Potpuno riješeni zadatci za vježbu PDF-varijanta - MIM-Sraga
Potpuno riješeni zadatci za vježbu PDF-varijanta - MIM-Sraga
Potpuno riješeni zadatci za vježbu PDF-varijanta - MIM-Sraga
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M-1-Kub zbroja i kub razlike<br />
Rješenja + kompletni postupak<br />
40.<br />
Koristeći se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
izračunaj:<br />
3 3<br />
( x+ y) ( x−<br />
y) ( y−x)<br />
1) 2) 3)<br />
( x−<br />
) ( −x) ( x+<br />
)<br />
3 3 3<br />
4) 1 5) 1 6) 1<br />
3 3<br />
( x−<br />
) ( −x) ( x+<br />
)<br />
7) 2 8) 2 9) 3<br />
3<br />
( a+ b)<br />
11) ( 2a−3b) 12) ( 3x−5y)<br />
10) 2<br />
3 3<br />
3 3<br />
( x+ y) ( ab−<br />
c) ( ab+<br />
c)<br />
13) 5 2 14) 5 15) 3 5<br />
( x + ) ( x − y ) ( x − y )<br />
16) 1 17) 18)<br />
2 3 2 3 3 3 3 3<br />
3 3<br />
⎛ 1⎞ ⎛ 1⎞ ⎛ 1⎞<br />
19) ⎜x− ⎟ 20) ⎜2x+<br />
⎟ 21) ⎜3x−<br />
⎟<br />
⎝ 3⎠ ⎝ 2⎠ ⎝ 3⎠<br />
3 3<br />
⎛1 2⎞ ⎛1 1 ⎞ ⎛ 2 3 1⎞<br />
22) ⎜ x+ ⎟ 23) ⎜ ab+<br />
c⎟ 24) ⎜a b + ⎟<br />
⎝2 3⎠ ⎝3 4 ⎠ ⎝ 3⎠<br />
3<br />
⎛ 2 3⎞ ⎛<br />
25) 26) 0.2<br />
3 4 5 ⎞<br />
⎜ x − y ⎟ ⎜ x+<br />
y⎟<br />
⎝4 3 ⎠ ⎝ 6 ⎠<br />
( x + )<br />
3 3 2 3<br />
( x+ ) ( x−<br />
y) ( x+<br />
y )<br />
2 3 3 3 4 3 4 2 3 3<br />
( xy + z ) ( xy −z ) ( x y −z<br />
)<br />
4 2 3 3 2 3 2 2 3 2<br />
( ab+ ab) ( xy+<br />
) ( xy−<br />
xy)<br />
2 3 2 3 2 3 5 3 2 3 4 3<br />
( a + a b ) ( a − b c ) ( x + y z )<br />
−1 3 −2 3 −3 2 3<br />
( x + x) ( x − x) ( x + x )<br />
m n 3 n n 3<br />
n n 3<br />
( − ) ( − )<br />
45) ( 2 + 3 )<br />
n n 3 n n+ 1 3<br />
n+<br />
1 n−1 3<br />
( a + b ) ( a −a ) ( a + a )<br />
3<br />
27) 0.1 10<br />
28) 2 0.1 29) 0.1 0.2 30) 5 0.4<br />
31) 32) 33)<br />
34) 35) 2 3 36) 3<br />
37) 2 38) 3 2 39) 5<br />
40) 41) 42)<br />
43) 2 2 44) 3 2<br />
46) 47) 48)<br />
( a− ) + ( a+ ) ( a− ) − ( a+<br />
)<br />
3 3 3 3<br />
49) 1 1 50) 1 1<br />
( x+ y) −( x− y) ( x+ y) + ( x−<br />
y)<br />
3 3 3 3<br />
51) 52)<br />
( x+ ) − ( x− ) ( x+ ) − ( −x)<br />
3 3 3 3<br />
53) 2 3 3 1 54) 3 1 5 2<br />
( x+ y) − ( x+ y) ( x− ) − x( x+<br />
)<br />
3 3 3 2<br />
55) 3 2 4 2 56) 1 2<br />
2 3 3 3 3 3<br />
( − ) − ( + ) ( + ) − ( − ) − ( + )<br />
57) x x 2 x 2 58) 2 x y 5 x y x y<br />
3<br />
3<br />
3<br />
3<br />
3<br />
3<br />
2<br />
1<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
+ kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
( )<br />
3 3 2 2 3 3 3 2<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+<br />
3AB<br />
3 3 2 2 3 3 2 2 3<br />
1) x+ y = x + 3⋅x ⋅ y+ 3⋅x⋅ y + y = x + 3x y+ 3xy + y<br />
−B<br />
2 3<br />
uputa:<br />
( )<br />
3 3 2 2 3<br />
x+ y = x + 3⋅x ⋅ y+ 3⋅x⋅ y + y =<br />
↓ ↓ ↑ ↑ ↑ ↑<br />
( )<br />
3 3 2 2 3<br />
A+ B = A + 3⋅A ⋅ B+ 3⋅AB ⋅ + B<br />
x + 3x y+ 3xy + y<br />
3 2 2 3<br />
( )<br />
( )<br />
3 3 2 2 3 3 2 2 3<br />
2) x− y = x −3⋅x ⋅ y+ 3⋅x⋅y − y = x − 3x y+ 3xy − y<br />
↓ ↓ ↑ ↑ ↑ ↑<br />
3 3 2 2 3<br />
A− B = A −3⋅A ⋅ B+ 3⋅AB ⋅ − B<br />
( )<br />
3 3 2 2 3 3 2 2 3<br />
3) y− x = y −3⋅y ⋅ x+ 3⋅y⋅x − x = y − 3y x+ 3yx −x<br />
( )<br />
3 3 2 2 3 3 2<br />
4) x− 1 = x −3⋅x ⋅ 1+ 3⋅x⋅1 − 1 = x − 3x + 3x−1<br />
( )<br />
5) 1− x = 1 − 3⋅1 ⋅ x + 3⋅1⋅x − x = 1− 3x+ 3x −x<br />
( )<br />
3 3 2 2 3 2 3<br />
↓ ↓ ↑ ↑ ↑ ↑<br />
3 3 2 2 3<br />
A− B = A −3⋅A ⋅ B+ 3⋅AB ⋅ − B<br />
( )<br />
3 3 2 2 3 3 2<br />
6) x+ 1 = x + 3⋅x ⋅ 1+ 3⋅x⋅ 1 + 1 = x + 3x + 3x+<br />
1<br />
( )<br />
( )<br />
3 3 2 2 3 3 2<br />
3 2<br />
7) x− 2 = x − 3⋅x ⋅ 2 + 3⋅x⋅2 − 2 = x − 6x + 3⋅4⋅x− 8 = x − 6x<br />
+ 12x−8<br />
↓ ↓ ↑ ↑ ↑ ↑<br />
3 3 2 2 3<br />
A− B = A −3⋅A ⋅ B+ 3⋅AB ⋅ − B<br />
( x)<br />
3 3<br />
8) 2 − = 2 −3⋅2<br />
⋅ x+ 32 ⋅ ⋅x − x = 8−34 ⋅ ⋅ x+ 6x − x = 8−12x+ 6x −x<br />
2 2 3 2 3<br />
2<br />
2 3<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
+ kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) 3 3<br />
( )<br />
( )<br />
3 3 2 2 3 3 3 2<br />
A+ B = A + A B+ AB + B A− B = A −3A B+ 3AB −B<br />
3 3 2 2 3<br />
9) x+ 3 = x + 3⋅x ⋅ 3 + 3⋅x⋅ 3 + 3 =<br />
9 3 9 27<br />
3 2<br />
= x + x + ⋅ ⋅ x+ =<br />
3 2<br />
= x + x + x+<br />
9 27 27<br />
2 3<br />
3 3 2<br />
( a+ b)<br />
= a + ⋅ a 2b 3 a ( 2b) ( 2b)<br />
10) 2 3<br />
2 3<br />
⋅ + ⋅ ⋅ + =<br />
3 2 2 2 3 3<br />
= a + 6a b+ 3a⋅2 ⋅ b + 2 ⋅ b =<br />
= a + 6a b+ 12ab + 8b<br />
3 2 2 3<br />
( a b) ( a) ( a) b a ( b) ( b)<br />
3 3 2 2 3<br />
11) 2 − 3 = 2 −3⋅ 2 ⋅ 3 + 3⋅2 ⋅ 3 − 3 =<br />
3 3 2 2 2 2 3 3<br />
= ⋅ − ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ − ⋅ =<br />
2 a 3 2 a 3 b 6 a 3 b 3 b<br />
8a 343a b 69a b 27b<br />
3 2 2 3<br />
= − ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ − =<br />
= 8a − 36a b+ 54ab −27b<br />
3 2 2 3<br />
( )<br />
3 3 2<br />
A − B = A − 3⋅A ⋅ B + 3⋅A⋅<br />
B<br />
2 3<br />
↑ ↑ ↓ ↓ ↓ ↓<br />
( x y) ( x) ( x) y x ( y) ( y)<br />
3 3 2 2 3<br />
12) 3 − 5 = 3 −3⋅ 3 ⋅ 5 + 3⋅3 ⋅ 5 − 5 =<br />
3 3 2 2 2 2 3 3<br />
= ⋅ − ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ − ⋅ =<br />
3 x 3 3 x 5 y 9 x 5 y 5 y<br />
3 2 2 3<br />
= − ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ − =<br />
27x 3 9 5 x y 9 25 x y 125y<br />
= 27x − 135x y+ 225xy −125y<br />
3 2 2 3<br />
−<br />
B<br />
( )<br />
3 3 2 2 3<br />
A + B = A + 3⋅A ⋅ B + 3⋅A⋅ B + B<br />
↑ ↑ ↓ ↓ ↓ ↓<br />
3 3 2<br />
( x+ y) = ( x) + ⋅( x) ⋅ y+ ⋅ x⋅ ( y)<br />
( 2y)<br />
13) 5 2 5 3 5 2 3 5 2<br />
2 3<br />
3 3 2 2 2 2 3 3<br />
= 5 ⋅ x + 3⋅5 ⋅x ⋅2⋅ y+ 15⋅x⋅2 ⋅ y + 2 ⋅ y =<br />
3 2 2 3<br />
= 125x + 3⋅25⋅2⋅x ⋅ y+ 15⋅4⋅x⋅ y + 8y<br />
=<br />
= 125x + 150x y+ 60xy + 8y<br />
3 2 2 3<br />
+ =<br />
3<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
+ kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+<br />
3AB<br />
( ab c) ( ab) ( ab) c ab ( c) ( c)<br />
3 3 2 2 3<br />
14) − 5 = −3⋅ ⋅ 5 + 3⋅ ⋅ 5 − 5 =<br />
ab 3 a b 5 c 3 ab5 c 5 c<br />
3 3 2 2 2 2 2 3 3<br />
= − ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ − ⋅ =<br />
3 3 2 2 2 2 3<br />
= ab −35 ⋅ ⋅a⋅b⋅ c + 325 ⋅ ⋅abc ⋅ ⋅ − 125c<br />
=<br />
= a b − 15a b c + 75abc −125c<br />
3 3 2 2 2 2 3<br />
−B<br />
2 3<br />
( ab c) ( ab) ( ab) c ab ( c) ( c)<br />
3 3 2 2 3<br />
15) 3 + 5 = 3 −3⋅ 3 ⋅ 5 + 3⋅3 ⋅ 5 + 5 =<br />
3 a b 3 9 a b 5 c 9 a b 25 c 5 c<br />
3 3 3 2 2 2 3 3<br />
= ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ =<br />
27a b 3 9 5 a b c 9 25 abc 125c<br />
3 3 2 2 2 3<br />
= − ⋅ ⋅ ⋅ + ⋅ ⋅ + =<br />
3 3 2 2<br />
27ab<br />
135abc<br />
2<br />
= − +<br />
25abc<br />
+ 125c<br />
2 3<br />
( ) ( ) ( )<br />
2 3 2 3 2 2 2 2 3<br />
16) x + 1 = x + 3⋅ x ⋅ 1+ 3⋅x<br />
⋅ 1 + 1 =<br />
23 ⋅ 22 ⋅ 2<br />
= + 3⋅ + 3⋅ ⋅ 1+ 1=<br />
x x x<br />
6 4 2<br />
= + + +<br />
x 3x 3x<br />
1<br />
( x y ) ( x ) ( x ) y x ( y ) ( y )<br />
2 3 3 2 3 2 2 3 2 3 2 3 3<br />
17) − = −3⋅ ⋅ + 3⋅ ⋅ − =<br />
23 ⋅ 22 ⋅ 3 2 32 ⋅ 33 ⋅<br />
= −3⋅ ⋅ + 3⋅ ⋅ − =<br />
x x y x y y<br />
= x − 3x y + 3x y − y<br />
6 4 3 2 6 9<br />
( x y ) ( x ) ( x ) y x ( y ) ( y )<br />
3 3 3 3 3 3 2 3 3 3 2 3 3<br />
18) − = −3⋅ ⋅ + 3⋅ ⋅ + =<br />
33 ⋅ 32 ⋅ 3 3 32 ⋅ 33 ⋅<br />
= x −3⋅x ⋅ y + 3⋅x ⋅ y + y =<br />
= x − 3x y + 3x y + y<br />
9 6 3 3 6 9<br />
3 2 3<br />
3 2<br />
⎛ 1⎞ 1 ⎛1⎞ ⎛1⎞<br />
19) ⎜x− ⎟ = x −3⋅x ⋅ + 3⋅x⋅⎜ ⎟ − ⎜ ⎟ =<br />
⎝ 3⎠ 3 ⎝3⎠ ⎝3⎠<br />
3<br />
3 1 2 1 1 1 3 3⋅1 1<br />
= x −3⋅ ⋅ x + 3⋅ ⋅x− = → 3⋅ = = =<br />
3<br />
3 9 3<br />
9 9 3⋅3 3<br />
3 2 1 1<br />
= x − x + x−<br />
3 27<br />
4<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
+ kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
3 2 3<br />
3 2<br />
⎛ 1⎞ 1 ⎛1⎞ ⎛1⎞<br />
20) ⎜2x+ ⎟ = ( 2x) + 3⋅( 2x)<br />
⋅ + 3⋅2x⋅ ⎜ ⎟ + ⎜ ⎟ =<br />
