Übungen zu den Potenzgesetzen
Übungen zu den Potenzgesetzen
Übungen zu den Potenzgesetzen
Erfolgreiche ePaper selbst erstellen
Machen Sie aus Ihren PDF Publikationen ein blätterbares Flipbook mit unserer einzigartigen Google optimierten e-Paper Software.
<strong>Übungen</strong> <strong>zu</strong> <strong>den</strong> <strong>Potenzgesetzen</strong><br />
Multiplikation und Division von Potenzen mit gleicher Basis<br />
1. a)<br />
e)<br />
2. a)<br />
e)<br />
3. a)<br />
e)<br />
4 5 2<br />
3 ⋅ 3 ⋅ 3<br />
3 5 2<br />
b) 12 ⋅ 12 ⋅ 12<br />
3 2<br />
c) x ⋅ x ⋅ x<br />
3 5 2 7<br />
k ⋅ k ⋅ m ⋅ m<br />
5<br />
f) x ⋅<br />
3 2<br />
y ⋅ x ⋅ y<br />
2 3<br />
g) a ⋅ b ⋅ b ⋅ a<br />
2 n<br />
x ⋅ x<br />
a<br />
b)<br />
5 2x<br />
2m<br />
m<br />
⋅ a<br />
f) z ⋅ z<br />
3 m−<br />
2<br />
x ⋅ x<br />
b)<br />
3<br />
b b<br />
m ⋅ c) y y<br />
a ⋅ d)<br />
g)<br />
5 x−<br />
7<br />
a ⋅ a<br />
c)<br />
3m<br />
2m<br />
a ⋅ a ⋅<br />
a<br />
m<br />
d)<br />
h)<br />
h)<br />
2m m−<br />
1<br />
y ⋅ y<br />
d)<br />
2x x+<br />
1 3x<br />
− 4 m+<br />
2 3m−<br />
4 2m<br />
+ 3<br />
p−<br />
1 3 p + 4 5 p−<br />
8<br />
a ⋅ a ⋅ a f) x ⋅ x ⋅ x g) ⋅ z ⋅ z<br />
z h)<br />
3 5<br />
d ⋅ d ⋅<br />
4 6<br />
p ⋅ q<br />
4<br />
d<br />
⋅ p ⋅ q<br />
m m<br />
x ⋅ x<br />
m<br />
x<br />
y<br />
3x<br />
⋅ m<br />
⋅ x<br />
4x<br />
p−<br />
4 p + 2<br />
4. a) x²(x³ + x 4 ) b) a³(a 5 + a 4 ) c) 3b³(4b² - 5b 5 )<br />
d) a m (a m+1 – a 3m-1 ) e) y 2a (y 3a+1 – y a-4 ) f) x n-3 (x 5 + x 4 )<br />
⋅ y<br />
⋅ m<br />
5<br />
2x<br />
⋅ y<br />
m−<br />
2 2m<br />
− 5 m+<br />
8<br />
5. a) (x² + x³)² b) (y 3 – y 4 )² c) (a 6 + a 4 )² d) (b 3 – b 7 )²<br />
e) (2a² + 3a³)² f) (4x 5 – 2x 6 )² g) (6d 5 – 3d 4 )² h) (3m² + 5m 7 )²<br />
6. a) (a² + a³)(a² - a³) b) (x 5 + y 4 )(x 5 – y 4 ) c) (m³ + n 5 )(m³ - n 5 )<br />
d) (3x 4 – 2y 5 )(3x 4 + 2y 5 ) e) (4y³ - 6x 7 )(4y³ + 6x 7 ) f) (3a 4 – 4b³)(3a 4 + 4b³)<br />
7. a) (a 3 + a 4 )(a² + a 5 ) b) (x² - x 5 )(x³ + x 6 ) c) (a 3 – b²)(a 5 + b³)<br />
d) (y 4 + y 5 )(y 3 – y 6 ) e) (2a 5 + 3b 3 )(2a 3 – 2b 4 ) f) (k m + k n )(k m+1 + k n+2 )<br />
8. Schreibe als Produkt von Potenzen.<br />
9. a)<br />
10. a)<br />
11. a)<br />
12. a)<br />
a) x 3+5<br />
e)<br />
5<br />
5<br />
8<br />
3<br />
a x<br />
a³<br />
x<br />
x<br />
x<br />
x<br />
v<br />
v<br />
m<br />
m−<br />
3<br />
n+<br />
3<br />
n+<br />
2<br />
7 x+<br />
4<br />
5x<br />
− 2<br />
b)<br />
b) a 3n+2<br />
8<br />
8<br />
9<br />
3<br />
x<br />
b)<br />
x<br />
y<br />
b)<br />
a<br />
a<br />
3m<br />
m−<br />
1<br />
b)<br />
f)<br />
a<br />
a<br />
x<br />
x<br />
2n<br />
− 1<br />
n−<br />
2<br />
3 p+<br />
1<br />
2 p+<br />
1<br />
c)<br />
c)<br />
c)<br />
12<br />
12<br />
c) 5 m+n<br />
13<br />
m<br />
5<br />
m<br />
x 3<br />
x<br />
z<br />
z<br />
3x<br />
x−<br />
4<br />
d)<br />
a<br />
a<br />
9<br />
5<br />
6m<br />
y<br />
3<br />
d) m<br />
y<br />
d)<br />
k<br />
k<br />
4a<br />
2a<br />
+ 3<br />
3x<br />
+ 4<br />
y<br />
c) x−<br />
2<br />
y<br />
g)<br />
z<br />
z<br />
k + 3x<br />
k + 2x<br />
d) z 5k+3m<br />
7<br />
y<br />
e) 6<br />
y<br />
e)<br />
2<br />
k m<br />
k<br />
3<br />
2b<br />
y<br />
e) b+<br />
3<br />
y<br />
13. a) (x 8 + x 6 – x 5 ) : x² b) (15a³ + 12a 6 – 3a 4 ) : 3a²<br />
c) (21b 8 – 28b 4 + 14b 5 ) : 7b³ d) (3x n+3 – 9x 2n-4 + 12x n+5 ) : 3x²<br />
e) (35y m+2 – 20y 2m+4 + 15y m+8 ) : 5y m<br />
f) (4z a+3 + 16z 2a+5 – 12z a+4 ) : 2z a<br />
14. 5<br />
15x<br />
y<br />
a)<br />
7<br />
21a<br />
b<br />
8<br />
5<br />
2x³<br />
y²<br />
: 10<br />
35a<br />
b<br />
6<br />
b)<br />
5<br />
6 p q<br />
3<br />
r²<br />
s<br />
4<br />
4<br />
3p<br />
q³<br />
: 7 5<br />
r s<br />
d)<br />
h)<br />
b<br />
b<br />
b<br />
b<br />
e) x m+4<br />
f)<br />
f)<br />
f)<br />
3m+<br />
4<br />
3m+<br />
3<br />
3m−<br />
6<br />
m−<br />
5<br />
k<br />
k<br />
9 7 5<br />
18a<br />
b 12a<br />
b³<br />
c) : 4 6<br />
35x³<br />
y²<br />
21x<br />
y<br />
23<br />
