 3 9 27 2) 9 81 3 3) 3 9 27<br />
5 3 2 3<br />
c)<br />
Pojednostavni i rezultati zapiši kao potenciju:<br />
1) 2 + 2 2) 3⋅2<br />
3 3 5<br />
4) 53 ⋅ −33 ⋅ 5) 35 ⋅ + 25 ⋅<br />
2 2 3 3<br />
+ 2 3) 6⋅2<br />
5 4<br />
−2⋅4<br />
4<br />
32.<br />
⎛9 6 5⎞ ⎛3 4 3⎞ ⎛4 7 3⎞ ⎛ 2 5 2⎞<br />
1) ⎜ x y ⎟: ⎜ x y ⎟ 2) ⎜ x y ⎟:<br />
⎜ x y ⎟<br />
⎝2 ⎠ ⎝2 ⎠ ⎝3 ⎠ ⎝15<br />
⎠<br />
⎛ 5 8 3⎞ ⎛ 1 3 3⎞ ⎛25 9 6 5⎞ ⎛5<br />
6 3 4⎞<br />
3) ⎜−<br />
xy⎟: ⎜−<br />
xy⎟ 4) ⎜ abc⎟:<br />
⎜ abc⎟<br />
⎝ 7 ⎠ ⎝ 7 ⎠ ⎝49 ⎠ ⎝7<br />
⎠<br />
33.<br />
16⋅2 4 16<br />
1) 2)<br />
8 32<br />
x+ 1 x− 1 x+<br />
2<br />
x−1 1−x<br />
3)<br />
x<br />
9 ⋅27<br />
x<br />
81<br />
x−1<br />
4)<br />
25 ⋅5<br />
125<br />
x+ 1 x+<br />
5<br />
+ 1 3−x<br />
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© M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
34.<br />
−2 −2 4<br />
2<br />
−2 2<br />
4<br />
2<br />
⎛ x ⎞ ⎛ x ⎞ ⎛ x ⎞ ⎛ x ⎞<br />
1) ⎜ 2)<br />
−3 ⎟ ⋅⎜ 2 ⎟ ⎜ ⋅<br />
−3 ⎟ ⎜ 2 ⎟<br />
⎝ y ⎠ ⎝ y ⎠ ⎝ y ⎠ ⎝ y ⎠<br />
3<br />
3<br />
2<br />
2<br />
3<br />
3<br />
2<br />
−2<br />
⎛ 3x ⎞ ⎛4x ⎞ ⎛ 3x ⎞ ⎛3x<br />
⎞<br />
3) ⎜ 4)<br />
4 ⎟ ⋅⎜ 3 ⎟ ⎜ 4 ⎟ ⋅⎜ 3 ⎟<br />
⎝2y ⎠ ⎝3y ⎠ ⎝2y ⎠ ⎝4y<br />
⎠<br />
7 2<br />
−2 2<br />
2<br />
2 3<br />
4<br />
6 2<br />
⎛ 5ab ⎞ ⎛ 5a ⎞ ⎛ 3xy ⎞ ⎛3xy<br />
5) ⎜− 6)<br />
3 ⎟ ⋅⎜ 3 2 ⎟ ⎜− 4 ⎟ ⋅⎜<br />
2<br />
⎝ 6c ⎠ ⎝3b c ⎠ ⎝ 5z ⎠ ⎝ 5z<br />
⎞<br />
⎟<br />
⎠<br />
−2<br />
35.<br />
1) ( x ) 2) ( 2x<br />
) 3) ( a b )<br />
2 3 4 2 2 5 2<br />
3<br />
( ) ( ) (<br />
2 3 4 2 2 3 2 2 3 2<br />
( ab − c ) ( a − bc )( a + bc )<br />
3 2 3 2 3 2 2<br />
( a + b ) ( x y + xy )<br />
4) 4ab 5) 2x y 6) 3x + y<br />
2 2 2 3 2 3 2<br />
7) 4 8)<br />
9) 2 10) 2 3<br />
35. b)<br />
1) ( x ) 2) ( y ) 3) ( a b )<br />
2 3 4 2 2 3 2<br />
( ab ) ab ⋅ ab ( x + y )<br />
2 3 3 5 2 2 3 3 5 2 3 3 5<br />
( ab −ab) ( ab −ab) ⋅ ( ab + ab)<br />
3 2 2 4 3<br />
( xy + xy)<br />
4) 5) 6)<br />
3 5 2 2 3 3 5 3 4 2<br />
7) 8)<br />
9)<br />
)<br />
autor: Mladen Sraga<br />
13
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
36 . I sada primjenom svih pravila za potencije riješite dopunske zadatke<br />
<strong>POTENCIJE</strong><br />
a<br />
a<br />
n<br />
m<br />
36.<br />
m<br />
(( ) )<br />