⎝ 2⎠ 2 ⎝2⎠ ⎝2⎠<br />
2 3<br />
3 3 2 2 1 1 1<br />
= 2 ⋅ x + 3⋅2 ⋅x<br />
⋅ + 6⋅x⋅ + =<br />
2 3<br />
2 2 2<br />
3 1 2 3 1 1<br />
1 2<br />
= 8x<br />
+⋅3⋅4⋅ ⋅ x + 6⋅<br />
⋅ x + = →34 ⋅ ⋅ = 3⋅<br />
4⋅<br />
2 4 8<br />
2<br />
3 2 3 1<br />
= 8x + 6x + x+<br />
2 8<br />
2<br />
1<br />
2<br />
1<br />
321 ⋅ ⋅<br />
= = 6<br />
1<br />
3 2 3<br />
3 2<br />
⎛ 1⎞ 1 ⎛1⎞ ⎛1⎞<br />
21) ⎜3x− ⎟ = ( 3x) −3⋅( 3x)<br />
⋅ + 3⋅3x⋅⎜ ⎟ − ⎜ ⎟ =<br />
⎝ 3⎠ 3 ⎝3⎠ ⎝3⎠<br />
2 3<br />
3 3 2 2 1 1 1<br />
= 3 ⋅x −3⋅3 ⋅x ⋅ + 9⋅x⋅ − =<br />
2 3<br />
3 3 3<br />
3 1 2 1 1<br />
1 1 119 ⋅⋅<br />
= 27x −3⋅ ⋅9⋅ x + 9⋅ ⋅x − = → 3⋅ ⋅ 9 = 3⋅<br />
⋅ 9 = = 9<br />
3 9 27<br />
3 3 1<br />
3 2 1<br />
= 27x − 9x + x−<br />
27<br />
( )<br />
3 3 2 2 3<br />
A + B = A + 3⋅A ⋅ B + 3⋅A⋅ B + B<br />
↑ ↑ ↓ ↓ ↓ ↓<br />
3 3 2 2 3<br />
⎛1 2⎞ ⎛1 ⎞ ⎛1 ⎞ 2 1 ⎛2⎞ ⎛2⎞<br />
22) ⎜ x+ ⎟ = ⎜ x⎟ + 3⋅⎜ x⎟ ⋅ + 3⋅ x⋅ ⎜ ⎟ + ⎜ ⎟ =<br />
⎝2 3⎠ ⎝2 ⎠ ⎝2 ⎠ 3 2 ⎝3⎠ ⎝3⎠<br />
3 2 2 3<br />
1 3 1 2 2 3 2 2<br />
= ⋅ x + 3⋅ ⋅x ⋅ + ⋅x⋅ + =<br />
3 2 2 3<br />
2 2 3 2 3 3<br />
1 3 1 2 2 3 4 8<br />
= x + 3⋅ ⋅ ⋅ x + ⋅ ⋅ x + =<br />
8 4 3 2 9 27<br />
1<br />
x<br />
8<br />
3<br />
= +<br />
3 ⋅<br />
1 2<br />
⋅<br />
4 3<br />
3 2<br />
= x + x + x+<br />
2<br />
1<br />
1<br />
2 3<br />
⋅ x +<br />
1 1 2 8<br />
8 2 3 27<br />
4<br />
2 ⋅<br />
1<br />
2<br />
9 3<br />
8<br />
⋅ x + =<br />
27<br />
5<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
+ kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
3 3 2 2<br />
⎛1 1 ⎞ ⎛1 ⎞ ⎛1 ⎞ 1 1 ⎛1 ⎞ ⎛1<br />
⎞<br />
23) ⎜ ab+ c⎟ = ⎜ ab⎟ + 3⋅⎜ ab⎟ ⋅ c + 3⋅ ab⋅ ⎜ c⎟ + ⎜ c⎟<br />
=<br />
⎝3 4 ⎠ ⎝3 ⎠ ⎝3 ⎠ 4 3 ⎝4 ⎠ ⎝4<br />
⎠<br />
3 2 2 3<br />
1 3 3 1 2 2 1 3 1 2 1 3<br />
= ⋅a ⋅ b + 3⋅ ⋅a ⋅b ⋅ ⋅ c + ⋅a⋅b⋅ ⋅ c + ⋅ c =<br />
3 2 2 3<br />
3 3 4 3 4 4<br />
1 3 3 1 1 2 2 1 2 1 3 1 1<br />
= a b + 3⋅ ⋅ ⋅a ⋅b ⋅ c + abc + c = →3⋅ ⋅ =<br />
27 9 4 16 64<br />
9 4<br />
1 1 1 1<br />
= ab + abc+ abc + c<br />
27 12 16 64<br />
3 3 2 2 2 3<br />
3<br />
3 ⋅11<br />
⋅<br />
3 ⋅34 ⋅<br />
=<br />
1<br />
12<br />
3 2 3<br />
2 3 1 2 3 3 2 3 2 1 2 3 1 1<br />
( ) ( )<br />
⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
24) ⎜ab + ⎟ = ab + 3⋅ ab ⋅ + 3⋅ab⋅ ⎜ ⎟ + ⎜ ⎟ =<br />
⎝ 3⎠ 3 ⎝3⎠ ⎝3⎠<br />
1 1 1<br />
( ) ( ) ( ) ( )<br />
3 3 3<br />
1 1<br />
a b a b 3 a b 9 27<br />
2 3<br />
2 3 3 3 2 2 3 2 2 3<br />
= a ⋅ b + 3⋅ ⋅ a ⋅ b + 3⋅a ⋅b<br />
⋅ + =<br />
2 3<br />
23 ⋅ 33 ⋅ 22 ⋅ 32 ⋅<br />
2 3<br />
= ⋅ + ⋅ + ⋅ ⋅ ⋅ + =<br />
1 1<br />
ab ab ab<br />
3 27<br />
6 9 4 6 2 3<br />
= + + +<br />
3 3 2 2 3<br />
⎛3 2 4 3⎞ ⎛3 2⎞ ⎛3 2⎞ 4 3 3 2 ⎛4 3⎞ ⎛4<br />
3⎞<br />
25) ⎜ x − y ⎟ = ⎜ x ⎟ −3⋅⎜ x ⎟ ⋅ y + 3⋅ x ⋅⎜ y ⎟ − ⎜ y ⎟ =<br />
⎝4 3 ⎠ ⎝4 ⎠ ⎝4 ⎠ 3 4 ⎝3 ⎠ ⎝3<br />
⎠<br />
3 2 2 3<br />
3 2 3 3 2 2 4 3 9 2 4 3 2 4 3 3<br />
= ⋅<br />
3 ( x ) −3 ⋅ ⋅<br />
2 ( x ) . ⋅ y + ⋅x ⋅ ⋅<br />
2 ( y ) − ⋅<br />
3 ( y ) =<br />
4 4 3 4 3 3<br />
27 23 ⋅ 9 4 22 ⋅ 3 9 16 2 32 ⋅ 64 33 ⋅<br />
= ⋅x −3⋅ ⋅ ⋅x ⋅ y + ⋅ ⋅x ⋅y<br />
− y =<br />
64 16 3 4 9 27<br />
27 3 ⋅9⋅4<br />
⋅x<br />
⋅ y +<br />
64 4⋅ 4⋅<br />
3<br />
6<br />
= x −<br />
4 3 9<br />
27 9 64<br />
= x − x y + 4x y − y<br />
64 4 27<br />
6 4 3 2 6 9<br />
⋅ 4 ⋅4<br />
2 6 64 9<br />
⋅x<br />
⋅y<br />
− y =<br />
4 ⋅ 9 27<br />
6<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
+ kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+<br />
3AB<br />
−B<br />
3<br />
3 3<br />
⎛ 5 ⎞ ⎛ 2 5 ⎞<br />
26) ⎜0.2x+ y⎟ = ⎜ x+ y⎟<br />
= → decimalni broj ( 0.2 ) pretvorimo u razlomak<br />
⎝ 6 ⎠ ⎝10 6 ⎠<br />
⎛1 5 ⎞<br />
= ⎜ x+ y⎟<br />
=<br />
⎝5 6 ⎠<br />
3<br />
3 2 2 3<br />
⎛1 ⎞ ⎛1 ⎞ 5 1 ⎛5 ⎞ ⎛5<br />
⎞<br />
= ⎜ x⎟ + 3⋅⎜ x⎟ ⋅ y+ 3⋅ x⋅ ⎜ y⎟ + ⎜ y⎟<br />
=<br />
⎝5 ⎠ ⎝5 ⎠ 6 5 ⎝6 ⎠ ⎝6<br />
⎠<br />
3 2 2 3<br />
1 3 1 2 5 3 5 2 5 3<br />
= ⋅ x + 3⋅ ⋅x ⋅ ⋅ y + ⋅x⋅ ⋅ y + ⋅ y =<br />
