Übungen zu den Potenzgesetzen

Übungen zu den Potenzgesetzen
Multiplikation und Division von Potenzen mit gleicher Basis
1. a)
e)
2. a)
e)
3. a)
e)
4 5 2
3 ⋅ 3 ⋅ 3
3 5 2
b) 12 ⋅ 12 ⋅ 12
3 2
c) x ⋅ x ⋅ x
3 5 2 7
k ⋅ k ⋅ m ⋅ m
5
f) x ⋅
3 2
y ⋅ x ⋅ y
2 3
g) a ⋅ b ⋅ b ⋅ a
2 n
x ⋅ x
a
b)
5 2x
2m
m
⋅ a
f) z ⋅ z
3 m−
2
x ⋅ x
b)
3
b b
m ⋅ c) y y
a ⋅ d)
g)
5 x−
7
a ⋅ a
c)
3m
2m
a ⋅ a ⋅
a
m
d)
h)
h)
2m m−
1
y ⋅ y
d)
2x x+
1 3x
− 4 m+
2 3m−
4 2m
+ 3
p−
1 3 p + 4 5 p−
8
a ⋅ a ⋅ a f) x ⋅ x ⋅ x g) ⋅ z ⋅ z
z h)
3 5
d ⋅ d ⋅
4 6
p ⋅ q
4
d
⋅ p ⋅ q
m m
x ⋅ x
m
x
y
3x
⋅ m
⋅ x
4x
p−
4 p + 2
4. a) x²(x³ + x 4 ) b) a³(a 5 + a 4 ) c) 3b³(4b² - 5b 5 )
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 )
⋅ y
⋅ m
5
2x
⋅ y
m−
2 2m
− 5 m+
8
5. a) (x² + x³)² b) (y 3 – y 4 )² c) (a 6 + a 4 )² d) (b 3 – b 7 )²
e) (2a² + 3a³)² f) (4x 5 – 2x 6 )² g) (6d 5 – 3d 4 )² h) (3m² + 5m 7 )²
6. a) (a² + a³)(a² - a³) b) (x 5 + y 4 )(x 5 – y 4 ) c) (m³ + n 5 )(m³ - n 5 )
d) (3x 4 – 2y 5 )(3x 4 + 2y 5 ) e) (4y³ - 6x 7 )(4y³ + 6x 7 ) f) (3a 4 – 4b³)(3a 4 + 4b³)
7. a) (a 3 + a 4 )(a² + a 5 ) b) (x² - x 5 )(x³ + x 6 ) c) (a 3 – b²)(a 5 + b³)
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 )
8. Schreibe als Produkt von Potenzen.
9. a)
10. a)
11. a)
12. a)
a) x 3+5
e)
5
5
8
3
a x
a³
x
x
x
x
v
v
m
m−
3
n+
3
n+
2
7 x+
4
5x
− 2
b)
b) a 3n+2
8
8
9
3
x
b)
x
y
b)
a
a
3m
m−
1
b)
f)
a
a
x
x
2n
− 1
n−
2
3 p+
1
2 p+
1
c)
c)
c)
12
12
c) 5 m+n
13
m
5
m
x 3
x
z
z
3x
x−
4
d)
a
a
9
5
6m
y
3
d) m
y
d)
k
k
4a
2a
+ 3
3x
+ 4
y
c) x−
2
y
g)
z
z
k + 3x
k + 2x
d) z 5k+3m
7
y
e) 6
y
e)
2
k m
k
3
2b
y
e) b+
3
y
13. a) (x 8 + x 6 – x 5 ) : x² b) (15a³ + 12a 6 – 3a 4 ) : 3a²
c) (21b 8 – 28b 4 + 14b 5 ) : 7b³ d) (3x n+3 – 9x 2n-4 + 12x n+5 ) : 3x²
e) (35y m+2 – 20y 2m+4 + 15y m+8 ) : 5y m
f) (4z a+3 + 16z 2a+5 – 12z a+4 ) : 2z a
14. 5
15x
y
a)
7
21a