17<br />
2<br />
d p<br />
d²<br />
m<br />
m<br />
4b<br />
2b<br />
+ 7
<strong>Übungen</strong> <strong>zu</strong> <strong>den</strong> <strong>Potenzgesetzen</strong><br />
Multiplikation und Division bei Potenzen mit gleichem Exponenten<br />
1. a)<br />
e)<br />
2. a)<br />
e)<br />
3 3<br />
5 ⋅ 2<br />
b)<br />
4 4<br />
4 ⋅ 3 ⋅<br />
0,<br />
25<br />
4<br />
f)<br />
x x<br />
5 ⋅ 4<br />
b)<br />
m m<br />
a ⋅ b<br />
f)<br />
2 2<br />
8 ⋅ 3<br />
c)<br />
1<br />
6<br />
3 3<br />
0, 5 ⋅ 4<br />
d)<br />
5<br />
6<br />
18 5<br />
⋅ h)<br />
25 3<br />
6 6<br />
6 ⋅ ( ) g) ( )³ ( )³ ⋅ ( )³<br />
a a<br />
12 ⋅ 3<br />
c)<br />
k k<br />
y ⋅ z<br />
g)<br />
x+<br />
1 x+<br />
1<br />
4 ⋅ 5<br />
d)<br />
( x +<br />
8<br />
y)<br />
⋅ ( x −<br />
8<br />
y)<br />
h)<br />
5 5<br />
0, 5 ⋅ 10 ⋅<br />
2<br />
3<br />
6<br />
10<br />
0,<br />
2<br />
4 4<br />
( ) ⋅ ( ) ⋅<br />
5<br />
5<br />
− 4 − 4<br />
3 ⋅ 6<br />
m m<br />
m<br />
( a + b)<br />
⋅ ( a − b)<br />
3. a) ( − 4)³<br />
⋅ ( − 0,<br />
5)³<br />
4 1 4<br />
b) ( − 3)<br />
⋅ ( )<br />
3<br />
c) ( − 4)²<br />
⋅ ( − 1,<br />
5)²<br />
d)<br />
e) a³ ⋅ ( − b)³<br />
5<br />
f) ( − x) ⋅ ( −<br />
5 5<br />
y)<br />
⋅ z g) ( − p)² ⋅ ( − r)²<br />
⋅ s²<br />
h) ( − −<br />
4. a)<br />
5. a)<br />
24<br />
3<br />
8<br />
3<br />
4<br />
2,<br />
6<br />
e) 4<br />
1,<br />
3<br />
e)<br />
6. a)<br />
e)<br />
6<br />
a<br />
6<br />
b<br />
8² ⋅ 3²<br />
6²<br />
27a³<br />
8b³<br />
( a²<br />
− b²)³<br />
( a + b)³<br />
Potenzen von Potenzen<br />
b)<br />
f)<br />
36<br />
18<br />
5<br />
5<br />
0,<br />
4²<br />
0,<br />
5²<br />
n<br />
x<br />
b) n<br />
y<br />
15³ ⋅ 3³<br />
f)<br />
9³<br />
b)<br />
25a²<br />
b²<br />
( 4a²<br />
− 9b²)<br />
f) 5<br />
( 2a<br />
− 3b)<br />
1. a) (2³)² b) (4²) 4<br />
e) (a 5 )³ f) (x³) m<br />
2. a) (x m ) n+1<br />
e) (p 2k+1 ) 3<br />
b) (a y ) x-1<br />
f) (b n-4 ) m<br />
5<br />
c)<br />
49<br />
3<br />
7<br />
3<br />
5<br />
3<br />
g ) 3 5<br />
( )<br />
2<br />
n+<br />
1<br />
x<br />
c) n+<br />
1<br />
y<br />
( 12x)<br />
g) m<br />
( 3x)<br />
c)<br />
m<br />
27x³<br />
1000y³<br />
( 16x²<br />
− 25y²)<br />
g) n<br />
( 4x<br />
− 5y)<br />
c) (0,2²) 4<br />
g) (a m ) n<br />
c) (x a+3 ) b<br />
g) (y p ) q-2<br />
3. a) (x²y³)² b) (a³b) 5<br />
c) (d 5 e³) 3<br />
e) (a³b 4 ) n<br />
f) (3x 5 y 2 )² g) (5a²b 7 ) 4<br />
4.<br />
a)<br />
2<br />
3 2 ⎛ ⎞<br />
4a b<br />
⎜ 4 3 ⎟<br />
⎝ 2x y ⎠<br />
10<br />
m n<br />
⎛ 5a b<br />
⎞<br />
b) ⎜ 7 3 ⎟<br />
⎝ 10p q<br />
⎠<br />
6 8 4<br />
( 6a<br />
b )<br />
c) 5 2 4<br />
( 3a<br />
b )<br />
n<br />
d)<br />
h)<br />
5<br />
5<br />
( − 5)<br />
⋅ ( − 0,<br />
1)<br />
⋅<br />
4<br />
m m<br />
a ) ⋅ ( b)<br />
27 2<br />
9²<br />
1<br />
( )³<br />
8<br />
1<br />
( )³<br />
4<br />
( − x)<br />
d) 4<br />
y<br />
4<br />
n−<br />
1<br />
( 48a)<br />
h) n−<br />
1<br />
( 12a)<br />
5<br />
32y<br />
d)<br />
5<br />
100000z<br />
n+<br />
( p²<br />
− 16q²)<br />
h) n+<br />
1<br />
( p + 4q)<br />
d) (10³) 5<br />
h) (y 2a ) b<br />
d) (z n-3 ) 4<br />
h) (k 2m+3 ) n<br />
d) (m 6 n 5 ) 8<br />
h) 5(m 4 n 5 ) 4<br />
5 6<br />
( 4x<br />
y )³<br />
d) 6 3<br />
( 2x<br />
y²)<br />
1<br />
m<br />
2<br />
5
Potenzrechnung – Vermischte Aufgaben 2<br />
1. Berechne.<br />
a) 0,2 3<br />
d)<br />
3 3<br />
2 3<br />
2. Berechne.<br />
⎛ 2 ⎞<br />
a) ⎜<br />
3<br />
⎟<br />
⎝ ⎠<br />
b) 3,1 3<br />
7 3<br />
⋅ e) 4 ⋅ 4<br />
3<br />
− 2<br />
⎛ 3 ⎞<br />
d) ⎜<br />
4<br />
⎟<br />
⎝ ⎠<br />
3. Löse die Klammern auf.<br />
4 2<br />
a) x ⋅ (x<br />
3<br />
x + )<br />
b)<br />
d)<br />
3 6 4<br />
x ⋅ (x x − )<br />
e)<br />
3<br />
⎛ 1 ⎞<br />
b) ⎜<br />
5<br />
⎟<br />
⎝ ⎠<br />
3<br />
⎛ 3 ⎞<br />
e) ⎜<br />
10<br />
⎟<br />
⎝ ⎠<br />
4 3<br />
c) 3 + 4 5⋅<br />
f) (-3) 5<br />
− 2<br />
⎛ 5 ⎞<br />
c) ⎜<br />
8<br />
⎟<br />
⎝ ⎠<br />
f)<br />
3 2 4<br />
2a ⋅ (a a − )<br />
c)<br />
3 2 4<br />
3a ⋅ (a a + )<br />
f)<br />
− 1<br />