3 2 2 3<br />
5 5 6 5 6 6<br />
1 1 5 3 25 125<br />
x 3 x y x y y<br />
125 25 6 5 36 216<br />
3 2 2 3<br />
= + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + =<br />
1 3⋅⋅<br />
15<br />
x<br />
125 5⋅ 5⋅<br />
3 ⋅2 ⋅ xy+<br />
3<br />
= +<br />
2 3<br />
1 1 5 125<br />
= x + x y+ xy + y<br />
125 10 12 216<br />
3 2 2 3<br />
⋅ 5 ⋅5<br />
2 125 3<br />
⋅ xy + y =<br />
5 ⋅ 3 ⋅12<br />
216<br />
( ) ( ) ( )<br />
3 3 2 2 3<br />
27) 0.1x+ 10 = 0.1x + 3⋅ 0.1x ⋅ 10 + 3⋅0.1x⋅ 10 + 10 =<br />
= ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ + =<br />
3 3 2 2<br />
0.1 x 3 0.1 x 10 0.3 x 100 1000<br />
= + ⋅ ⋅ ⋅ + ⋅ ⋅ + =<br />
3 2<br />
0.001x 3 0.01 10 x 0.3 100 x 1000<br />
= + + +<br />
3 2<br />
0.001x 0.3x 30x<br />
1000<br />
( )<br />
II način da decimalni broj 0.1 pretvorimo u razlomak<br />
3 ⎛ 1 ⎞<br />
0.1 + 10 = ⎜ x+ 10⎟<br />
=<br />
⎝10<br />
⎠<br />
( x )<br />
3<br />
3 2<br />
⎛ 1 ⎞ ⎛ 1 ⎞ 1 2 3<br />
= x + ⋅ x ⋅ + ⋅ x⋅ + =<br />
⎜ ⎟ 3 ⎜ ⎟ 10 3 10 10<br />
⎝10 ⎠ ⎝10 ⎠ 10<br />
3<br />
1 3 1 2 1<br />
3<br />
= ⋅ x + 3⋅ ⋅x ⋅ 10 + 3⋅ ⋅100⋅ x+ 1000 =<br />
10 100 10<br />
1 3 3⋅⋅ 1 10 2 3⋅10⋅10<br />
= x + ⋅ x + ⋅ x + 1000 =<br />
1000 10⋅10 10<br />
1 3 2<br />
x 30 1000<br />
1000 10 x x<br />
3<br />
= +<br />
+ + 7<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
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40. Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
3 3 2 2 3<br />
3<br />
( x+ ) = ( x) + ⋅( x) ⋅ + ⋅ x⋅ + =<br />
28) 2 0.1 2 3 2 0.1 3 2 0.1 0.1<br />
= ⋅ x + ⋅<br />
3 3 2 2<br />
2 3 2<br />
⋅x<br />
⋅ 0.1 + 6⋅x⋅ 0.01 + 0.001 =<br />
3 2<br />
= 8x + 3⋅4⋅0.1⋅ x + 6⋅0.01⋅ x + 0.001 =<br />
= x + x + x +<br />
3 2<br />
8 1.2 0.06 0.001<br />
→ 0.1 = 0.1⋅0.<br />
1⋅ 0.1 = 0.001<br />
( x y) ( x) ( x) y x ( y) ( y)<br />
3 3 2 2 3<br />
29) 0.1 − 0.2 = 0.1 − 3⋅ 0.1 ⋅ 0.2 + 3⋅0.1 ⋅ 0.2 − 0.2 =<br />
0.1 x 3 0.1 x 0.2 y 0.3 x 0.2 y 0.2 y<br />
3 3 2 2 2 2 3 3<br />
= ⋅ − ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ − ⋅ =<br />
3 2<br />
x x y<br />
= 0.001 − 3⋅0.01⋅0.2⋅ ⋅<br />
0.3 0.04 x y 0.008 y<br />
2 3<br />
+ ⋅ ⋅ ⋅ − =<br />
= 0.001x − 0.006x y + 0.012xy − 0.008 y<br />
3 2 2 3<br />
3<br />
2 3 ⎛<br />
2⎞<br />
30) 5x 0.4y 5x y<br />
0.4<br />
4 4 2<br />
+ = ⎜ + ⎟ = → pretvorimo u: =<br />
⎝ 10 ⎠<br />
10 5<br />
( )<br />
3<br />
⎛ 2 2 ⎞<br />
= 5x+ y =<br />
⎜<br />
⎝<br />
5<br />
⎟<br />
⎠<br />
2 3<br />
3 2 2 2 ⎛2 2⎞ ⎛2<br />
2⎞<br />
( ) ( )<br />
= 5x + 3⋅ 5x ⋅ y + 3⋅5x⋅ ⎜ y ⎟ + ⎜ y ⎟ =<br />
5 ⎝5 ⎠ ⎝5<br />
⎠<br />
2<br />
3<br />
3 3 2 2 2 2 2 2 2 2 2 3<br />
= 5 ⋅ x + 3⋅5 ⋅x ⋅ ⋅ y + 15⋅x⋅<br />
⋅ ( y<br />
2 ) + ⋅<br />
3 ( y ) =<br />
5 5 5<br />
3 2 2 2 4 2⋅2 8 2⋅3<br />
= 125x + 3⋅25⋅ ⋅x ⋅ y + 15⋅ ⋅x⋅ y + ⋅ y =<br />
5 25 125<br />
125x<br />
3<br />
= +<br />
35 ⋅ ⋅5⋅2<br />
2 2 35 ⋅ ⋅4<br />
4 8 6<br />
⋅ xy + ⋅ xy + y =<br />
5<br />
5 ⋅5<br />
125<br />
12 8<br />
= 125x + 30x y + xy + y<br />
5 125<br />
3 2 2 4 6<br />
( xy z ) ( xy ) ( xy ) z xy ( z ) ( z )<br />
⋅ ⋅<br />
x ( y ) 3 x ( y ) z 3xy z z<br />
2 3 3 2 3 2 2 3 2 3 2 3 3<br />
31) + = + 3⋅ ⋅ + 3⋅ ⋅ + =<br />
3 2 3 2 2 2 3 2 3 2 3 3<br />
= ⋅ + ⋅ ⋅ ⋅ + ⋅ + =<br />
x y 3x y z 3xy z z<br />
3 2⋅3 2 2⋅2 3 2 6 9<br />
= ⋅ + ⋅ ⋅ + + =<br />
= xy + 3xyz + 3xyz + z<br />
3 6 2 4 3 2 6 9<br />
8<br />
© M I M <strong>Sraga</strong> 2005<br />
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M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
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40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
( xy z ) ( xy ) ( xy ) z xy ( z ) ( z )<br />
3 3 3 2 3 2 4 3 4⋅2 4<br />
= x ⋅( y ) −3⋅x ⋅( y ) ⋅ z + 3xy ⋅z −z<br />
3 4 3 3 3 3 2 4 3 4 2 4 3<br />
32) − = −3⋅ ⋅ + 3⋅ ⋅ − =<br />
xy 3x y z 3xyz z<br />
3 3⋅3 2 3⋅2 4 3 8 12<br />
= − ⋅ ⋅ + − =<br />