b
8
5
2x³
y²
: 10
35a
b
6
b)
5
6 p q
3
r²
s
4
4
3p
q³
: 7 5
r s
d)
h)
b
b
b
b
e) x m+4
f)
f)
f)
3m+
4
3m+
3
3m−
6
m−
5
k
k
9 7 5
18a
b 12a
b³
c) : 4 6
35x³
y²
21x
y
23
17
2
d p
d²
m
m
4b
2b
+ 7

Übungen zu den Potenzgesetzen
Multiplikation und Division bei Potenzen mit gleichem Exponenten
1. a)
e)
2. a)
e)
3 3
5 ⋅ 2
b)
4 4
4 ⋅ 3 ⋅
0,
25
4
f)
x x
5 ⋅ 4
b)
m m
a ⋅ b
f)
2 2
8 ⋅ 3
c)
1
6
3 3
0, 5 ⋅ 4
d)
5
6
18 5
⋅ h)
25 3
6 6
6 ⋅ ( ) g) ( )³ ( )³ ⋅ ( )³
a a
12 ⋅ 3
c)
k k
y ⋅ z
g)
x+
1 x+
1
4 ⋅ 5
d)
( x +
8
y)
⋅ ( x −
8
y)
h)
5 5
0, 5 ⋅ 10 ⋅
2
3
6
10
0,
2
4 4
( ) ⋅ ( ) ⋅
5
5
− 4 − 4
3 ⋅ 6
m m
m
( a + b)
⋅ ( a − b)
3. a) ( − 4)³
⋅ ( − 0,
5)³
4 1 4
b) ( − 3)
⋅ ( )
3
c) ( − 4)²
⋅ ( − 1,
5)²
d)
e) a³ ⋅ ( − b)³
5
f) ( − x) ⋅ ( −
5 5
y)
⋅ z g) ( − p)² ⋅ ( − r)²
⋅ s²
h) ( − −
4. a)
5. a)
24
3
8
3
4
2,
6
e) 4
1,
3
e)
6. a)
e)
6
a
6
b
8² ⋅ 3²
6²
27a³
8b³
( a²
− b²)³
( a + b)³
Potenzen von Potenzen
b)
f)
36
18
5
5
0,
4²
0,
5²
n
x
b) n
y
15³ ⋅ 3³
f)
9³
b)
25a²
b²
( 4a²
− 9b²)
f) 5
( 2a
− 3b)
1. a) (2³)² b) (4²) 4
e) (a 5 )³ f) (x³) m
2. a) (x m ) n+1
e) (p 2k+1 ) 3
b) (a y ) x-1
f) (b n-4 ) m
5
c)
49
3
7
3
5
3
g ) 3 5
( )
2
n+
1
x
c) n+
1
y
( 12x)
g) m
( 3x)
c)
m
27x³
1000y³
( 16x²
− 25y²)
g) n
( 4x
− 5y)
c) (0,2²) 4
g) (a m ) n
c) (x a+3 ) b
g) (y p ) q-2
3. a) (x²y³)² b) (a³b) 5
c) (d 5 e³) 3
e) (a³b 4 ) n
f) (3x 5 y 2 )² g) (5a²b 7 ) 4
4.
a)
2
3 2 ⎛ ⎞
4a b
⎜ 4 3 ⎟
⎝ 2x y ⎠
10
m n
⎛ 5a b
⎞
b) ⎜ 7 3 ⎟
⎝ 10p q
⎠
6 8 4
( 6a
b )
c) 5 2 4
( 3a
b )
n
d)
h)
5
5
( − 5)
⋅ ( − 0,
1)
⋅
4
m m
a ) ⋅ ( b)
27 2
9²
1
( )³
8
1
( )³
4
( − x)
d) 4
y
4
n−
1
( 48a)
h) n−
1
( 12a)
5
32y
d)
5
100000z
n+
( p²
− 16q²)
h) n+
1
( p + 4q)
d) (10³) 5
h) (y 2a ) b
d) (z n-3 ) 4
h) (k 2m+3 ) n
d) (m 6 n 5 ) 8
h) 5(m 4 n 5 ) 4
5 6
( 4x
y )³
d) 6 3
( 2x
y²)
1
m
2
5

Potenzrechnung – Vermischte Aufgaben 2
1. Berechne.
a) 0,2 3
d)