( )<br />
ca ⋅ ± da ⋅ = c± d⋅a<br />
n n n<br />
n n<br />
n m n+<br />
m n n n n ⎛a⎞<br />
a<br />
1<br />
a ⋅ a = a ( abc)<br />
= a b c = a = a<br />
n<br />
⎜ ⎟<br />
⎝b<br />
⎠<br />
−n n n<br />
m<br />
n m n−m n n⋅m<br />
⎛a⎞ ⎛b⎞<br />
b<br />
−1<br />
:<br />
( )<br />
n<br />
1<br />
a a = a a = a ⎜ ⎟ = ⎜ ⎟ = a =<br />
⎝b⎠ ⎝a⎠<br />
a<br />
a<br />
z<br />
n m n−m n n⋅m⋅z 0<br />
−n<br />
: 1<br />
= a a = a a = a a = a =<br />
Izračunaj:<br />
1 2<br />
1) 2x ⋅ x + 5x⋅x 2) x ⋅x − x : x 3) a ⋅a<br />
3 5<br />
b<br />
( )<br />
2 3 4 7 2 17 8 3 4 2<br />
( a ⋅a ⋅a ) ( a a ) ( a a )<br />
3 2 2 4<br />
( ) ( ) ( ) ( )<br />
4) 5) : 6) :<br />
2 4 5 3 8 4 2 7 5 6<br />
7) : 8) : 9) 25 :5<br />
2 2 7 3 9 4 7 2<br />
a a x x x y x y<br />
2m+<br />
3<br />
5 6⎞<br />
2 3 4 4 3 4 2<br />
⎛ 1 1<br />
10) ⎜ x⋅7 x: x ⎟<br />
11) 2abc ⋅3abc⋅4abc<br />
⎝47 7 ⎠<br />
4 3 2 5 6 5 4 7 2<br />
12) a ⋅ a + 3a ⋅ a + 4a⋅a 13) 2x ⋅ x + 5x ⋅x<br />
−3x<br />
14) 7x ⋅ x + 2 x : x −3x ⋅y 15) 3x y z ⋅2x y z<br />
5 3 10 2 3 5 4 2b m m−2 3b<br />
2<br />
16) x y : x y 17) x ⋅ x : x<br />
m+ 3 4n− 1 m+ 2 1− 2n m+ n−3 2m− 3n+ 2 m+ 2n−1<br />
x<br />
x<br />
3x 2 4x 3x 2 4x<br />
( a ) ⋅( a ) ⋅( a ) ( a ) ⋅( a ) ( a )<br />
x<br />
3x<br />
2 4x<br />
7 2 3 3<br />
( a ) + ( a ) − ( a ) ( x y x y )<br />
7 2 3 3 2 3 2 4<br />
( xy ) ( xy) ( ab) ⋅( ab )<br />
( x )<br />
4 6 3 4 6 3<br />
18) 19) :<br />
4 6 3 4<br />
20) 3 2 21) :<br />
2 3 4 3<br />
22) : 23)<br />
24) 1<br />
0 0<br />
0 0 0<br />
+ + 3 − 4x 25) ( x + 2) ⋅ ( x+<br />
2)<br />
0 0 2 3 0 0<br />
( x y −xy ) ⋅( xy − x y ) x + y −( xy)<br />
0 0<br />
26) 27) 2 3 5<br />
⋅ y<br />
3 6<br />
0 0<br />
2 0 1<br />
3xy 3+<br />
x x y 3<br />
0 0<br />
0<br />
2 − 2<br />
28) 29) 30)<br />
x y y xy<br />
( )<br />
−5 2 −4 2 −3 −2<br />
31) ⋅<br />
32) : 33) ⋅ :<br />
x x x x<br />
34) x : x 35) 12 x :3x<br />
8 −3 −9 −2<br />
1<br />
n<br />
a<br />
x x x<br />
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<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
Rješenja su od 16.strane na dalje…<br />
Iz naše ponude izdvajamo<br />
Zbirke potpuno riješenih zadataka priručnici za samostalno učenje:<br />
Matematika-1-prvo polugodište:<br />