⎛ 9 ⎞<br />
⎜<br />
10<br />
⎟<br />
⎝ ⎠<br />
3<br />
(x + y)<br />
2<br />
(m n) −<br />
−<br />
4. Faktorisiere.<br />
2 4 3<br />
a) 16x y + 32x yz<br />
5 3 2<br />
40x −y z<br />
3 6<br />
b) 1,4u v −<br />
2 4<br />
0,7u v<br />
5 5<br />
2,8u −v<br />
c)<br />
4 2 5<br />
3a − 2a 6a+<br />
d)<br />
5. Vereinfache die folgen<strong>den</strong> Terme.<br />
7 3 a) 4 4 −<br />
⋅ b)<br />
d)<br />
4<br />
16a ⋅ 2a<br />
e)<br />
2 3 3 2 2<br />
9a b + 6a b 12a+ b<br />
3 5<br />
3,5x ⋅ 2x<br />
c)<br />
5 2<br />
4x 3x −<br />
⋅ f)<br />
2 2<br />
25x ⋅ y<br />
9a ⋅ 2a<br />
6. Vereinfache die folgen<strong>den</strong> Terme.<br />
3 2 a) x ⋅ x<br />
2 3<br />
b) a a<br />
5<br />
a −<br />
⋅ ⋅<br />
3<br />
c) b ⋅ b<br />
3 2<br />
d) a ⋅ b<br />
2<br />
a ⋅<br />
3<br />
b ⋅<br />
2<br />
e) a ⋅ a<br />
f) a ⋅ a<br />
− 3 3 −<br />
7. Vereinfache die folgen<strong>den</strong> Terme.<br />
2 a) x ⋅ x<br />
3 2 b) x ⋅ x<br />
5<br />
x ⋅<br />
4 3<br />
c) a ⋅ x<br />
2<br />
a ⋅<br />
2<br />
x ⋅<br />
2 2<br />
d) 2a ⋅ b<br />
3<br />
4a⋅ 4<br />
b ⋅<br />
2 − 3<br />
e) 5a ⋅ b<br />
4<br />
b ⋅<br />
3<br />
2a ⋅<br />
−<br />
f) 9 ⋅ a<br />
4 2<br />
3b⋅ 5<br />
a<br />
3<br />
b ⋅ ⋅<br />
8. Berechne.<br />
3 3<br />
x ⋅ x<br />
a)<br />
4<br />
x ⋅ x<br />
d)<br />
12a b<br />
2<br />
4ab<br />
2 2<br />
b)<br />
e)<br />
16x y<br />
2<br />
4xy<br />
x<br />
x<br />
2n+ 3<br />
3n− 4<br />
2 2<br />
c)<br />
f)<br />
5 2<br />
a ⋅ a<br />
a ⋅ a<br />
2 4<br />
3 2<br />
12a ⋅ b<br />
− −<br />
6 5<br />
3<br />
2a b<br />
9. Vereinfache die folgen<strong>den</strong> Terme.<br />
6 3<br />
a) (12a ) : ( − 3a )<br />
3 3<br />
b) (3x) ⋅ (2y)<br />
c) ( ) 2<br />
3<br />
(3x)<br />
d)<br />
6 2<br />
4x y z<br />
3 5<br />
2x y z<br />
e)<br />
(15a )<br />
5<br />
(5a)<br />
2 5<br />
f)<br />
x y<br />
(xy)<br />
10. Schreibe mit positiven Exponenten.<br />
2 a) 2 x −<br />
⋅<br />
3 3<br />
b) x a −<br />
⋅ c) 9 10 −<br />
⋅<br />
3 3<br />
d) a x<br />
5<br />
b −<br />
⋅ ⋅<br />
3<br />
e) (5b) −<br />
f) 3x y<br />
8 4<br />
3<br />
4<br />
− 2 5 −
11. Schreibe ohne Bruch.<br />
2<br />
a<br />
a)<br />
3<br />
x<br />
a<br />
d)<br />
10000<br />
12. Schreibe ohne Bruch.<br />
a)<br />
d)<br />
1<br />
2<br />
x<br />
2<br />
3<br />
x − e)<br />
1<br />
b) 3 4<br />
a ⋅ b<br />
9<br />
e) 5 2<br />
a b<br />
1<br />
b) 2 a ⋅ b<br />
3<br />
5<br />
5<br />
Potenzrechnung – Vermischte Aufgaben 3<br />
1<br />
3<br />
7<br />
c) 3<br />
10<br />
f)<br />
c)<br />
5x<br />
4<br />
y<br />
2<br />
3<br />
x<br />
5<br />
1 1<br />
3 2<br />
f)<br />
13 ⋅ 5<br />
1. Vereinfache.<br />
a) 4a³ + 3x² - 5z 4 + 2a³ + z 4 – 2x³ b) (15a 4 – 12a 3+n + 9a 1-n ) : 3a²<br />
c) (8x³ - 28x² - 12x + 2) : ( 4x + 2) d) (x 4 – 1) : (x – 1)<br />
e) (6a²b 4 c³ + 9a 5 c³ + 9a 5 b²c 6 ) 2<br />
f) 3x²(4x³ – 5x 4 )<br />
2. Vereinfache<br />
3 7<br />
a ⋅ b<br />
a)<br />
2 4<br />
a ⋅ b<br />
b)<br />
z ⋅ z<br />
m<br />
z<br />
n m− n<br />
c)<br />
4z ⋅ 8y<br />
6 3<br />
2y ⋅ z<br />
5 7<br />
5 5<br />
2 3<br />
5 6<br />
3 2 2<br />
x+ b<br />
⎛ x ⋅ y ⎛ ⎞x<br />
y ⋅ ⎞ ⎛ 2x ⋅ y x⎞ 2y ⎛ ⋅ 250a<br />
d) ⎜ :<br />
2 3 ⎟3<br />
5<br />
a ⋅ b<br />
⎜<br />
a b ⋅<br />
⎟e)<br />
:<br />
f)<br />
⎞<br />
⎜ 2 3 ⎟2 2⎜<br />
x b⎟<br />
⎝ ⎝ ⎠<br />
⎠ ⎝ 3a ⋅ 2b 2a⎠ 3b ⎝ ⋅ 75a ⋅ a<br />
⎠<br />
3. Vereinfache.<br />
a)<br />
d)<br />
4 3 6<br />
6x 9y 0,5z 3x<br />
⋅ ⋅ ⋅<br />
3 2<br />
1,5y ⋅ 18z<br />
4<br />
3x ⋅<br />
n+ 2<br />
x +<br />
n 1<br />
2x<br />
n<br />
x<br />
n+<br />
x −<br />
4. Vereinfache.<br />
6 5<br />
x + x<br />
a)<br />
4 3<br />
x + x<br />
5n− 1 1 5n + 5 n<br />
+<br />
d) a ⋅ b a b⋅<br />
⋅ e)<br />
b)<br />
2<br />
(3x +<br />
2 3<br />
6x ) x<br />
4 2<br />
x ⋅ y<br />
5<br />
⋅y ⋅<br />
c)<br />
5<br />
1− x 1<br />
e) + f)<br />
7 2<br />
x x<br />
b)<br />
22x y − 121x y<br />
3 4<br />
11x y<br />
77x y+<br />
5 6 4 5 6 7<br />
c)<br />
(a ⋅ x )<br />
(a ⋅ x )<br />
− 3 5 2 −<br />
2 − 3 4<br />
a + a<br />
a + a<br />