3 3
2 3
2. Berechne.
⎛ 2 ⎞
a) ⎜
3
⎟
⎝ ⎠
b) 3,1 3
7 3
⋅ e) 4 ⋅ 4
3
− 2
⎛ 3 ⎞
d) ⎜
4
⎟
⎝ ⎠
3. Löse die Klammern auf.
4 2
a) x ⋅ (x
3
x + )
b)
d)
3 6 4
x ⋅ (x x − )
e)
3
⎛ 1 ⎞
b) ⎜
5
⎟
⎝ ⎠
3
⎛ 3 ⎞
e) ⎜
10
⎟
⎝ ⎠
4 3
c) 3 + 4 5⋅
f) (-3) 5
− 2
⎛ 5 ⎞
c) ⎜
8
⎟
⎝ ⎠
f)
3 2 4
2a ⋅ (a a − )
c)
3 2 4
3a ⋅ (a a + )
f)
− 1
⎛ 9 ⎞
⎜
10
⎟
⎝ ⎠
3
(x + y)
2
(m n) −
−
4. Faktorisiere.
2 4 3
a) 16x y + 32x yz
5 3 2
40x −y z
3 6
b) 1,4u v −
2 4
0,7u v
5 5
2,8u −v
c)
4 2 5
3a − 2a 6a+
d)
5. Vereinfache die folgenden Terme.
7 3 a) 4 4 −
⋅ b)
d)
4
16a ⋅ 2a
e)
2 3 3 2 2
9a b + 6a b 12a+ b
3 5
3,5x ⋅ 2x
c)
5 2
4x 3x −
⋅ f)
2 2
25x ⋅ y
9a ⋅ 2a
6. Vereinfache die folgenden Terme.
3 2 a) x ⋅ x
2 3
b) a a
5
a −
⋅ ⋅
3
c) b ⋅ b
3 2
d) a ⋅ b
2
a ⋅
3
b ⋅
2
e) a ⋅ a
f) a ⋅ a
− 3 3 −
7. Vereinfache die folgenden Terme.
2 a) x ⋅ x
3 2 b) x ⋅ x
5
x ⋅
4 3
c) a ⋅ x
2
a ⋅
2
x ⋅
2 2
d) 2a ⋅ b
3
4a⋅ 4
b ⋅
2 − 3
e) 5a ⋅ b
4
b ⋅
3
2a ⋅
−
f) 9 ⋅ a
4 2
3b⋅ 5
a
3
b ⋅ ⋅
8. Berechne.
3 3
x ⋅ x
a)
4
x ⋅ x
d)
12a b
2
4ab
2 2
b)
e)
16x y
2
4xy
x
x
2n+ 3
3n− 4
2 2
c)
f)
5 2
a ⋅ a
a ⋅ a
2 4
3 2
12a ⋅ b
− −
6 5
3
2a b
9. Vereinfache die folgenden Terme.
6 3
a) (12a ) : ( − 3a )
3 3
b) (3x) ⋅ (2y)
c) ( ) 2
3
(3x)
d)
6 2
4x y z
3 5
2x y z
e)
(15a )
5
(5a)
2 5
f)
x y
(xy)
10. Schreibe mit positiven Exponenten.
2 a) 2 x −
⋅
3 3
b) x a −
⋅ c) 9 10 −
⋅
3 3
d) a x
5
b −
⋅ ⋅
3
e) (5b) −
f) 3x y
8 4
3
4
− 2 5 −

11. Schreibe ohne Bruch.
2
a
a)
3
x
a
d)
10000
12. Schreibe ohne Bruch.
a)
d)
1
2
x
2
3
x − e)
1
b) 3 4
a ⋅ b
9
e) 5 2
a b
1
b) 2 a ⋅ b
3
5
5
Potenzrechnung – Vermischte Aufgaben 3
1
3
7
c) 3
10
f)
c)
5x
4
y
2
3
x
5
1 1
3 2
f)