Sve dodatne informacije o ovim zbirkama<br />
zatražite na mail: mim-sraga@zg.htnet.hr<br />
ili na naše telefone 01-4578-431 , 4579-130<br />
Dodatne informacije i PDF ogledne primjerke<br />
potražite na našoj web-stranici : www.mim-sraga.com<br />
autor: Mladen Sraga<br />
15
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
Novo: MALA ŠKOLA MATEMATIKE –1 na<br />
BESPLATNA video poduka i instrukcije<br />
UČIMO ZAJEDNO<br />
<strong>POTENCIJE</strong><br />
ALGEBARSKI IZRAZI<br />
ALGEBARSKI RAZLOMCI<br />
link: http://www.mim-sraga.com/Mala-skola-matematike--video.htm<br />
www.mim-sraga.com 16 © M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
Rješenja svih zadataka s kompletnim postupkom i uputama<br />
2 3<br />
n<br />
21. Koristimo pravila: aa ⋅ = a , aaa ⋅ ⋅ = a , aaa ⋅ ⋅ ⋅...<br />
⋅ a=<br />
a<br />
2−se ponavlja pet<br />
puta pa u eksponent<br />
pišemo 5<br />
n−puta<br />
Uputa: Prebrojite koliko se puta ponavlja isti faktor i taj broj stavite u eksponent:<br />
1) 22222 ⋅ ⋅ ⋅ ⋅<br />
<br />
1) 22222 ⋅ ⋅ ⋅ ⋅ = 2<br />
2)<br />
3)<br />
4)<br />
5)<br />
6)<br />
7)<br />
8)<br />
xxx ⋅ ⋅ = x<br />
3<br />
5<br />
= →<br />
( xy) ⋅( xy) ⋅ ( xy) = ( xy)<br />
5<br />
2<br />
( x+ y) ⋅ ( x+ y) ⋅ ( x+ y) ⋅ ( x+ y) = ( x+<br />
y)<br />
3<br />
čitamo: dva na petu<br />
grupiramo iste faktore...<br />
4<br />
<br />
xyxyxzxyxxz ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = xxxxxxyyyzz ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = x⋅y⋅ z = xyz<br />
⎛ x⎞ ⎛ x⎞<br />
⎛ x⎞ ⎛ x⎞ ⎛ x⎞<br />
⎜ ⎟⋅⎜ ⎟⋅⎜ ⎟⋅ ⎜ ⎟ = ⎜ ⎟<br />
⎝ y⎠ ⎝ y⎠<br />
⎝ y⎠ ⎝ y⎠ ⎝ y⎠<br />
x− y ⋅ x+ y ⋅ x− y ⋅ x− y ⋅ x+ y =<br />
( ) ( ) ( ) ( ) ( )<br />
4<br />
6 3 2 6 3 2<br />
( x y) ( x y) ( x y) ( x y) ( x y) ( x y) ( x y)<br />
⎛a−b⎞ ⎛a−b⎞ ⎛a−b⎞ ⎛a−b⎞<br />
⎜ ⎟⋅⎜ ⎟⋅ ⎜ ⎟ = ⎜ ⎟<br />
⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠<br />
3<br />
Pogledaj dodatnu VIDEO uputu uz ovaj zadatak<br />
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ili<br />
direktni link na : video uputu …uz 21. zadatak<br />
3 2<br />
= − ⋅ − ⋅ − ⋅ + ⋅ + = − ⋅ +<br />
2 3<br />
n<br />
22. Koristimo pravila: a = a⋅ a , a = a⋅a⋅ a , a = a⋅a⋅a⋅...<br />
⋅a<br />
<br />
n−puta<br />
2<br />