n+ 1 n 2 +<br />
n n + 1<br />
x y<br />
y x<br />
2n+ 1 3n 1 +<br />
3n 2n −1<br />
6 5 n 4<br />
(s s ) s −<br />
− ⋅ f) (x²y³ + xy 4 ) 2<br />
5. Vereinfache<br />
5 5<br />
a) 1,2xy z ⋅ (0,5x²yz<br />
8<br />
0,8xy²z −<br />
7<br />
1,2xyz )<br />
4 5<br />
b) + (x y −<br />
3 4<br />
x y<br />
5 3 2<br />
x y + ) : (xy)<br />
c)<br />
4 5<br />
x ⋅ x 5<br />
: 3<br />
8 2x b³<br />
4<br />
⎛ 2 5 ⎞<br />
e) ⎜ ⎟<br />
7 ⎟<br />
⎝ ⎠<br />
2 2<br />
⎛<br />
d) ⎜<br />
⎝<br />
5r − 7s<br />
7c + 2d<br />
⋅<br />
21c ⎞<br />
5r<br />
⎟<br />
⎠<br />
6d ⎛<br />
7s<br />
⎜<br />
⎝<br />
+<br />
−<br />
⎞<br />
⎟<br />
⎠<br />
4 6 2 3<br />
2x ⋅ 5x 5x 4x<br />
f) : 9 8<br />
4y 8y<br />
⋅
Multiplikation und Division von Potenzen mit gleicher Basis – Lösungen<br />
1. 4 5 2<br />
a) 3 ⋅ 3 ⋅ 3<br />
3 5 2<br />
b) 12 ⋅ 12 ⋅ 12<br />
3 2<br />
c) x ⋅ x ⋅ x<br />
= 3 11<br />
= 12 10<br />
= x 6<br />
3 5 2 7<br />
e) k ⋅ k ⋅ m ⋅ m<br />
5<br />
f) x ⋅<br />
3 2<br />
y ⋅ x ⋅ y<br />
2 3<br />
g) a ⋅ b ⋅ b ⋅ a<br />
2. a)<br />
3. a)<br />
= k 8 m 9<br />
2 n<br />
x ⋅ x<br />
= x 7 y 4<br />
b)<br />
= x 2+n<br />
= b m+3<br />
5 2x<br />
e) a ⋅ a<br />
2m<br />
m<br />
f) z ⋅ z<br />
= a 2x+5<br />
= z 3m<br />
= x m+1<br />
e)<br />
3 m−<br />
2<br />
x ⋅ x<br />
b)<br />
a<br />
= a 6x-3<br />
⋅ a<br />
⋅ a<br />
2x x+<br />
1 3x<br />
− 4<br />
= a³b 4<br />
d)<br />
= d 12<br />
3<br />
b b<br />
m ⋅ c) y y<br />
a ⋅ d)<br />
= y a+1<br />
g)<br />
= a 6m<br />
5 x−<br />
7<br />
a ⋅ a<br />
c)<br />
= a x-2<br />
f)<br />
m+<br />
x ⋅ x<br />
= x 6m+1<br />
⋅ x<br />
2 3m−<br />
4 2m<br />
+ 3<br />
3m<br />
2m<br />
a ⋅ a ⋅<br />
a<br />
m<br />
h)<br />
2m m−<br />
1<br />
y ⋅ y<br />
d)<br />
= p 5 q 11<br />
= x 2m<br />
h) m<br />
= m 9x<br />
= y 3m-1<br />
= x 2p-2<br />
p−<br />
1 3 p + 4 5 p−<br />
8<br />
g) z ⋅ z ⋅ z<br />
m<br />
h) y<br />
= z 9p-5<br />
3 5<br />
d ⋅ d ⋅<br />
d<br />
4<br />
4 6<br />
p ⋅ q ⋅ p ⋅<br />
m m<br />
x ⋅ x<br />
x<br />
= y 4m+1<br />
3x<br />
⋅ m<br />
⋅ x<br />
4x<br />
p−<br />
4 p + 2<br />
4. a) x²(x³ + x 4 ) b) a³(a 5 + a 4 ) c) 3b³(4b² - 5b 5 )<br />
= x 5 + x 6<br />
= a 8 + a 7<br />
= 12b 5 – 15b 8<br />
d) a m (a m+1 – a 3m-1 ) e) y 2a (y 3a+1 – y a-4 ) f) x n-3 (x 5 + x 4 )<br />
= a 2m+1 – a 4m-1<br />
= y 5a+1 – y 3a-4<br />
= x n+2 + x n+1<br />
⋅ y<br />
q<br />
⋅ m<br />
5<br />
2x<br />
⋅ y<br />
− 2 2m<br />
− 5 m+<br />
8<br />
5. a) (x² + x³)² b) (y 3 – y 4 )² c) (a 6 + a 4 )² d) (b 3 – b 7 )²<br />
= x 4 + 2x 5 + x 6<br />
= y 6 – 2y 7 + y 8<br />
= a 12 + 2a 10 + a 8<br />
= b 6 – 2b 10 + b 14<br />
e) (2a² + 3a³)² f) (4x 5 – 2x 6 )² g) (6d 5 – 3d 4 )² h) (3m² + 5m 7 )²<br />
= 4a 4 + 12a 5 + 9a 6<br />
= 16x 10 – 16x 11 + 4x 12<br />
= 36d 10 – 36d 9 + 9d 8<br />
= 9m 4 +30m 9 + 25m 14<br />
6. a) (a² + a³)(a² - a³) b) (x 5 + y 4 )(x 5 – y 4 ) c) (m³ + n 5 )(m³ - n 5 )<br />
= a 4 – a 6<br />
= x 10 – y 8<br />
= m 6 – n 10<br />
d) (3x 4 – 2y 5 )(3x 4 + 2y 5 ) e) (4y³ - 6x 7 )(4y³ + 6x 7 ) f) (3a 4 – 4b³)(3a 4 + 4b³)<br />
= 9x 8 – 4y 10<br />
= 16y 6 – 36x 14<br />
= 9a 8 – 16b 6<br />
7. a) (a 3 + a 4 )(a² + a 5 ) b) (x² - x 5 )(x³ + x 6 ) c) (a 3 – b²)(a 5 + b³)<br />
= a 5 + a 8 + a 6 + a 9<br />
= x 5 – x 11<br />
= a 8 + a 3 b 3 – a 5 b² - b 5<br />
d) (y 4 + y 5 )(y 3 – y 6 ) e) (2a 5 + 3b 3 )(2a 3 – 2b 4 ) f) (k m + k n )(k m+1 + k n+2 )<br />
= y 7 – y 10 + y 8 – y 11<br />
= 4a 8 – 4a 5 b 4 + 6a³b³ - 6b 7<br />
= k 2m+1 +k m+n+2 +k m+n+1 +k 2n+2<br />
8. Schreibe als Produkt von Potenzen.<br />
9. a)<br />
10. a)<br />
a) x 3+5<br />
=<br />
3<br />
x<br />
5<br />
x ⋅<br />
= 5 5<br />
5<br />
5<br />
= a x-3<br />
8<br />
3<br />
a x<br />
a³<br />
b)<br />
b) a 3n+2<br />
=<br />
3n<br />
a<br />
2<br />
a ⋅<br />
= 8 6<br />
8<br />
8<br />
9<br />
3<br />
x<br />
b)<br />
x<br />
y<br />
= x y-1<br />
c)<br />
= 12 8<br />
c)<br />
= x 2m<br />
12<br />
12<br />
c) 5 m+n<br />
=<br />
m<br />
5<br />
n<br />
5 ⋅<br />
13<br />
5<br />
m<br />
x 3<br />
x<br />