13 ⋅ 5
1. Vereinfache.
a) 4a³ + 3x² - 5z 4 + 2a³ + z 4 – 2x³ b) (15a 4 – 12a 3+n + 9a 1-n ) : 3a²
c) (8x³ - 28x² - 12x + 2) : ( 4x + 2) d) (x 4 – 1) : (x – 1)
e) (6a²b 4 c³ + 9a 5 c³ + 9a 5 b²c 6 ) 2
f) 3x²(4x³ – 5x 4 )
2. Vereinfache
3 7
a ⋅ b
a)
2 4
a ⋅ b
b)
z ⋅ z
m
z
n m− n
c)
4z ⋅ 8y
6 3
2y ⋅ z
5 7
5 5
2 3
5 6
3 2 2
x+ b
⎛ x ⋅ y ⎛ ⎞x
y ⋅ ⎞ ⎛ 2x ⋅ y x⎞ 2y ⎛ ⋅ 250a
d) ⎜ :
2 3 ⎟3
5
a ⋅ b
⎜
a b ⋅
⎟e)
:
f)
⎞
⎜ 2 3 ⎟2 2⎜
x b⎟
⎝ ⎝ ⎠
⎠ ⎝ 3a ⋅ 2b 2a⎠ 3b ⎝ ⋅ 75a ⋅ a
⎠
3. Vereinfache.
a)
d)
4 3 6
6x 9y 0,5z 3x
⋅ ⋅ ⋅
3 2
1,5y ⋅ 18z
4
3x ⋅
n+ 2
x +
n 1
2x
n
x
n+
x −
4. Vereinfache.
6 5
x + x
a)
4 3
x + x
5n− 1 1 5n + 5 n
+
d) a ⋅ b a b⋅
⋅ e)
b)
2
(3x +
2 3
6x ) x
4 2
x ⋅ y
5
⋅y ⋅
c)
5
1− x 1
e) + f)
7 2
x x
b)
22x y − 121x y
3 4
11x y
77x y+
5 6 4 5 6 7
c)
(a ⋅ x )
(a ⋅ x )
− 3 5 2 −
2 − 3 4
a + a
a + a
n+ 1 n 2 +
n n + 1
x y
y x
2n+ 1 3n 1 +
3n 2n −1
6 5 n 4
(s s ) s −
− ⋅ f) (x²y³ + xy 4 ) 2
5. Vereinfache
5 5
a) 1,2xy z ⋅ (0,5x²yz
8
0,8xy²z −
7
1,2xyz )
4 5
b) + (x y −
3 4
x y
5 3 2
x y + ) : (xy)
c)
4 5
x ⋅ x 5
: 3
8 2x b³
4
⎛ 2 5 ⎞
e) ⎜ ⎟
7 ⎟
⎝ ⎠
2 2
⎛
d) ⎜
⎝
5r − 7s
7c + 2d
⋅
21c ⎞
5r
⎟
⎠
6d ⎛
7s
⎜
⎝
+
−
⎞
⎟
⎠
4 6 2 3
2x ⋅ 5x 5x 4x
f) : 9 8
4y 8y
⋅

Multiplikation und Division von Potenzen mit gleicher Basis – Lösungen
1. 4 5 2
a) 3 ⋅ 3 ⋅ 3
3 5 2
b) 12 ⋅ 12 ⋅ 12
3 2
c) x ⋅ x ⋅ x
= 3 11
= 12 10
= x 6
3 5 2 7
e) k ⋅ k ⋅ m ⋅ m
5
f) x ⋅
3 2
y ⋅ x ⋅ y
2 3
g) a ⋅ b ⋅ b ⋅ a
2. a)
3. a)
= k 8 m 9
2 n
x ⋅ x
= x 7 y 4
b)
= x 2+n
= b m+3
5 2x
e) a ⋅ a
2m
m
f) z ⋅ z
= a 2x+5
= z 3m
= x m+1
e)
3 m−
2
x ⋅ x
b)
a
= a 6x-3
⋅ a
⋅ a
2x x+
1 3x
− 4
= a³b 4
d)
= d 12
3
b b
m ⋅ c) y y
a ⋅ d)
= y a+1
g)
= a 6m
5 x−
7
a ⋅ a
c)
= a x-2
f)
m+
x ⋅ x
= x 6m+1
⋅ x
2 3m−
4 2m
+ 3
3m
2m
a ⋅ a ⋅
a
m
h)
2m m−
1
y ⋅ y
d)
= p 5 q 11
= x 2m
h) m
= m 9x
= y 3m-1
= x 2p-2