= xy − 3xyz + 3xyz −z<br />
3 9 2 6 4 3 8 12<br />
⋅3<br />
=<br />
( xy z) ( xy) ( xy) z xy ( z) ( z)<br />
( ) ( ) 3 ( ) ( ) 3<br />
4 2 3 3 4 2 3 4 2 2 3 4 2 3 2 3 3<br />
33) − = −3⋅ ⋅ + 3⋅ ⋅ − =<br />
4 3 2 3 4 2 2 2 3 4 2 3⋅2 3⋅3<br />
= ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ − =<br />
x y x y z x y z z<br />
x y 3 x y z 3x y z z<br />
43 ⋅ 23 ⋅ 42 ⋅ 22 ⋅ 3 4 2 6 9<br />
= ⋅ − ⋅ ⋅ ⋅ + − =<br />
= x y − 3x y z + 3x y z<br />
12 6 8 4 3 4 2 6<br />
− z<br />
9<br />
4 2 3 3 4 3 4 2 2 3 4 2 3 2 2 3 3<br />
( ab + a b ) = ( ab ) + ⋅( ab ) ⋅ a b + ⋅ab ⋅ ( a b ) + ( a b ) =<br />
= a ⋅ ( b ) + 3⋅a ⋅( b ) ⋅a⋅2⋅ b + 3⋅a⋅b ⋅( a ) ⋅ ( b ) + ( a ) ⋅( b )<br />
34) 3 3<br />
3 4 3 2 4 2 3 4 2 2 3 2 2 3 3 3<br />
3 4⋅3 2 1 4⋅2 3 1 2⋅2 4 3⋅2 2⋅3 3⋅3<br />
= a ⋅ b + ⋅a ⋅a ⋅b ⋅ b + ⋅a ⋅a ⋅b ⋅ b + a ⋅ b =<br />
3 12 2+ 1 8+ 3 1+ 4 4+<br />
6 6 9<br />
= a b + 3⋅a ⋅ b + 3⋅a ⋅ b + a b =<br />
= ab + 3ab<br />
3 12 3 11<br />
3 3<br />
+ 3ab<br />
+ ab<br />
5 10 6 9<br />
=<br />
( ) ( ) ( )<br />
( ) ( ) ( ) ( )<br />
2 3 2 2 3 3 2 3 2 2 3 2 3<br />
35) 2xy+ 3 = 2xy + 3⋅ 2xy ⋅ 3+ 3⋅2xy⋅ 3 + 3 =<br />
3 2 3 3 3 2 2 2 3 2 2 3<br />
= 2 ⋅ ⋅ + 3⋅2 ⋅ ⋅ ⋅ 3+ 6⋅9⋅ + 27 =<br />
x y x y x y<br />
23 ⋅ 33 ⋅ 22 ⋅ 32 ⋅ 2 3<br />
= 8⋅ ⋅ + 3⋅4⋅3⋅ ⋅ + 54 + 27 =<br />
x y x y x y<br />
= xy+ xy + xy+<br />
6 9 4 6 2 3<br />
8 36 54 27<br />
2 3 2 2 2 3 3 2 3 2 2 2 3 2 2 2 3<br />
( xy− xy) = ( xy) − ⋅( xy)<br />
⋅ xy + ⋅xy⋅( xy) − ( 3xy<br />
) =<br />
( x ) ( y ) ( x ) ( y ) x y x y x ( y )<br />
36) 3 3 3 3 3<br />
= ⋅ −33 ⋅ ⋅ ⋅ ⋅ ⋅ + 3⋅ ⋅ ⋅3⋅ ⋅ −3<br />
= x ⋅y −9⋅x ⋅x ⋅y ⋅ y + 3⋅9⋅x ⋅x ⋅y ⋅y −27⋅x ⋅y<br />
2⋅3 3⋅3 2⋅2 1 3⋅2 2 2 2 3 2⋅2 3 2⋅3<br />
6 9 4+ 1 6+ 2 2+ 2 3+<br />
4 3 6<br />
= − ⋅ ⋅ + ⋅ ⋅ − =<br />
xy 9 x y 27 x y 27xy<br />
= xy − 9xy+ 27xy −27xy<br />
6 9 5 8 4 7 3 6<br />
( y )<br />
2 3 3 3 2 2 3 2 2 2 3 2 2 2 2 3<br />
⋅ 3 ⋅ 2 3 =<br />
=<br />
x<br />
9<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
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40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
2 3 2 3 2 3 2 2 3 2 2 3 2 2 3 2 3<br />
( a + ab ) = ( a ) + ⋅( a ) ⋅ ab + ⋅a ⋅ ( ab ) + ( ab ) =<br />
23 22 3 2 2<br />
a ⋅<br />
⋅<br />
= + 32 ⋅ ⋅a ⋅a ⋅ b + 3⋅ a ⋅2 ⋅( a ) ⋅ ( b ) + 2 ⋅( a ) ⋅( b )<br />
37) 2 3 2 3 2 2<br />
2 3 2 2 2 3 3 3 2 3<br />
6 4+ 3 2 2 3⋅2 2⋅2 3⋅3 2⋅3<br />
= + 6⋅ ⋅ + 3⋅4⋅ ⋅ ⋅ + 8⋅ ⋅ =<br />
a a b a a b a b<br />
6 7 2 2+<br />
6 4 9 6<br />
= + 6 + 12⋅ ⋅ + 8 =<br />
a a b a b a b<br />
= a + 6a b + 12a b + 8a b<br />
6 7 2 8 4 9 6<br />
=<br />
2 3 5 3 2 3 2 2 3 5 2 3 5 2 3 5 3<br />
( a − bc ) = ( a ) − ⋅( a ) ⋅ bc + ⋅ a ⋅( bc ) − ( bc ) =<br />
3 2 3 2 2 2 3 5 2 2 3 2 5 2 3 3<br />
= 3 ⋅( a ) −3⋅3 ⋅2⋅( a ) ⋅b ⋅ c + 9⋅a<br />
⋅2<br />
⋅( b ) ⋅( c ) − 2 ⋅( b )<br />
38) 3 2 3 3 3 2 3 3 2 2<br />
2⋅3 2⋅2 3 5 2 3⋅2 5⋅2 3⋅2 5⋅2<br />
= 27⋅ −3⋅9⋅2⋅ ⋅ ⋅ + 9⋅4⋅ ⋅ ⋅ − 8⋅ ⋅ =<br />
a a b c a b c b c<br />
= 27a − 54a b c + 36a b c −8b c<br />
6 4 3 5 2 6 10 6 10<br />
( c )<br />
⋅ =<br />
2 5 2<br />
2 3 4 3 2 3 2 2 3 4 2 3 4 2 3 4 3<br />
( x + yz) = ( x) + ⋅( x) ⋅ yz + ⋅ x⋅ ( yz) + ( yz)<br />
=<br />
= 5 ⋅ ( x ) + 3⋅5 ⋅( x ) . y ⋅ z + 15⋅x ⋅( y ) ⋅ ( z ) + ( y ) ⋅( z )<br />
39) 5 5 3 5 3 5<br />
3 2 3 2 2 2 3 4 2 3 2 4 2 3 3 4 3<br />
2⋅3 2⋅2 3 4 2 3⋅2 4⋅2 3⋅3 4⋅3<br />
= 125⋅ x + 3⋅25⋅x ⋅y ⋅ z + 15⋅x ⋅y ⋅ z + y ⋅ z =<br />
= 125x + 75x y z + 15x y z + y z<br />
6 4 3 4 2 6 8 9 12<br />
=<br />
3 3 2<br />
1 3 ⎛1 ⎞ ⎛1⎞ ⎛1⎞<br />
1 2 3<br />
−<br />
( )<br />
40) x + x = ⎜ + x⎟ = ⎜ ⎟ + 3⋅⎜ ⎟ ⋅ x+ 3⋅ ⋅ x + x =<br />
⎝ x ⎠ ⎝ x⎠ ⎝ x⎠<br />
x<br />
1 1<br />
x x<br />
3 2 2<br />
3<br />
= + 3⋅ ⋅ x+ 3⋅ + x =<br />
3 2<br />
1<br />
1⋅<br />
x<br />
= 3<br />
3<br />
x<br />
+ ⋅ x ⋅ x<br />
x<br />
x<br />
+ 3⋅<br />
x⋅<br />
x<br />
x<br />
1 3<br />
= + + 3x<br />
+ x<br />
3<br />
x x<br />