4) ( − 1) = ( −1) ⋅( − 1)<br />
= 1<br />
⎫<br />
parni<br />
3<br />
⎪<br />
( )<br />
5) ( − 1) = ( −1) ⋅( −1) ⋅( − 1)<br />
= −1<br />
⎬ Zaključak:<br />
na neparni<br />
4<br />
⎪<br />
( −1)<br />
6) ( − 1) = ( −1) ⋅( −1) ⋅( −1) ⋅( − 1)<br />
= 1 ⎪ ⎭<br />
2 3 4<br />
1) 5 = 5⋅ 5 = 25 2) 2 = 2⋅2⋅ 2 = 8 3) 3 = 3⋅3⋅3⋅ 3 = 81<br />
3<br />
( ) ( ) ( ) ( )<br />
7) − 2 = −2 ⋅ −2 ⋅ − 2 = −8<br />
<br />
4<br />
( − ) = ( − ) ⋅( − ) ⋅( − ) ⋅( − )<br />
Imamo paran broj "minusa" pa<br />
je produkt pozitivan broj<br />
na eksponent<br />
− 1 = 1<br />
eksponent<br />
=−1<br />
Imamo neparan broj "minusa" pa će i umnožak biti negativno tj. imati će predznak minus<br />
8) x x x x x =<br />
<br />
<br />
↓<br />
4<br />
x<br />
autor: Mladen Sraga<br />
17
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
22. Koristimo pravila:<br />
2 3<br />
n<br />
a = a⋅ a , a = a⋅a⋅ a , a = a⋅a⋅a⋅...<br />
⋅a<br />
<br />
n−puta<br />
9)<br />
10)<br />
2<br />
⎛1⎞<br />
1 1 1<br />
⎜ ⎟ = ⋅ =<br />
⎝3⎠<br />
3 3 9<br />
3<br />
⎛3⎞ 3 3 3 27<br />
⎜ ⎟ = ⋅ ⋅ =<br />
⎝5⎠<br />
5 5 5 125<br />
Pogledaj dodatnu VIDEO uputu uz ovaj zadatak<br />
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ili<br />
direktni link na : video uputu …uz 22. zadatak<br />
11)<br />
5<br />
⎛2⎞ 2 2 2 2 2 32<br />
⎜ ⎟ = ⋅ ⋅ ⋅ ⋅ =<br />
⎝3⎠<br />
3 3 3 3 3 243<br />
12)<br />
2<br />
⎛ 2⎞ ⎛ 2⎞ ⎛ 2⎞<br />
22 ⋅ 4<br />
⎜− ⎟ = ⎜− ⎟⋅⎜− ⎟ = =<br />
⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠<br />
3⋅3 9<br />
13)<br />
3<br />
⎛ 2 ⎞ ⎛ 2⎞ ⎛ 2⎞ ⎛ 2⎞<br />
222 ⋅ ⋅ 8<br />
⎜− ⎟ = ⎜−<br />
⎟⋅⎜− ⎟⋅⎜− ⎟ = − = −<br />
⎝ 3 ⎠ ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠<br />
333 ⋅ ⋅ 27<br />
14)<br />
4<br />
⎛3⎞<br />
3 3 3 3 81<br />
⎜ ⎟ = ⋅ ⋅ ⋅ =<br />
⎝4⎠<br />
4 4 4 4 256<br />
15)<br />
2<br />
⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞<br />
44 ⋅ 16<br />
⎜− ⎟ = ⎜− ⎟⋅⎜− ⎟ = + =<br />
⎝ 5⎠ ⎝ 5⎠ ⎝ 5⎠<br />
5⋅5 25<br />
16)<br />
3<br />
⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞<br />
444 ⋅ ⋅ 64<br />
⎜− ⎟ = ⎜− ⎟⋅⎜− ⎟⋅⎜− ⎟ = − = −<br />
⎝ 5⎠ ⎝ 5⎠ ⎝ 5⎠ ⎝ 5⎠<br />
5⋅5⋅5 125<br />
17)<br />
4<br />
⎛ 4⎞<br />
⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞<br />
4444 ⋅ ⋅ ⋅ 256<br />