p−
1 3 p + 4 5 p−
8
g) z ⋅ z ⋅ z
m
h) y
= z 9p-5
3 5
d ⋅ d ⋅
d
4
4 6
p ⋅ q ⋅ p ⋅
m m
x ⋅ x
x
= y 4m+1
3x
⋅ m
⋅ x
4x
p−
4 p + 2
4. a) x²(x³ + x 4 ) b) a³(a 5 + a 4 ) c) 3b³(4b² - 5b 5 )
= x 5 + x 6
= a 8 + a 7
= 12b 5 – 15b 8
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 )
= a 2m+1 – a 4m-1
= y 5a+1 – y 3a-4
= x n+2 + x n+1
⋅ y
q
⋅ m
5
2x
⋅ y
− 2 2m
− 5 m+
8
5. a) (x² + x³)² b) (y 3 – y 4 )² c) (a 6 + a 4 )² d) (b 3 – b 7 )²
= x 4 + 2x 5 + x 6
= y 6 – 2y 7 + y 8
= a 12 + 2a 10 + a 8
= b 6 – 2b 10 + b 14
e) (2a² + 3a³)² f) (4x 5 – 2x 6 )² g) (6d 5 – 3d 4 )² h) (3m² + 5m 7 )²
= 4a 4 + 12a 5 + 9a 6
= 16x 10 – 16x 11 + 4x 12
= 36d 10 – 36d 9 + 9d 8
= 9m 4 +30m 9 + 25m 14
6. a) (a² + a³)(a² - a³) b) (x 5 + y 4 )(x 5 – y 4 ) c) (m³ + n 5 )(m³ - n 5 )
= a 4 – a 6
= x 10 – y 8
= m 6 – n 10
d) (3x 4 – 2y 5 )(3x 4 + 2y 5 ) e) (4y³ - 6x 7 )(4y³ + 6x 7 ) f) (3a 4 – 4b³)(3a 4 + 4b³)
= 9x 8 – 4y 10
= 16y 6 – 36x 14
= 9a 8 – 16b 6
7. a) (a 3 + a 4 )(a² + a 5 ) b) (x² - x 5 )(x³ + x 6 ) c) (a 3 – b²)(a 5 + b³)
= a 5 + a 8 + a 6 + a 9
= x 5 – x 11
= a 8 + a 3 b 3 – a 5 b² - b 5
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 )
= y 7 – y 10 + y 8 – y 11
= 4a 8 – 4a 5 b 4 + 6a³b³ - 6b 7
= k 2m+1 +k m+n+2 +k m+n+1 +k 2n+2
8. Schreibe als Produkt von Potenzen.
9. a)
10. a)
a) x 3+5
=
3
x
5
x ⋅
= 5 5
5
5
= a x-3
8
3
a x
a³
b)
b) a 3n+2
=
3n
a
2
a ⋅
= 8 6
8
8
9
3
x
b)
x
y
= x y-1
c)
= 12 8
c)
= x 2m
12
12
c) 5 m+n
=
m
5
n
5 ⋅
13
5
m
x 3
x
m
d)
= a 4
a
a
9
5
6m
y
3
d) m
y
= y 3m
d) z 5k+3m
=
5k
z
3m
z ⋅
7
y
e) 6
y
x m+4
m 4
= x x ⋅
f)
= y = k 6
e)
2
k m
k
= k 2m-3
3
f)
k
k
23
17
2
d p
d²
= d 2p-2

11. a)
12 a)
x
x
m
m−
3
b)
a
a
= x³ = a 2m+1
x
x
n+
3
n+
2
3m
m−
1
b)
a
a
= x = a n+1
7 x+
4
v
e)
5x
− 2
v
x
f)
x
= v 2x+6
= x p
2n
− 1
n−
2
3 p+
1