3<br />
3<br />
+ x =<br />
10<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
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40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
3 3 2<br />
2 3 ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ 1 2 3<br />
2 2 2 2<br />
−<br />
( )<br />
41) x − x = ⎜ − x⎟ = ⎜ ⎟ −3⋅⎜ ⎟ ⋅ x+ 3⋅ ⋅x − x = → x<br />
⎝ x ⎠ ⎝ x ⎠ ⎝ x ⎠ x<br />
3<br />
= 1<br />
( x )<br />
1<br />
− 3⋅ ⋅ x + 3⋅<br />
x<br />
( x )<br />
2 3 2 2<br />
2 2<br />
3<br />
2<br />
x<br />
3<br />
− =<br />
1 1<br />
3<br />
= − 3⋅ ⋅ x + 3⋅1− x =<br />
23 ⋅<br />
22 ⋅<br />
x x<br />
1 x<br />
3<br />
x 1⋅<br />
x<br />
= − 3⋅ + 3 − x = → 3⋅ = 3⋅<br />
6 4 4<br />
x x x x⋅<br />
x<br />
1 3<br />
= + + 3 + x<br />
6 3<br />
x x<br />
x<br />
−2<br />
1<br />
=<br />
2<br />
x<br />
1 3<br />
= 3⋅ =<br />
x x<br />
3 3 3<br />
42)<br />
3 2 3 ⎛ 1 2⎞<br />
x + x = ⎜ + x =<br />
3 ⎟<br />
⎝ x ⎠<br />
−<br />
( )<br />
( x ) ( x )<br />
3<br />
3 2<br />
2 2 2 2 3<br />
3 x 3 x x<br />
3 3 3<br />
⎛ 1 ⎞ ⎛ 1 ⎞ 1<br />
= ⎜ ⎟ + ⋅⎜ ⎟ ⋅ + ⋅ ⋅ ( ) + ( ) =<br />
⎝ x ⎠ ⎝ x ⎠ x<br />
3 2<br />
1 1 2 3 2⋅2 2⋅3<br />
3 x x x<br />
3 3 3 2<br />
3<br />
= + ⋅ ⋅ + ⋅ + =<br />
x<br />
2 4 4<br />
1 31 ⋅⋅x<br />
3⋅x<br />
6<br />
3⋅x<br />
= + + + x = → vidi : pod<br />
33 ⋅ 32 ⋅<br />
() 1<br />
3 3<br />
x x x<br />
x<br />
+<br />
1 3⋅x 3⋅x 6<br />
3⋅x<br />
= + + + x =<br />
→ vidi : pod ( 2)<br />
9 6 3 6<br />
x x x x<br />
2 3 1<br />
1 3⋅x 3⋅x ⋅x<br />
6<br />
= + + + x =<br />
9 2+<br />
4 3<br />
x x x<br />
2<br />
1 3⋅<br />
x<br />
= +<br />
9 2<br />
x x ⋅x<br />
2 3 1 2<br />
4<br />
3<br />
3⋅<br />
x ⋅x<br />
+<br />
3<br />
x<br />
1 3<br />
= + + 3x<br />
+ x<br />
9 4<br />
x x<br />
6<br />
1<br />
x<br />
6<br />
+ =<br />
() 1<br />
( 2)<br />
3⋅x 3⋅x 3⋅x ⋅x 3⋅<br />
x<br />
= = =<br />
3 3 3<br />
x x x x<br />
4 3+<br />
1 3 1 1 3<br />
2 2 2 1 2<br />
3⋅x 3⋅x 3⋅x 3⋅<br />
x<br />
= = =<br />
x x x x ⋅x<br />
6 2+<br />
4 2 4 2<br />
⋅<br />
1<br />
x<br />
3<br />
1<br />
4<br />
⋅x<br />
1<br />
1<br />
31 ⋅⋅x<br />
= = 3x<br />
1<br />
31 ⋅ 3<br />
= =<br />
4 4<br />
1⋅<br />
x x<br />
11<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
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40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
m n m m n m n n<br />
( ) ( ) ( ) ( ) ( )<br />
3 3 2 2 3<br />
− = − ⋅ ⋅ + ⋅ ⋅ + =<br />
43) 2 2 2 3 2 2 3 2 2 2<br />
m⋅3 m⋅2 n m n⋅2 n⋅3<br />
= − ⋅ ⋅ + ⋅ ⋅ + =<br />
2 3 2 2 3 2 2 2<br />
= −<br />
3m<br />
2m n m 2n 3n<br />
2 3⋅ 2 ⋅ 2 + 3⋅ 2 ⋅ 2 + 2 =<br />
3m 2m+ n m+<br />
2n 3n<br />
= 2 −3⋅ 2 + 3⋅ 2 + 2 = → možemo ostaviti u ovom obliku ili nastaviti dalje:<br />
2 3 2 3 2 2<br />
3⋅ m 2m+ n m+ 2n 3⋅n<br />
= − ⋅ + ⋅ + =<br />
3 2 + + 2 3<br />
( 2 ) 3 2 3 2 ( 2 )<br />
m<br />
= 8 −3⋅2<br />
<br />
<br />
primjeni primjeni<br />
a ⋅ a = a a ⋅ a = a<br />
m n m+ n m n m+<br />
n<br />
m m n m n n<br />
= − ⋅ + ⋅ + =<br />
+ 32 ⋅ + 8<br />
2m+ n m+<br />
2n n<br />
n n 3 n 3 n 2 n n n 2 n 3<br />
n<br />
( − ) = ( ) − ⋅( ) ⋅ + ⋅ ⋅( ) − ( ) = → vidi : pod ()( 1 )<br />
44) 3 2 3 3 3 2 3 3 2 2 3<br />
n⋅3 n⋅2 n n n⋅2 n⋅3<br />
= 3 −3⋅3 ⋅ 2 + 3⋅3 ⋅2 − 2 =<br />
3⋅n 2⋅n n n 2⋅n 3⋅n<br />
= 3 −3⋅3 ⋅ 2 + 3⋅3 ⋅2 − 2 =<br />
3 2 2 3<br />
( 3 ) 3 ( 3 ) 2 3 3 ( 2 ) ( 2 )<br />
n n n n n n<br />
= − ⋅ ⋅ + ⋅ ⋅ − =<br />
n n n n n n<br />
= 27 −3⋅ 9<br />
<br />
⋅ 2 + 3⋅3 ⋅4 − 8 =<br />
n<br />
n<br />
( )<br />
( )<br />
a ⋅ b = ab<br />
n<br />
n<br />
n<br />
= 27 −3⋅ 9⋅ 2 + 3⋅ 3⋅4 − 8 =<br />
n<br />
( )<br />
= 27 −3⋅ 18 + 3⋅12 −8<br />
n n n n<br />
n<br />
3<br />
n n 3 n 3 n 2 n n n 2 n 3<br />
( + ) = ( ) + ⋅( ) ⋅ + ⋅ ⋅ ( ) + ( ) = → vidi : pod ( 2)<br />
( ) 3<br />
45) 2 3 2 3 2 3 3 2 3 3 2<br />
n⋅3 n⋅2 n n n⋅2 n⋅3<br />
= 2 + 3⋅2 ⋅ 3 + 3⋅2 ⋅ 3 + 3 =<br />