⎜− ⎟ = ⎜−<br />
⎟⋅⎜− ⎟⋅⎜− ⎟⋅⎜− ⎟ = + =<br />
⎝ 5⎠<br />
⎝ 5⎠ ⎝ 5⎠ ⎝ 5⎠ ⎝ 5⎠<br />
5555 ⋅ ⋅ ⋅ 625<br />
2<br />
18) 0,2 0,2 0,2 0,04<br />
= ⋅ =<br />
2<br />
( ) ( ) ( )<br />
19) − 0,2 = −0,2 ⋅ − 0,2 = + 0,2⋅ 0,2 = 0,04<br />
3<br />
20) 0,2 0,2 0,2 0,2 0,04 0,2 0,008<br />
= ⋅ ⋅ = ⋅ =<br />
3<br />
( ) ( ) ( ) ( ) ( )<br />
21) − 0,2 = −0,2 ⋅ −0,2 ⋅ − 0,2 = − 0,2⋅0,2⋅ 0,2 = −0,008<br />
( − )<br />
2<br />
( ) ( ) ( )<br />
22) 2,5 = −2,<br />
5 ⋅− 2,5 =+ 2,5⋅ 2,5 = 6, 25<br />
www.mim-sraga.com 18<br />
© M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
22.<br />
n<br />
a = a⋅ a a = a⋅a⋅ a a = a⋅a⋅a⋅ ⋅a<br />
<br />
2 3<br />
Koristimo pravila: , , ...<br />
3<br />
( ) ( ) ( ) ( ) ( )<br />
n−puta<br />
23) − 2,5 = −2,5 ⋅ −2,5 ⋅ − 2,5 = − 2,5⋅2,5⋅ 2,5 = 15,625<br />
4<br />
) ( ) ( )( )( )( ) ( )<br />
24 − 2,5 = −2,5 ⋅ −2,5 ⋅ −2,5 ⋅ − 2,5 = + 2,5⋅2,5⋅2,5⋅ 2,5 = 39,0625<br />
2 3 4 5<br />
( ) ( ) ( ) ( ) ( ) ( ) ( )<br />
25) − 1 + − 1 + − 1 + − 1 = + 1+ − 1 + + 1 + − 1 = 0+ 0 = 0<br />
<br />
0 0<br />
20 30 45<br />
( ) ( ) ( ) ( ) ( )<br />
26) − 1 + − 1 + − 1 = + 1⋅ + 1 ⋅ − 1 = −1<br />
( ) ( ) ( ) ( )<br />
1 2 3 4<br />
27) − 2 + − 2 + − 2 + − 2 = − 2+ 4− 8+ 16 = 4+ 16−2− 8 = 10<br />
5 2<br />
28) 2 − 3 = 22222 ⋅ ⋅ ⋅ ⋅ −33 ⋅ = 32− 9=<br />
23<br />
4 5<br />
29) 3 − 2 = 3333 ⋅ ⋅ ⋅ −22222 ⋅ ⋅ ⋅ ⋅ = 81− 32=<br />
49<br />
3 5<br />
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )<br />
30) − 2 + − 2 = −2 ⋅ −2 ⋅ − 2 + −2 ⋅ −2 ⋅ −2 ⋅ −2 ⋅ −2 −8− 32 = −40<br />
31)<br />
2<br />
3<br />
⎛1⎞ ⎛ 1⎞ 1 1 ⎛ 1⎞⎛ 1⎞⎛ 1⎞<br />
1 1 1−<br />
2 1<br />
⎜ ⎟ + ⎜−<br />
⎟ = ⋅ + ⎜− ⎟⋅⎜− ⎟⋅⎜− ⎟ = − = = −<br />
⎝4⎠ ⎝ 2⎠ 4 4 ⎝ 2⎠⎝ 2⎠⎝ 2⎠<br />
16 8 16 16<br />
2 2<br />
2 3<br />
( ) ( ) ( ) ( ) ( ) ( ) ( )<br />
32) ⎡ − 0,2 + − 0,2 ⎤ = ⎡⎣<br />
−0,2 ⋅ − 0,2 + −0,2 ⋅ −0,2 ⋅ − 0,2 ⎤⎦<br />
=<br />
⎣<br />
⎦<br />
( )<br />
2 2<br />
= 0,04− 0,008 = 0,032 = 0,001024<br />
Pogledaj dodatnu VIDEO uputu uz ovaj zadatak<br />
http://www.mim-sraga.com/potencije.htm<br />
ili<br />
direktni link na : video uputu …uz 22. zadatak<br />
autor: Mladen Sraga<br />
19
M-1- univerzalna zbirka potpuno riješenih zadataka ⋅ + ⋅ = ( + ) ⋅ ⋅ − ⋅ = ( − ) ⋅<br />
Koristimo pravila:<br />
( ) ( )<br />
23.<br />