2 p+
1
c)
z
z
= z 2x+4
3x
x−
4
d)
k
k
= k 2a-3
4a
2a
+ 3
3x
+ 4
y
c) x−
2
y
= y 2x+6
z
g)
z
= z x
k + 3x
k + 2x
2b
y
e) b+
3
y
= y b-3
d)
b
b
= b
3
b
h)
b
= b 2m-1
13. a) (x 8 + x 6 – x 5 ) : x² b) (15a³ + 12a 6 – 3a 4 ) : 3a²
= x 6 + x 4 – x 3
= 5a + 4a 4 – a²
c) (21b 8 – 28b 4 + 14b 5 ) : 7b³ d) (3x n+3 – 9x 2n-4 + 12x n+5 ) : 3x²
= 3b 5 – 4b + 2b² = x n+1 – 3x 2n-6 + 4x n+3
e) (35y m+2 – 20y 2m+4 + 15y m+8 ) : 5y m
f) (4z a+3 + 16z 2a+5 – 12z a+4 ) : 2z a
= 7y² - 4y m+4 + 3y 8
= 2z 3 + 8z a+5 – 6z 4
14. a)
=
5
15x
y
7
21a
b
8
5
6
25 ² ³
2x³
y²
: 10
35a
b
6
b)
5
6 p q
3
r²
s
4
4
3p
q³
: 7 5
r s
f)
3m+
4
3m+
3
m−
6
m−
5
m
m
= m 2b-7
9 7 5
18a
b 12a
b³
c) : 4 6
35x³
y²
21x
y
x y a b
2
= 2pqr 5 s² = 0,9a 4 b 4 xy 4
15. Löse die folgenden Exponentialgleichungen:
a)
5
a
a
a²
x 12
1 y
= b) y 8
y
x+
=
x = 3 x = 3 x = 3
x+
5
d) z =
3x
+ 5
z
3x
− 15
z
2x
+ n
e) a =
3x
+ 2n
a
3n
a
f) a
x = 15 x = 2n x = 2
c)
2 1
a x−
x+
1
=
a
a
15
10
a
= m
a
m+
1
− 2
4b
2b
+ 7

Multiplikation und Division von Potenzen mit gleichem Exponenten -
Lösungen
1. a)
2. a)
3 3
5 ⋅ 2
b)
2 2
8 ⋅ 3
c)
3 3
0, 5 ⋅ 4
d)
5 5
0, 5 ⋅ 10 ⋅
= 10³ = 1000 = 24² = 576 = 2³ = 8 = 1 5 = 1
4 4 4
e) 4 ⋅ 3 ⋅ 0,
25
6 1 6
f) 6 ⋅ ( )
6
5 18 5
g) ( )³ ⋅ ( )³ ⋅ ( )³
6 25 3
2 4 6 4
h) ( ) ⋅ ( ) ⋅ 5
3 10
= 3 4 = 81 = 1 6 = 1 = 1³ = 1 = 2 4 = 16
= 20 x
e)
x x
5 ⋅ 4
b)
m m
a ⋅ b
f)
= (ab) m
= 36 a
a a
12 ⋅ 3
c)
= (yz) k
k k
y ⋅ z
g)
4
= 20 x+1
x+
1 x+
1
⋅ 5
( x − y
= (x² - y²) 8
8
8
+ y)
⋅ ( x ) h)
d) 3
= 18 m-4
− 4 − 4
⋅ 6
m m
0,
2
( −
m
m
a + b)
⋅ ( a b)
= (a² - b²) m
3. a) ( − 4)³
⋅ ( − 0,
5)³
4 1 4
b) ( − 3)
⋅ ( )
3
c) ( − 4)²
⋅ ( − 1,
5)²
5
5
d) ( − 5)
⋅ ( − 0,
1)
⋅ 2
= 2³ = 8 = (-1) 4 = 1 = 6² = 36 = 1 5 = 1
e) a³ ⋅ ( − b)³
5
f) ( − x) ⋅ ( −
5 5
y)
⋅ z g) ( − p)² ⋅ ( − r)²
⋅ s²
m m