3⋅n 2⋅n n n 2⋅n 3⋅n<br />
= 2 + 3⋅2 ⋅ 3 + 3⋅2 ⋅ 3 + 3 =<br />
3 2 2 3<br />
( 2 ) 3 ( 2 ) 3 3 2 ( 3 ) ( 3 )<br />
n n n n n n<br />
= + ⋅ ⋅ + ⋅ ⋅ + =<br />
n n n n n n<br />
= 8 + 3⋅4 ⋅ 3 + 3⋅2 ⋅ 9 + 27 =<br />
n<br />
n<br />
= 8 + 3⋅( 4⋅ 3)<br />
+ 3⋅( )<br />
= 8 + 3⋅ 12 + 3⋅ 18 + 27<br />
n n n n<br />
n<br />
n<br />
29 ⋅ + 27 =<br />
n<br />
() ( ) 3 n⋅ 3 3 ⋅n ( 3 n n n<br />
1 3 3 3 3 ) 27 ( 2) ( 2 ) 3 n⋅ 2 3 2<br />
3 ⋅n<br />
= = = = = = = (<br />
2 = 8<br />
3n n<br />
)<br />
n<br />
12<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja + kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
n n n n n n n n<br />
( a b ) ( a ) ( a ) b a ( b ) ( b )<br />
46) 3 3<br />
3 3 2 2 3<br />
+ = + ⋅ ⋅ + ⋅ ⋅ + =<br />
n⋅3 n⋅2 b n n⋅2 n⋅3<br />
= a + 3⋅a ⋅ b + 3⋅a ⋅ b + b =<br />
3n<br />
= a +<br />
3a b + 3a b + b<br />
2n n n 2n 3n<br />
n n+ n n n+ n n+ n+<br />
( a a ) ( a ) ( a ) a a ( a ) ( a )<br />
1 3 3 2 1 1 2 1 3<br />
− = − ⋅ ⋅ + ⋅ ⋅ − =<br />
47) 3 3<br />
( n+ 1) ⋅ 2 ( n+ 1) ⋅ 3 ( n+ 1) ⋅3 3⋅ ( n+<br />
1)<br />
= a −3⋅a ⋅ a + 3⋅a ⋅a − a = → a = a = a<br />
n⋅3 n⋅ 2 n+ 1 n 3n+<br />
3<br />
3n 2n n+ 1 n 2n+ 2 3n+<br />
3<br />
= a −3⋅a ⋅ a + 3⋅a ⋅a − a = ⎪<br />
3n 2n+ n+ 1 n+ 2n+ 2 3n+<br />
3<br />
= a −3⋅ a + 3⋅a − a = ⎪⎭<br />
= a − 3a + 3a −a<br />
3n 3n+ 1 3n+ 2 3n+<br />
3<br />
⎫<br />
⎬<br />
→ a ⋅ a = a<br />
= a<br />
n 2n+ 2 n+<br />
2n+ 2 3n+<br />
2<br />
n+ n− n+ n+ n− n+ n− n−<br />
( a a ) ( a ) ( a ) a a ( a ) ( a )<br />
1 1 3 1 3 1 2 1 1 1 2 1 3<br />
48) + = + 3⋅ ⋅ + 3⋅ ⋅ + =<br />
( n ) ( n ) n− 1 n+<br />
1 ( n ) ( n )<br />
+ 1 ⋅ 3 + 1 ⋅2 −1 ⋅2 −1 ⋅3<br />
= a + 3⋅a ⋅ a + 3⋅a ⋅ a + a =<br />
3n+ 3 2n+ 2 n− 1 n+ 1 2n−2 3n−3<br />
= + ⋅ ⋅ + ⋅ ⋅ + =<br />
a 3 a a 3 a a a<br />
= a + 3⋅ a + 3⋅ a + a<br />
3n+ 3 2n+ 2+ n− 1 n+ 1+ 2n−2 3n−3<br />
3n+ 3 2n+ n+ 2− 1 n+ 2n+ 1−2 3n−3<br />
= a + 3⋅ a + 3⋅ a + a =<br />
= a + 3a + 3a + a<br />
3n+ 3 3n+ 1 3n−1 3n−3<br />
=<br />
( ) ( )<br />
3 3 3 2 2 3 3 2 2 3<br />
49) a− 1 + a+ 1 = a −3⋅a ⋅ 1+ 3⋅a⋅1 − 1 + a + 3⋅a ⋅ 1+ 3⋅a⋅ 1 + 1 =<br />
3 2 3 2<br />
= a − 3a + 3a− 1+ a + 3a + 3a+ 1=<br />
3 3 2 2<br />
= a + a − 3a + 3a + 3a+ 3a− 1+ 1=<br />
<br />
= +<br />
3<br />
2a<br />
6<br />
a<br />
( )<br />
2<br />
= − 3+ 3 a = 0<br />
( a−<br />
)<br />
50) 1<br />
3<br />
3 3 2 2 3 3 2 2 3<br />
−( a+ 1) = a −3⋅a ⋅ 1+ 3⋅a⋅1 −1 − ( a + 3⋅a ⋅ 1+ 3⋅a⋅ 1 + 1 ) =<br />
( 3 3) a<br />
2<br />
( )<br />
3 2 3 2<br />
= a − 3a + 3a−1− a + 3a + 3a+ 1 =<br />
3 2 3 2<br />
= a − 3a + 3a−1−a −3a −3a− 1=<br />
3 3 2 2<br />
= a −a −<br />
3a − 3a + 3a−3a−1− 1=<br />
= 0 = − −<br />
= 0<br />
=− −<br />
2<br />
6a<br />
2<br />
13<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
+ kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2 3<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+ 3AB −B<br />
3 3 3 2 2 3 3 2 2 3<br />
( x y) ( x y) x x y xy y ( x x y xy y )<br />
51) + − − = + 3 + 3 + − − 3 + 3 − =<br />
= + 3 + 3 + − + 3 −3<br />
3 2 2 3 3 2<br />
2 3<br />
x x y xy y x x y xy<br />
3 3 2 2 2 2 3 3<br />
= x − x + 3x y+ 3x y+ 3xy − 3xy + y + y =<br />
<br />
= 0<br />
= + = 0<br />
= + y<br />
= 6xy+<br />
2y<br />
2 3<br />
( 3 3) x y<br />
( 1 1)<br />
2 3<br />
+ y =<br />
( ) ( )<br />
3 3 3 2 2 3 3 2 2 3<br />
52) x+ y + x− y = x + 3x y+ 3xy + y + x − 3x y+ 3xy − y =<br />
3 3 2 2 2 2 3 3<br />
= x + x + 3x y− 3x y+ 3xy + 3xy + y − y =<br />
= 2x<br />
+ 6xy<br />
3 2<br />
<br />
<br />
= 0 = 0<br />
( x )<br />
3<br />
( ) ( ) (<br />
( ) (<br />
3 3 2 2 3 3 2<br />
53) 2 + 3 −3<br />
x− 1 = 2⋅ x + 3⋅x ⋅ 3+ 3⋅x⋅ 3 + 3 −3⋅ x −3⋅x ⋅ 1+ 3⋅x⋅1<br />
( 2 3) x ( 18 9)<br />
3 2<br />
)<br />
( )<br />
3 2 3 2<br />
= 2⋅ x + 9x + 27x+ 27 −3⋅ x − 3x + 3x− 1 =<br />
= ⋅ + ⋅ + ⋅ + ⋅ − ⋅ − ⋅ − − ⋅<br />
3 2 3 2<br />