n n n n n n<br />
5⋅ x⎫ 1) 2 3 ( 2 3)<br />
⎪<br />
x+ x = + ⋅ x = 5⋅ x = 5x<br />
ili ⎬<br />
5x<br />
⎪⎭<br />
( )<br />
2) x+ 2x+ 4x = 1+ 2 + 4 ⋅ x = 7x<br />
ca da c d a ca da c d a<br />
ca ⋅ + da ⋅ = c+ d⋅a ca ⋅ −da ⋅ = c−d⋅a<br />
ili taj isti zadatak na malo duži ali sigurniji naćin:<br />
( )<br />
x+ 2x+ 4x = 1x+ 2x+ 4x = 1+ 2+ 4 ⋅ x = 7x Dakle: x = 1x<br />
je potpuno isti izraz...<br />
Praksa je pokazala da velika većina đaka radi istu grešku: uzimate da je:<br />
dakle vi kada računate u glavi grešite na ovaj način:<br />
x + 2x+ 4x = 6x ili x+ 2x+ 4x = ( 0+ 2+ 4)<br />
⋅ x = 6x<br />
Jednom zauvjek treba zapamtiti<br />
x = 1x<br />
( )<br />
x+ 2x+ 4x = 1x+ 2x+ 4x = 1+ 2+ 4 ⋅ x = 7x<br />
što nije točno!!!<br />
x = 0 x<br />
pa to u zadatku treba izgledati ovako:<br />
što nije točno!!<br />
( )<br />
3) 7a− 2a = 7 −2 ⋅ a = 5⋅ a = 5a<br />
( )<br />
4) 9y− 2y+ 3y− y = 9 − 2 + 3 −1 ⋅ y = 9⋅ y = 9y<br />
ili taj isti zadatak na malo duži ali sigurniji naćin:<br />
( )<br />
9y− 2y+ 3y− y = 9y− 2y+ 3y− 1y = 9− 2+ 3−1 ⋅ y = 9y<br />
5) 2x+ 3a− x+ 5a+ 7x− 2a<br />
=<br />
( )<br />
( )<br />
( ) ( )<br />
= 2x − x+ 7x+ 3a+ 5a− 2a = 2− 1+ 7 ⋅ x+ 3+ 5−2 ⋅ a = 8⋅ x+ 6⋅ a = 8x+<br />
6a<br />
6) 2xy + 3xy + xy = 2 + 3 + 1 ⋅ xy = 6xy<br />
7) 2ab − 4ab + ab = 2 − 4 + 1 ⋅ ab = −1⋅ab<br />
=−ab<br />
( )<br />
8) 4xy + 2xy − 9xy = 4 + 2 −9 ⋅ xy = −3⋅ xy = −3xy<br />
2 2 2 2 2 2<br />
2 2 2<br />
9) y+ 3x y−4z−5x y− 2y+ 8z− 3y+ 8x y =<br />
2 2 2<br />
= y−2y− 3y+ 8z− 4z+ 3x y− 5x y+ 8x y =<br />
( 1 2 3) ( 8 4) ( 3 5 8)<br />
= − − ⋅ + − ⋅ + − + ⋅ =<br />
2<br />
y z x y<br />
2<br />
=−4⋅ y+ 4⋅ z+ 6⋅ x y = ⎪<br />
2<br />
⎬<br />
=− 4y+ 4z+ 6x y<br />
⎪⎭<br />
⎫<br />
Ovo je potpuno isti izraz<br />
www.mim-sraga.com 20 © M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
( )<br />
( ) ( )<br />
( ) ( )<br />
ca ⋅ + da ⋅ = c+ d⋅a ca ⋅ −da ⋅ = c−d⋅a<br />
23. Koristimo pravila:<br />
n n n n n n<br />
ca ⋅ + da ⋅ = c+ d⋅a ca ⋅ −da ⋅ = c−d⋅a<br />
10) 7xy − 2xy + 4xy = 7 − 2 + 4 ⋅ xy = 9⋅ xy = 9xy<br />
3 3 3 3 3 3<br />
( ) ( ) ( ) ( ) ( )<br />
11) 2 − − 3 + = 2 −3 ⋅ + = −1⋅ + = − −<br />
2 2 2 2 2<br />
x y x y x y x y x y<br />
( x+ y) + ( x+ y) − ( x+ y) = ( + − ) ⋅ ( x+ y) = ( x+<br />