m<br />
d)<br />
= a 4<br />
a<br />
a<br />
9<br />
5<br />
6m<br />
y<br />
3<br />
d) m<br />
y<br />
= y 3m<br />
d) z 5k+3m<br />
=<br />
5k<br />
z<br />
3m<br />
z ⋅<br />
7<br />
y<br />
e) 6<br />
y<br />
x m+4<br />
m 4<br />
= x x ⋅<br />
f)<br />
= y = k 6<br />
e)<br />
2<br />
k m<br />
k<br />
= k 2m-3<br />
3<br />
f)<br />
k<br />
k<br />
23<br />
17<br />
2<br />
d p<br />
d²<br />
= d 2p-2
11. a)<br />
12 a)<br />
x<br />
x<br />
m<br />
m−<br />
3<br />
b)<br />
a<br />
a<br />
= x³ = a 2m+1<br />
x<br />
x<br />
n+<br />
3<br />
n+<br />
2<br />
3m<br />
m−<br />
1<br />
b)<br />
a<br />
a<br />
= x = a n+1<br />
7 x+<br />
4<br />
v<br />
e)<br />
5x<br />
− 2<br />
v<br />
x<br />
f)<br />
x<br />
= v 2x+6<br />
= x p<br />
2n<br />
− 1<br />
n−<br />
2<br />
3 p+<br />
1<br />
2 p+<br />
1<br />
c)<br />
z<br />
z<br />
= z 2x+4<br />
3x<br />
x−<br />
4<br />
d)<br />
k<br />
k<br />
= k 2a-3<br />
4a<br />
2a<br />
+ 3<br />
3x<br />
+ 4<br />
y<br />
c) x−<br />
2<br />
y<br />
= y 2x+6<br />
z<br />
g)<br />
z<br />
= z x<br />
k + 3x<br />
k + 2x<br />
2b<br />
y<br />
e) b+<br />
3<br />
y<br />
= y b-3<br />
d)<br />
b<br />
b<br />
= b<br />
3<br />
b<br />
h)<br />
b<br />
= b 2m-1<br />
13. a) (x 8 + x 6 – x 5 ) : x² b) (15a³ + 12a 6 – 3a 4 ) : 3a²<br />
= x 6 + x 4 – x 3<br />
= 5a + 4a 4 – a²<br />
c) (21b 8 – 28b 4 + 14b 5 ) : 7b³ d) (3x n+3 – 9x 2n-4 + 12x n+5 ) : 3x²<br />
= 3b 5 – 4b + 2b² = x n+1 – 3x 2n-6 + 4x n+3<br />
e) (35y m+2 – 20y 2m+4 + 15y m+8 ) : 5y m<br />
f) (4z a+3 + 16z 2a+5 – 12z a+4 ) : 2z a<br />
= 7y² - 4y m+4 + 3y 8<br />
= 2z 3 + 8z a+5 – 6z 4<br />
14. a)<br />
=<br />
5<br />
15x<br />
y<br />
7<br />
21a<br />
b<br />
8<br />
5<br />
6<br />
25 ² ³<br />
2x³<br />
y²<br />
: 10<br />
35a<br />
b<br />
6<br />
b)<br />
5<br />
6 p q<br />
3<br />
r²<br />
s<br />
4<br />
4<br />
3p<br />
q³<br />
: 7 5<br />
r s<br />
f)<br />
3m+<br />
4<br />
3m+<br />
3<br />
m−<br />
6<br />
m−<br />
5<br />
m<br />
m<br />
= m 2b-7<br />
9 7 5<br />
18a<br />
b 12a<br />
b³<br />
c) : 4 6<br />
35x³<br />
y²<br />
21x<br />
y<br />
x y a b<br />
2<br />
= 2pqr 5 s² = 0,9a 4 b 4 xy 4<br />
15. Löse die folgen<strong>den</strong> Exponentialgleichungen:<br />
a)<br />
5<br />
a<br />
a<br />
a²<br />
x 12<br />
1 y<br />
= b) y 8<br />
y<br />
x+<br />
=<br />
x = 3 x = 3 x = 3<br />
x+<br />
5<br />
d) z =<br />
3x<br />
+ 5<br />
z<br />
3x<br />
− 15<br />
z<br />
2x<br />
+ n<br />
e) a =<br />
3x<br />
+ 2n<br />
a<br />
3n<br />
a<br />
f) a<br />
x = 15 x = 2n x = 2<br />
c)<br />
2 1<br />
a x−<br />
x+<br />
1<br />
=<br />
a<br />
a<br />
15<br />
10<br />
a<br />
= m<br />
a<br />
m+<br />
1<br />
− 2<br />
4b<br />
2b<br />
+ 7
Multiplikation und Division von Potenzen mit gleichem Exponenten -<br />
Lösungen<br />
1. a)<br />
2. a)<br />
3 3<br />
5 ⋅ 2<br />
b)<br />
2 2<br />
8 ⋅ 3<br />
c)<br />
3 3<br />
0, 5 ⋅ 4<br />
d)<br />
5 5<br />
0, 5 ⋅ 10 ⋅<br />
= 10³ = 1000 = 24² = 576 = 2³ = 8 = 1 5 = 1<br />
4 4 4<br />
e) 4 ⋅ 3 ⋅ 0,<br />
25<br />
6 1 6<br />
f) 6 ⋅ ( )<br />
6<br />
5 18 5<br />
g) ( )³ ⋅ ( )³ ⋅ ( )³<br />
6 25 3<br />
2 4 6 4<br />
h) ( ) ⋅ ( ) ⋅ 5<br />
3 10<br />
= 3 4 = 81 = 1 6 = 1 = 1³ = 1 = 2 4 = 16<br />
= 20 x<br />
e)<br />
x x<br />
5 ⋅ 4<br />
b)<br />
m m<br />
a ⋅ b<br />
f)<br />
= (ab) m<br />
= 36 a<br />
a a<br />
12 ⋅ 3<br />
c)<br />
= (yz) k<br />
k k<br />
y ⋅ z<br />
g)<br />
4<br />
= 20 x+1<br />
x+<br />
1 x+<br />
1<br />
⋅ 5<br />
( x − y<br />
= (x² - y²) 8<br />
8<br />
8<br />
+ y)<br />
⋅ ( x ) h)<br />
d) 3<br />
= 18 m-4<br />
− 4 − 4<br />
⋅ 6<br />
m m<br />
0,<br />
2<br />
( −<br />
m<br />
m<br />
a + b)<br />
⋅ ( a b)<br />
= (a² - b²) m<br />
3. a) ( − 4)³<br />
⋅ ( − 0,<br />
5)³<br />
4 1 4<br />
b) ( − 3)<br />
⋅ ( )<br />
3<br />
c) ( − 4)²<br />
⋅ ( − 1,<br />
5)²<br />
5<br />
5<br />
d) ( − 5)<br />
⋅ ( − 0,<br />
1)<br />
⋅ 2<br />
= 2³ = 8 = (-1) 4 = 1 = 6² = 36 = 1 5 = 1<br />
e) a³ ⋅ ( − b)³<br />
5<br />