2 x 2 9x 2 27x 2 27 3 x 3 3x<br />
3 3<br />
3 2 3 2<br />
= 2x + 18x + 54x+ 54− 3x + 9x − 9x+ 3=<br />
3 3 2 2<br />
= 2 x − 3x + 18x + 9x + 54x−9x<br />
+ 54+ 3 =<br />
= − = +<br />
3 2<br />
=− x + 27x + 45x+<br />
57<br />
x<br />
( 54 9)<br />
= −<br />
x<br />
)<br />
− 1 =<br />
2 3<br />
( )<br />
x−3⋅ − 1 =<br />
3 3 3 2 2 3 3 2<br />
( x ) ( x) ( x x x ) ( x<br />
3 2 2 3<br />
3 ( x 3x 3x 1) 5 ( 8 12x 6x x )<br />
54) 3 + 1 −5 2− = 3⋅ + 3⋅ ⋅ 1+ 3⋅ ⋅ 1 + 1 −5⋅ 2 −3⋅2 ⋅ + 3⋅2⋅x<br />
= ⋅ + + + − ⋅ − + − =<br />
( )<br />
3 2 2<br />
= ⋅ + ⋅ + ⋅ + ⋅ − ⋅ − ⋅ − − ⋅ −<br />
3 x 3 3x 3 3x 3 1 5 8 5 12x 5 6x<br />
3 2 2 3<br />
= + + + − + − + =<br />
3x 9x 9x 3 40 60x 30x 5x<br />
= 3x<br />
+ 5x + 9x − 30x + 9x+ 60x+ 3− 40 =<br />
3 3 2 2<br />
= − + −<br />
3 2<br />
8x 21x 69x<br />
37<br />
2 3<br />
)<br />
− x =<br />
3<br />
( x )<br />
5⋅ − =<br />
14<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
+ kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+<br />
3AB<br />
−B<br />
2 3<br />
( x y) ( x y)<br />
3 3<br />
55) 3 2 + − 4 + 2 =<br />
3 2 2 3 3 2<br />
( x) ( x) y x y y x x y x ( y)<br />
= 3⋅ ⎡ 2 + 3⋅ 2 ⋅ + 3⋅2 ⋅ + ⎤−4⋅ ⎡ + 3⋅ ⋅ 2 + 3⋅ ⋅ 2<br />
⎣ ⎦ ⎣<br />
( ) (<br />
3 2 2 3 3 2 2 3<br />
( x x y xy y ) ( x x y xy y )<br />
3 2 x 3 4x y 6xy y 4 x 6x y 3 x 4y 2 y<br />
3 3 2 2 3 3 2 2 3 3<br />
= ⋅ + ⋅ ⋅ + + − ⋅ + + ⋅ ⋅ + =<br />
= 3⋅ 8 + 12 + 6 + −4⋅ + 6 + 12 + 8 =<br />
3 2 2 3 3 2 2<br />
38 312 36 3 4 46 412 48<br />
= ⋅ x + ⋅ x y+ ⋅ xy + ⋅y − ⋅x − ⋅ x y− ⋅ xy − ⋅<br />
24x 36x y 18xy 3y 4x 24x y 48xy 32y<br />
3 2 2 3 3 2 2 3<br />
= + + + − − − − =<br />
3 3<br />
24 4 36<br />
= x − x + x<br />
= 20x 12x y−30xy −29y<br />
3 2 2 3<br />
y− 24x y+ 18xy − 48xy + 3y − 32y<br />
=<br />
2 2 2 2 3 3<br />
)<br />
y<br />
3<br />
( 2y)<br />
2 3<br />
=<br />
+<br />
⎤ =<br />
⎦<br />
3 2 3 2 2 3 2 2<br />
( x ) x( x ) x x x x ( x x )<br />
3 2 2<br />
= x − 3x + 3x−1−x⋅ ( x + 4x+ 4)<br />
=<br />
56) −1 − + 2 = −3⋅ ⋅ 1+ 3⋅ ⋅1 −1 − ⋅ + 2⋅ ⋅ 2+ 2 =<br />
= − 3 + 3 −1− ⋅ − ⋅4 − ⋅ 4 =<br />
3 2 2<br />
x x x x x x x x<br />
= − 3 + 3 −1− −4 − 4 =<br />
3 2 1+ 2 1+<br />
1<br />
x x x x x x<br />
3 2 3 2<br />
= x − 3x + 3x−1−x −4x − 4x<br />
=<br />
3 3 2 2<br />
= x −x −3x − 4x + 3x−4x− 1=<br />
2<br />
7x<br />
x 1<br />
=− − −<br />
2 3 2 2 3 2<br />
( ) ( ) ( ) (<br />
2 3 2<br />
x ( x 4x 4) ( x 6x 3 4 x 8)<br />
57) x x−2 − x+ 2 = x⋅ x −2⋅x⋅ 2+ 2 − x + 3⋅x ⋅ 2+ 3⋅x⋅ 2 + 2<br />
= ⋅ − + − + + ⋅ ⋅ + =<br />
( )<br />
2 3 2<br />
= xx ⋅ + x⋅ − 4x+ x⋅4−x−6x −12x− 8 =<br />
1+ 2 1+<br />
1 3 2<br />
= x − 4x + 4x−x −6x −12x− 8 =<br />
3 3 2 2<br />
= x −x −<br />
4x − 6x + 4x−12x− 8 =<br />
( ) x<br />
2<br />
= 0 =−4−6<br />
2<br />
10 8 8<br />
=− x − x−<br />
( 412)<br />
= −<br />
x<br />
2 3<br />
)<br />
=<br />
15<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com
M-1-Kub zbroja i kub razlike<br />
Rješenja<br />
+ kompletni postupak<br />
40.<br />
Koristimo se formulama <strong>za</strong>:<br />
kub zbroja i kub razlike<br />
( ) ( )<br />
3 3 2 2 3 3 3 2 2<br />
A+ B = A + 3A B+ 3AB + B A− B = A − 3A B+<br />
3AB<br />
3<br />
−B<br />
3 3 3 3<br />
( x+ y) − ( x− y) − ( x + y ) =<br />
3 2 2 3 3 2 2 3 3 3<br />
= 2⋅ ( x + 3x y+ 3xy + y ) −5⋅( x − 3x y+ 3xy − y ) −x − y =<br />
3 2 2 3 3 2 2<br />
= 2⋅ x + 2⋅ 3 23 2 5 5( 3 ) 53 5(<br />
58) 2 5<br />
xy+ ⋅ xy + ⋅y− ⋅x− ⋅ − xy− ⋅ xy − ⋅ −y<br />
3 2 2 3 3 2 2 3 3 3<br />
= 2x + 6x y+ 6xy + 2y − 5x + 15x y− 15xy + 5y −x − y =<br />
3 3 3 2 2 2 2 3 3 3<br />
= 2x −5x − x + 6x y+ 15x y+ 6xy − 15xy + 2y + 5y − y =<br />
<br />
( 2 5 1) 3<br />
( 615) ( 615) ( 251)<br />
= − − x = + x y = − xy = + − y<br />
=−4x + 21x y− 9xy + 6y<br />
3 2 2 3<br />
2 2 3<br />
)<br />
−x− y =<br />
3 3 3<br />
16<br />
© M I M <strong>Sraga</strong> 2005<br />
www mim sraga com