h) ( − a ) ⋅ ( − b)
= (-ab)³ = (xyz) 5
= (prs)² = (ab) m
4. 3
24
a)
3
8
5
36
b)
5
18
3
49
c)
3
7
27
d)
9²
2
= 3³ = 27 = 2 5 = 32 = 7³ = 343 = 3² = 9
4
2,
6
e) 4
1,
3
0,
4²
f)
0,
5²
5
3
g ) 3 5
( )
2
1
( )³
8
h)
1
( )³
4
= 2 4 = 16 = 0,8² = 0,64 = 2 5 = 32 = 0,5³ = 0,125
5. a)
6. a)
a
b
6
6
6
⎛ a ⎞
= ⎜
b
⎟
⎝ ⎠
8² ⋅ 3²
e)
6²
n
x
b) n
y
n
=
⎛
⎜
⎝
x
y
⎞
⎟
⎠
15³ ⋅ 3³
f)
9³
= 4² = 16 = 5³ = 125 = 4 m
27a³
8b³
3
=
⎛
⎜
⎝
3a
2b
⎞
⎟
⎠
( a²
− b²)³
e)
( a + b)³
b)
25a²
b²
2
⎛ 5a ⎞
= ⎜
b
⎟
⎝ ⎠
( 4a²
− 9b²)
f) 5
( 2a
− 3b)
= (a – b)³ = (2a + 3b) 5
5
c)
n+ 1
x
( − x)
n+ 1
d) 4
y
y
n+ 1
=
⎛
⎜
⎝
x
y
⎞
⎟
⎠
( 12x)
g) m
( 3x)
c)
m
27x³
1000y³
⎛ 3x ⎞
= ⎜ ⎟
⎝ 10y ⎠
( 16x²
− 25y²)
g) n
( 4x
− 5y)
3
= (4x + 5y) n
n
4
4
=
⎛
⎜
⎝
x
−
y
⎞
⎟
⎠
n−
1
( 48a)
h) n−
1
( 12a)
= 4 n-1
d)
5
32y
100000z
5
5
⎛ 2y ⎞
= ⎜
10x
⎟
⎝ ⎠
n+
( p²
− 16q²)
h) n+
1
( p + 4q)
= (p – 4q) n+1
5
4
1
5

Potenzen von Potenzen - Lösungen
1. a) (2³)² b) (4²) 4
= 2 6
= 4 8
e) (a 5 )³ f) (x³) m
= a 15
= x 3m
2. a) (x m ) n+1
= x mn+m
e) (p 2k+1 ) 3
= p 6k+3
b) (a y ) x-1
= a xy-y
f) (b n-4 ) m
= b mn-4m
c) (0,2²) 4
= 0,2 8
g) (a m ) n
= a mn
c) (x a+3 ) b
= x ab+3b
g) (y p ) q-2
= y pq-2p
3. a) (x²y³)² b) (a³b) 5
c) (d 5 e³) 3
= x 4 y 6
= a 15 b 5
= d 15 e 9
e) (a³b 4 ) n
f) (3x 5 y 2 )² g) (5a²b 7 ) 4
= a 3n b 4n
= 9x 10 y 4
= 625a 8 b 28
4.
2
3 2 ⎛ ⎞
4a b
a) ⎜ 4 3 ⎟
⎝ 2x y ⎠
6 4
4a
b
= 8 6
x y
10
m n
⎛ 5a b
⎞
b) ⎜ 7 3 ⎟
⎝ 10p q
⎠
10m 10n
a b
= 10 70 30
2 p q
Potenzrechnung – Vermischte Aufgaben 2 – Lösungen
6 8 4
( 6a
b )
c) 5 2 4
( 3a
b )
= 16a 4 b 24
1. Berechne.
a) 0,2 3 = 0,008 b) 3,1 3 = 29,791 c)
d)
3 3
2 3
2. Berechne.
d) (10³) 5
= 10 15
h) (y 2a ) b
= y 2ab
d) (z n-3 ) 4
= z 4n-12
h) (k 2m+3 ) n
= k 2mn+3n
d) (m 6 n 5 ) 8
= m 48 n 40
h) 5(m 4 n 5 ) 4
= 5m 16 n 20
5 6
( 4x
y )³
d) 6 3
( 2x
y²)