f) ( − x) ⋅ ( −<br />
5 5<br />
y)<br />
⋅ z g) ( − p)² ⋅ ( − r)²<br />
⋅ s²<br />
m m<br />
h) ( − a ) ⋅ ( − b)<br />
= (-ab)³ = (xyz) 5<br />
= (prs)² = (ab) m<br />
4. 3<br />
24<br />
a)<br />
3<br />
8<br />
5<br />
36<br />
b)<br />
5<br />
18<br />
3<br />
49<br />
c)<br />
3<br />
7<br />
27<br />
d)<br />
9²<br />
2<br />
= 3³ = 27 = 2 5 = 32 = 7³ = 343 = 3² = 9<br />
4<br />
2,<br />
6<br />
e) 4<br />
1,<br />
3<br />
0,<br />
4²<br />
f)<br />
0,<br />
5²<br />
5<br />
3<br />
g ) 3 5<br />
( )<br />
2<br />
1<br />
( )³<br />
8<br />
h)<br />
1<br />
( )³<br />
4<br />
= 2 4 = 16 = 0,8² = 0,64 = 2 5 = 32 = 0,5³ = 0,125<br />
5. a)<br />
6. a)<br />
a<br />
b<br />
6<br />
6<br />
6<br />
⎛ a ⎞<br />
= ⎜<br />
b<br />
⎟<br />
⎝ ⎠<br />
8² ⋅ 3²<br />
e)<br />
6²<br />
n<br />
x<br />
b) n<br />
y<br />
n<br />
=<br />
⎛<br />
⎜<br />
⎝<br />
x<br />
y<br />
⎞<br />
⎟<br />
⎠<br />
15³ ⋅ 3³<br />
f)<br />
9³<br />
= 4² = 16 = 5³ = 125 = 4 m<br />
27a³<br />
8b³<br />
3<br />
=<br />
⎛<br />
⎜<br />
⎝<br />
3a<br />
2b<br />
⎞<br />
⎟<br />
⎠<br />
( a²<br />
− b²)³<br />
e)<br />
( a + b)³<br />
b)<br />
25a²<br />
b²<br />
2<br />
⎛ 5a ⎞<br />
= ⎜<br />
b<br />
⎟<br />
⎝ ⎠<br />
( 4a²<br />
− 9b²)<br />
f) 5<br />
( 2a<br />
− 3b)<br />
= (a – b)³ = (2a + 3b) 5<br />
5<br />
c)<br />
n+ 1<br />
x<br />
( − x)<br />
n+ 1<br />
d) 4<br />
y<br />
y<br />
n+ 1<br />
=<br />
⎛<br />
⎜<br />
⎝<br />
x<br />
y<br />
⎞<br />
⎟<br />
⎠<br />
( 12x)<br />
g) m<br />
( 3x)<br />
c)<br />
m<br />
27x³<br />
1000y³<br />
⎛ 3x ⎞<br />
= ⎜ ⎟<br />
⎝ 10y ⎠<br />
( 16x²<br />
− 25y²)<br />
g) n<br />
( 4x<br />
− 5y)<br />
3<br />
= (4x + 5y) n<br />
n<br />
4<br />
4<br />
=<br />
⎛<br />
⎜<br />
⎝<br />
x<br />
−<br />
y<br />
⎞<br />
⎟<br />
⎠<br />
n−<br />
1<br />
( 48a)<br />
h) n−<br />
1<br />
( 12a)<br />
= 4 n-1<br />
d)<br />
5<br />
32y<br />
100000z<br />
5<br />
5<br />
⎛ 2y ⎞<br />
= ⎜<br />
10x<br />
⎟<br />
⎝ ⎠<br />
n+<br />
( p²<br />
− 16q²)<br />
h) n+<br />
1<br />
( p + 4q)<br />
= (p – 4q) n+1<br />
5<br />
4<br />
1<br />
5
Potenzen von Potenzen - Lösungen<br />
1. a) (2³)² b) (4²) 4<br />
= 2 6<br />
= 4 8<br />
e) (a 5 )³ f) (x³) m<br />
= a 15<br />
= x 3m<br />
2. a) (x m ) n+1<br />
= x mn+m<br />
e) (p 2k+1 ) 3<br />
= p 6k+3<br />
b) (a y ) x-1<br />
= a xy-y<br />
f) (b n-4 ) m<br />
= b mn-4m<br />
c) (0,2²) 4<br />
= 0,2 8<br />
g) (a m ) n<br />
= a mn<br />
c) (x a+3 ) b<br />
= x ab+3b<br />
g) (y p ) q-2<br />
= y pq-2p<br />
3. a) (x²y³)² b) (a³b) 5<br />
c) (d 5 e³) 3<br />
= x 4 y 6<br />
= a 15 b 5<br />
= d 15 e 9<br />
e) (a³b 4 ) n<br />
f) (3x 5 y 2 )² g) (5a²b 7 ) 4<br />
= a 3n b 4n<br />
= 9x 10 y 4<br />
= 625a 8 b 28<br />
4.<br />
2<br />
3 2 ⎛ ⎞<br />
4a b<br />
a) ⎜ 4 3 ⎟<br />
⎝ 2x y ⎠<br />
6 4<br />
4a<br />
b<br />
= 8 6<br />
x y<br />
10<br />
m n<br />
⎛ 5a b<br />
⎞<br />
b) ⎜ 7 3 ⎟<br />
⎝ 10p q<br />
⎠<br />
10m 10n<br />
a b<br />
= 10 70 30<br />
2 p q<br />
Potenzrechnung – Vermischte Aufgaben 2 – Lösungen<br />
6 8 4<br />
( 6a<br />
b )<br />
c) 5 2 4<br />
( 3a<br />
b )<br />
= 16a 4 b 24<br />
1. Berechne.<br />
a) 0,2 3 = 0,008 b) 3,1 3 = 29,791 c)<br />
d)<br />
3 3<br />
2 3<br />
2. Berechne.<br />
d) (10³) 5<br />
= 10 15<br />
h) (y 2a ) b<br />
= y 2ab<br />
d) (z n-3 ) 4<br />
= z 4n-12<br />
h) (k 2m+3 ) n<br />
= k 2mn+3n<br />
d) (m 6 n 5 ) 8<br />
= m 48 n 40<br />
h) 5(m 4 n 5 ) 4<br />
= 5m 16 n 20<br />
5 6<br />
( 4x<br />
y )³<br />
d) 6 3<br />
( 2x<br />
y²)<br />
8y<br />
= 3<br />
x<br />
12<br />
4 3<br />
3 + 4 5⋅<br />
= 581<br />
7 3<br />
⋅ = 216 e) 4 ⋅ 4 = 4 10 f) (-3) 5 = –243<br />
3<br />
⎛ 2 ⎞ 8<br />
a) ⎜<br />
3<br />
⎟=<br />
⎝ ⎠ 27<br />
− 2 2<br />
⎛ 3 ⎞ ⎛ 4 ⎞ 16<br />
d) ⎜ =<br />
4<br />
⎟ = ⎜<br />
3<br />
⎟<br />
⎝ ⎠ ⎝ ⎠ 9<br />
3. Löse die Klammern auf.<br />
4 2 3<br />
x ⋅ (x + x )<br />
a)<br />
6 7<br />
= x + x<br />
d)<br />
3 6 4<br />
x ⋅ (x − x )<br />
= x − x<br />
9 7<br />
3<br />
⎛ 1 ⎞<br />
b) ⎜ =<br />
5<br />
⎟<br />