y)<br />
12) 3 4 3 4 1 6<br />
ili taj isti zadatak na malo duži ali sigurniji naćin:<br />
( x y) ( x y) ( x y) ( x y) ( x y) ( x y)<br />
= ( 3+ 4−1) ⋅ ( x+ y)<br />
=<br />
= 6( x+<br />
y)<br />
3 + + 4 + − + = 3 + + 4 + − 1 + =<br />
2 3 2 3 2 3<br />
13) 5 2 2 7 3 3<br />
x y + z− x y + z+ x y − z =<br />
2 3 2 3 2 3<br />
5 2 3 2 7<br />
= x y − x y + x y + z+ z =<br />
2 3<br />
( 5 2 3) x y ( 2 7)<br />
= − + ⋅ + + ⋅ z =<br />
2 3<br />
6 9<br />
= ⋅ x y + ⋅ z =<br />
2 3<br />
6 9<br />
= x y +<br />
2 2<br />
( x x ) ( x x )<br />
2<br />
= ( 3 4) ( x 3x<br />
5)<br />
2<br />
= −1⋅( x − 3x+ 5)<br />
=<br />
14) 3 − 3 + 5 −4 − 5 + 1 =<br />
x<br />
z<br />
− ⋅ − + =<br />
3x<br />
5<br />
2<br />
=− + −<br />
( x+ y− ) − ( x+ y− ) + ( x+ y− ) = ( − + ) ⋅ ( x+ y− ) = − ( x+ y−<br />
)<br />
15) 3 1 6 1 1 3 6 1 1 2 1<br />
ili taj isti zadatak na malo duži ali sigurniji naćin:<br />
( x y ) ( x y ) ( x y ) ( x y ) ( x y ) ( x y )<br />
= ( 3− 6+ 1) ⋅ ( x+ y− 1)<br />
=<br />
=− 2( x+ y−1)<br />
3 + −1 − 6 + − 1 + + − 1 = 3 + −1 − 6 + − 1 + 1 + − 1 =<br />
autor: Mladen Sraga<br />
21
M-1- univerzalna zbirka potpuno riješenih zadataka<br />
24.<br />
n<br />
a<br />
Koristimo pravila: a ⋅ a = a a : a = a = a : a = a<br />
m<br />
a<br />
n m n+ m n m n−m n m n−m<br />
1) 2 ⋅ 2 = 2 = 2<br />
5 3 5+<br />
3 8<br />
2) 5 ⋅ 5 = 5 = 5<br />
2 7 2+<br />
7 9<br />
Pogledaj dodatnu VIDEO uputu uz ovaj zadatak<br />
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ili<br />
direktni link na : video uputu …uz 24. zadatak<br />
3) 3 ⋅ 3 = 3 = 3<br />
x 2x x+<br />
2x 3x<br />
4) 7 ⋅ 7 = 7 = 7 = 7<br />
2m−5 m−2 2m− 5+ m− 2 2m+ m−5−2 3m−7<br />
5) 2 ⋅2 ⋅2 ⋅ 2 = 2 = 2<br />
2 6 4 8 2+ 6+ 4+<br />
8 20<br />
6) 3⋅a ⋅2⋅ a = 3⋅2⋅a ⋅ a = 6⋅ a = 6a<br />
4 7 4 7 4+<br />
7 11<br />
7)<br />
x ⋅ x = x = x<br />
2 3 2+<br />
3 5<br />
8)<br />
xx ⋅ ⋅ x = x⋅x⋅ x = x = x Pazi x=<br />
x<br />
3 5 1 3 5 1+ 3+<br />
5 9 1<br />
9)<br />
x ⋅x ⋅ x = x = x<br />
2 4 6 2+ 4+<br />
6 12<br />
10)<br />
2 5 2 5 2⋅ 2+ 2⋅ 6+ 5⋅ 3 4+ 12+<br />
15 31<br />
+ 2+<br />
3 2 2 3 2 6 6 6<br />
x ⋅x ⋅ x = x = x = x = x<br />
11)<br />
x ⋅x ⋅x ⋅ x = x = x = x<br />
2 2+ a a− 2 3 2+ 2+ a+ a− 2+ 3 a+ a+ 2+ 2− 2+ 3 2a+<br />