⎝ ⎠<br />
3<br />
1<br />
125<br />
⎛ 3 ⎞ 27<br />
e) ⎜ =<br />
10<br />
⎟<br />
⎝ ⎠ 1000<br />
b)<br />
4. Faktorisiere.<br />
2 4 3<br />
16x<br />
y + 32x yz<br />
a)<br />
3<br />
= 8x²y( 2y 4xz<br />
5 3 2<br />
40x<br />
−y<br />
z<br />
3 2 2<br />
5x + y z ) −<br />
3 2 4<br />
2a ⋅ (a a − )<br />
5 7<br />
= 2a 2a−<br />
− 2 2<br />
⎛ 5 ⎞ ⎛ 8 ⎞ 64<br />
c) ⎜<br />
=<br />
8<br />
⎟ = ⎜<br />
5<br />
⎟<br />
⎝ ⎠ ⎝ ⎠ 25<br />
f)<br />
− 1<br />
⎛ 9 ⎞ 10<br />
⎜ =<br />
10<br />
⎟<br />
⎝ ⎠ 9<br />
3<br />
(x + y)<br />
c)<br />
3 3<br />
= x 3x²y + 3xy² + y<br />
+<br />
3 2 4<br />
3a ⋅ (a a + )<br />
e)<br />
5 7<br />
= 3a 3a+<br />
f)<br />
(m −<br />
− 2<br />
n)<br />
=<br />
1<br />
(m − n) ² m²<br />
1<br />
=<br />
2mn − n²<br />
+<br />
3 6 2 4 5 5<br />
1,4u v − 0,7u<br />
v 2,8u<br />
−v<br />
b)<br />
4<br />
= 0,7u²v<br />
(2uv²<br />
1 4u³v)<br />
− −
4 2 5<br />
3a − 2a 6a+<br />
c)<br />
3<br />
= a²(3a² 2 6a − ) +<br />
5. Vereinfache die folgen<strong>den</strong> Terme.<br />
7 3<br />
4 4<br />
a)<br />
4<br />
4<br />
−<br />
⋅<br />
=<br />
3,5x ⋅ 2x<br />
b)<br />
8<br />
= 7x<br />
4<br />
16a ⋅ 2a<br />
d)<br />
5<br />
= 32a<br />
5 − 2<br />
4x ⋅ 3x<br />
e)<br />
3<br />
= 12x<br />
6. Vereinfache die folgen<strong>den</strong> Terme.<br />
3 2<br />
x ⋅ x<br />
a)<br />
5<br />
= x<br />
a a a<br />
b)<br />
0<br />
a 1<br />
−<br />
⋅ ⋅<br />
= =<br />
3 2 2<br />
a ⋅ b a ⋅<br />
d)<br />
5 5<br />
= a b ⋅<br />
3<br />
b ⋅<br />
2<br />
a ⋅ a<br />
e)<br />
3<br />
= a<br />
3 5<br />
2 3 5<br />
7. Vereinfache die folgen<strong>den</strong> Terme.<br />
2<br />
x ⋅ x<br />
a)<br />
3<br />
= x<br />
3 2<br />
x ⋅ x<br />
b)<br />
10<br />
= x<br />
5<br />
x ⋅<br />
2 2 3<br />
2a ⋅ b 4a⋅ d)<br />
5 6<br />
= 8a b ⋅<br />
4<br />
b ⋅<br />
2 3 4<br />
5a b b<br />
e)<br />
1<br />
10a b<br />
3<br />
2a<br />
−<br />
⋅ ⋅<br />
=<br />
⋅<br />
⋅<br />
8. Berechne.<br />
3 3<br />
x ⋅ x<br />
4<br />
x ⋅ x<br />
a)<br />
6<br />
x<br />
= = x 5<br />
x<br />
12a b<br />
2<br />
4ab<br />
d)<br />
= 3a<br />
2 2<br />
b)<br />
e)<br />
16x y<br />
2<br />
4xy<br />
=<br />
x<br />
x<br />
4x<br />
2n+ 3<br />
3n− 4<br />
2 2<br />
2 3 3 2 2<br />
9a b + 6a b + 12a b<br />
d)<br />
= 3a²b(3b² 2ab + + 4)<br />
− −<br />
2n+ 3 (3n − 4) − 7 − n<br />
= x x =<br />
9. Vereinfache die folgen<strong>den</strong> Terme.<br />
6 3<br />
(12a ) : ( − 3a )<br />
a)<br />
3<br />
= 4a −<br />
3 3<br />
(3x) ⋅ (2y)<br />
b)<br />
3 3 3<br />
= 6 x ⋅ y ⋅<br />
d)<br />
6 2<br />
4x y z<br />
3 5<br />
2x y z<br />
=<br />
2x<br />
3 3<br />
y −<br />
e)<br />
(15a )<br />
5<br />
(5a)<br />
=<br />
2 5<br />
(3a)<br />
10. Schreibe mit positiven Exponenten.<br />
− 2<br />
2 ⋅ x<br />
x ⋅ a<br />
a)<br />
=<br />
2<br />
x²<br />
b)<br />
5<br />
3 − 3<br />
x³<br />
=<br />
a³<br />
c)<br />
f)<br />
25x ⋅ y<br />
=<br />
2 2<br />
25x²y²<br />
− 3 3 −<br />
9a ⋅ 2a<br />
=<br />
18a<br />
3<br />
b ⋅ b<br />
c)<br />
4<br />
= b<br />
5 2<br />
a ⋅ a<br />
f)<br />
7<br />
= a<br />
− 6<br />
4 3 2<br />
a ⋅ x a ⋅<br />
c)<br />
6 5<br />
= a x ⋅<br />
2<br />
x ⋅<br />
− 4 2<br />
9 ⋅ a 3b⋅ f)<br />
− 1<br />
= 27a<br />
b ⋅<br />
5<br />
a<br />
−3<br />
b ⋅ ⋅<br />
2 4<br />
a ⋅ a<br />
3 2<br />
a ⋅ a<br />
c)<br />
6<br />
a<br />
= = a 5<br />
a<br />
f)<br />
c)<br />
f)<br />
c)<br />
12a ⋅ b<br />
3<br />
2a b<br />
=<br />
6 5<br />
3 4<br />
6a b<br />
2<br />
3 ( (3x) )<br />
6 6 6<br />
= (3x)<br />
3 x=<br />
8 4<br />
x y<br />
(xy)<br />
3<br />
8 4<br />
x y 5<br />
= x y=<br />
3 3<br />
x y<br />
9 ⋅<br />
10<br />
9<br />
=<br />
10<br />
− 4<br />
4
a<br />
x b −<br />
⋅ ⋅<br />
3 3 5<br />
d) a³ ⋅ x³<br />
= 5<br />
b<br />
11. Schreibe ohne Bruch.<br />
2<br />
a<br />
3 a) x<br />
= a² a<br />
− 3<br />
x ⋅<br />
d) 10000<br />
5<br />
a 10 −<br />
= ⋅<br />
12. Schreibe ohne Bruch.<br />
1<br />
a) 2 x = x<br />
d)<br />
x<br />
2<br />
−<br />
3<br />
=<br />
3<br />
1<br />
x²<br />