5<br />
12)<br />
13)<br />
a ⋅ a = a = a<br />
a<br />
3 6 3+<br />
6 9<br />
⋅a<br />
4 2<br />
⋅ a = a ⋅a ⋅ a = a = a Pazi a = a<br />
4 2 1 4+ 2+<br />
1 7 1<br />
Dosta često radite ovakve greške:<br />
4 2 4+<br />
2 6<br />
a ⋅a ⋅ a = a = a ⎫⎪<br />
što nije točno jer je:<br />
4 2 4+ 2+<br />
0 6⎬<br />
a ⋅a ⋅ a = a = a ⎪⎭<br />
a<br />
= a<br />
1<br />
14)<br />
a ⋅a ⋅a ⋅ a = a = a<br />
2 5 7 3 2+ 5+ 7+<br />
3 17<br />
15)<br />
a ⋅a ⋅ a = a = a<br />
3x x+ 2 2x 3x+ x+ 2+ 2x 6x+<br />
2<br />
16)<br />
⋅<br />
⋅<br />
x+<br />
y 2x<br />
x+ 3y x+ y+ 2x+ x+ 3y x+ 2x+ x+ y+ 3y 4x+<br />
4y<br />
a a a<br />
= a = a = a<br />
17) −2⋅x⋅6⋅x ⋅3⋅ x =−2⋅6⋅3⋅x ⋅x ⋅ x =−36⋅ x =−36<br />
x<br />
3 4 1 3 4 1+ 3+<br />
4 8<br />
www.mim-sraga.com 22<br />
© M.I.M-Sraga centar za poduku<br />
mim-sraga@zg.htnet.hr<br />
tel-01-4578-431
<strong>POTENCIJE</strong><br />
M-1- rješenja i kompletni postupak<br />
Rješenja svih zadataka šaljemo besplatno<br />
u obliku PDF dokumenta mailom<br />
dovoljno je da nam pošaljete mail na<br />
našu adresu: mim-sraga@zg.htnet.hr s porukom da trebate<br />
kompletna rješenja zadataka iz potencija …<br />
24.<br />
n<br />
a<br />
Koristimo pravila: a ⋅ a = a a : a = a = a : a = a<br />
m<br />
a<br />
n m n+ m n m n−m n m n−m<br />
24.rješenj od 18) do 32) zadatka narućite preko maila !<br />
33)<br />
( x y) ( x y) ( x y) ( x y) ( x y) ( x y) ( x y) ( x y)<br />
2 3 2m−1 m−3 2 2m−1 3 m−3<br />
+ ⋅ − ⋅ + ⋅ − = + ⋅ + ⋅ − ⋅ − =<br />
m<br />
( x y) ( x y)<br />
2+ 2 − 1 3+ m−3<br />
= + ⋅ − =<br />
2m+ 2−1<br />
( x y) ( x y)<br />
= + ⋅ −<br />
2m+<br />
1<br />
( x y) ( x y)<br />
= + ⋅ −<br />
m<br />
m+ 3−3<br />
=<br />
34)<br />
2<br />
3x−2y 2<br />
4x−y 2<br />
x−y 2<br />
3y−2x 2<br />
3x− 2y+ 4x− y+ x− y+ 3y−2x<br />
⎛ab ⎞ ⎛ab ⎞ ⎛ab ⎞ ⎛ab ⎞ ⎛ab<br />
⎞<br />
⎜ ⎟ ⋅⎜ ⎟ ⋅⎜ ⎟ ⋅ ⎜ ⎟ = ⎜ ⎟<br />
=<br />
⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠<br />
3x+ 4x+ x−2x−2y−y− y+<br />
3y<br />
2<br />
⎛ab<br />
⎞<br />
= ⎜ ⎟<br />
=<br />
⎝ c ⎠<br />
2<br />
⎛ab<br />
⎞<br />
= ⎜ ⎟<br />
⎝ c ⎠<br />
6x−<br />
y<br />
autor: Mladen Sraga<br />
23
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autor: Mladen Sraga<br />
25