13. Vereinfache.<br />
a)<br />
d)<br />
2a ⋅ 4,5a<br />
= 9a² 3a<br />
=<br />
27<br />
3<br />
= 9 3 =<br />
e)<br />
b)<br />
( 5b)<br />
− 3<br />
1<br />
= 3<br />
(5b)<br />
1<br />
3 4<br />
a ⋅ b<br />
=<br />
− 3<br />
a<br />
9<br />
b<br />
4<br />
⋅<br />
−<br />
5<br />
e) a b<br />
2<br />
= 9<br />
− 5<br />
a⋅<br />
2<br />
b<br />
−<br />
⋅<br />
1<br />
b) 2 3 a ⋅ b = a ⋅<br />
1<br />
3<br />
e) 5 5<br />
b)<br />
5 = 5³<br />
=<br />
6<br />
a<br />
a³<br />
5<br />
27<br />
e) 13,5<br />
Potenzrechnung – Vermischte Aufgaben 3<br />
5<br />
5<br />
= 2 3=<br />
2<br />
3 b<br />
f)<br />
3x<br />
y<br />
=<br />
− 2 5 −<br />
3<br />
x²<br />
y<br />
5<br />
7<br />
3<br />
c) 10<br />
= 7 ⋅<br />
f)<br />
5<br />
5x<br />
4<br />
y<br />
3<br />
10 −<br />
5 − 4<br />
= 5x y ⋅<br />
2<br />
c) 3 3<br />
x = x²<br />
1 1<br />
3 2 13 ⋅ 5<br />
f)<br />
=<br />
3<br />
13<br />
⋅ 5<br />
c)<br />
f)<br />
3 x<br />
=<br />
6<br />
x<br />
0,4<br />
5 ⎛ ⎞<br />
8<br />
⎜ x ⎟<br />
⎝ ⎠<br />
=<br />
1<br />
4 x =<br />
4<br />
x<br />
1. Vereinfache.<br />
a) 4a³ + 3x² - 5z 4 + 2a³ + z 4 – 2x³ b) (15a 4 – 12a 3+n + 9a 1-n ) : 3a²<br />
= 6a³ + 3x² - 4z 4 – 2x³ = 5a² – 4a 1+n + 3a -1-n<br />
c) (8x³ - 28x² - 12x + 2) : ( 4x + 2) d) (x 4 – 1) : (x – 1)<br />
= 2x² - 8x + 1 = x³ + x² + x + 1<br />
e) (6a²b 4 c³ + 9a 5 c³ + 9a 5 b²c 6 ) 2<br />
= 36a 4 b 8 c 6 + 108a 7 b 4 c 6 + 108a 7 b 6 c 9 + 81a 10 c 6 + 126a 10 b²c 9 + 81a 10 b 4 c 12<br />
f) 3x²(4x³ – 5x 4 )<br />
= 12x 5 – 15x 6<br />
2. Vereinfache<br />
3 7<br />
a ⋅ b<br />
a) 2 4<br />
a ⋅ b<br />
= a b³<br />
b)<br />
z ⋅ z<br />
m<br />
z<br />
= 1<br />
n m− n<br />
c)<br />
4z ⋅ 8y<br />
6 3<br />
2y ⋅ z<br />
5 7<br />
=<br />
16yz²
5 6<br />
5 5<br />
3 2<br />
2<br />
2<br />
3<br />
x+ b<br />
⎛ x ⋅ y ⎛ ⎞x<br />
y ⋅ ⎞ ⎛ 2x ⋅ y x⎞ 2y ⎛ ⋅ 250a ⎞<br />
⎜ :<br />
2 3 ⎟3<br />
5<br />
d) a ⋅ b<br />
⎜<br />
a ⋅ b<br />
⎟ ⎜ :<br />
2 3 ⎟2 2⎜<br />
x b⎟<br />
⎝ ⎝ ⎠ ⎠e)<br />
3a ⋅ 2b<br />
2a 3b ⋅ 75a ⋅ a<br />
⎝ ⎠ ⎝ ⎠ f)<br />
20 25 5 10<br />
10<br />
= x y a b<br />
= y ⋅ 3a²<br />
=<br />
3<br />
3. Vereinfache.<br />
a)<br />
d)<br />
4 3 6<br />
6x 9y 0,5z 3x<br />
=<br />
⋅ ⋅ ⋅<br />
3 2<br />
1,5y ⋅ 18z<br />
4<br />
3x<br />
⋅<br />
xz<br />
4<br />
n+ 2<br />
x +<br />
n 1<br />
2x<br />
n<br />
x<br />
n+<br />
x −<br />
= x² + 2x<br />
1 −<br />
4. Vereinfache.<br />
a)<br />
x + x<br />
x + x<br />
= x²<br />
6 5<br />
4 3<br />
5n− 1 1 5n + 5+<br />
n<br />
a ⋅ b a b⋅<br />
⋅<br />
d)<br />
5n 6+<br />
6n<br />
= a b<br />
5. Vereinfache<br />
b)<br />
e)<br />
2<br />
(3x +<br />
2<br />
6x )<br />
4 2<br />
x ⋅ y<br />
3<br />
x<br />
5<br />
⋅y ⋅<br />
=<br />
9xy³<br />
5<br />
1− x 1<br />
+ 7 2<br />
x x<br />
=<br />
1<br />
7<br />
x<br />
(a ⋅ x )<br />
2 − 3 4<br />
(a ⋅ x )<br />
c)<br />
x²<br />
=<br />
a²<br />
n 1<br />
a +<br />
n 2<br />
a<br />
f) n<br />
a +<br />
n + 1<br />
a<br />
= a<br />
b)<br />
x y<br />
5 6 4 5<br />
22x y − 121x y<br />
6 7<br />
77x y+<br />
c) y x<br />
3 4<br />
11x<br />
y<br />
= x²<br />
y<br />
= 2x²y²<br />
11xy− 7x³y³ +<br />
(s − s ) ⋅ s<br />
e)<br />
n+ 2 n + 1<br />
= s s −<br />
6 5 n − 4<br />
f)<br />
− 3 5 2 −<br />
+ +<br />
2n+ 1 3n 1 +<br />
3n 2n −1<br />
4<br />
(x²y³ − xy )²<br />
4 6 7 8<br />
= x y 2x³y<br />
+ x²y +<br />
5 5 8 7<br />
4 5 3 4 5 3 2<br />
1,2xy z ⋅ (0,5x²yz 0,8xy²z − 1,2xyz ) + (x y − x y + x y ) : ( xy)<br />
a)<br />
b)<br />
6 6 7 9 6 8<br />
= 0,6x³y z 0,96x²y − z 1,44x² y z + = x²y³ x y² − x³y +<br />
c)<br />
4 5<br />
x ⋅ x 5<br />
: 3<br />
8 2x b³<br />
=<br />
b³x<br />
20<br />
12<br />
4<br />
⎛ 2 5<br />
⎜ 7<br />
e) ⎝<br />
⎞<br />
⎟<br />
⎠ f)<br />
=<br />
400<br />
49<br />
2 2<br />
⎛<br />
d) ⎜<br />
⎝<br />
5r − 7s<br />
7c + 2d<br />
⋅<br />
21c ⎞<br />
5r<br />
⎟<br />
⎠<br />
6d ⎛<br />
7s<br />
⎜<br />
⎝<br />
+<br />
−<br />
⎞<br />
⎟<br />
⎠<br />
= 9<br />
2x ⋅<br />
5x 5x ⋅ 4x<br />
: 9 8<br />
4y 8y<br />
=<br />
4 6 2 3<br />
5<br />
x<br />
y