8y
= 3
x
12
4 3
3 + 4 5⋅
= 581
7 3
⋅ = 216 e) 4 ⋅ 4 = 4 10 f) (-3) 5 = –243
3
⎛ 2 ⎞ 8
a) ⎜
3
⎟=
⎝ ⎠ 27
− 2 2
⎛ 3 ⎞ ⎛ 4 ⎞ 16
d) ⎜ =
4
⎟ = ⎜
3
⎟
⎝ ⎠ ⎝ ⎠ 9
3. Löse die Klammern auf.
4 2 3
x ⋅ (x + x )
a)
6 7
= x + x
d)
3 6 4
x ⋅ (x − x )
= x − x
9 7
3
⎛ 1 ⎞
b) ⎜ =
5
⎟
⎝ ⎠
3
1
125
⎛ 3 ⎞ 27
e) ⎜ =
10
⎟
⎝ ⎠ 1000
b)
4. Faktorisiere.
2 4 3
16x
y + 32x yz
a)
3
= 8x²y( 2y 4xz
5 3 2
40x
−y
z
3 2 2
5x + y z ) −
3 2 4
2a ⋅ (a a − )
5 7
= 2a 2a−
− 2 2
⎛ 5 ⎞ ⎛ 8 ⎞ 64
c) ⎜
=
8
⎟ = ⎜
5
⎟
⎝ ⎠ ⎝ ⎠ 25
f)
− 1
⎛ 9 ⎞ 10
⎜ =
10
⎟
⎝ ⎠ 9
3
(x + y)
c)
3 3
= x 3x²y + 3xy² + y
+
3 2 4
3a ⋅ (a a + )
e)
5 7
= 3a 3a+
f)
(m −
− 2
n)
=
1
(m − n) ² m²
1
=
2mn − n²
+
3 6 2 4 5 5
1,4u v − 0,7u
v 2,8u
−v
b)
4
= 0,7u²v
(2uv²
1 4u³v)
− −

4 2 5
3a − 2a 6a+
c)
3
= a²(3a² 2 6a − ) +
5. Vereinfache die folgenden Terme.
7 3
4 4
a)
4
4
−
⋅
=
3,5x ⋅ 2x
b)
8
= 7x
4
16a ⋅ 2a
d)
5
= 32a
5 − 2
4x ⋅ 3x
e)
3
= 12x
6. Vereinfache die folgenden Terme.
3 2
x ⋅ x
a)
5
= x
a a a
b)
0
a 1
−
⋅ ⋅
= =
3 2 2
a ⋅ b a ⋅
d)
5 5
= a b ⋅
3
b ⋅
2
a ⋅ a
e)
3
= a
3 5
2 3 5
7. Vereinfache die folgenden Terme.
2
x ⋅ x
a)
3
= x
3 2
x ⋅ x
b)
10
= x
5
x ⋅
2 2 3
2a ⋅ b 4a⋅ d)
5 6
= 8a b ⋅
4
b ⋅
2 3 4
5a b b
e)
1
10a b
3
2a
−
⋅ ⋅
=
⋅
⋅
8. Berechne.
3 3
x ⋅ x
4
x ⋅ x
a)
6
x
= = x 5
x
12a b
2
4ab
d)
= 3a
2 2
b)
e)
16x y
2
4xy
=
x
x
4x
2n+ 3
3n− 4
2 2
2 3 3 2 2
9a b + 6a b + 12a b
d)
= 3a²b(3b² 2ab + + 4)
− −
2n+ 3 (3n − 4) − 7 − n
= x x =
9. Vereinfache die folgenden Terme.
6 3
(12a ) : ( − 3a )
a)
3
= 4a −
3 3
(3x) ⋅ (2y)
b)
3 3 3
= 6 x ⋅ y ⋅
d)
6 2
4x y z
3 5
2x y z
=
2x
3 3
y −
e)
(15a )
5
(5a)
=
2 5
(3a)
10. Schreibe mit positiven Exponenten.
− 2
2 ⋅ x
x ⋅ a
a)
=
2
x²
b)
5
3 − 3
x³
=
a³
c)
f)
25x ⋅ y
=
2 2
25x²y²
− 3 3 −
9a ⋅ 2a
=
18a
3
b ⋅ b
c)
4
= b
5 2
a ⋅ a
f)
7
= a
− 6
4 3 2
a ⋅ x a ⋅
c)
6 5
= a x ⋅
2
x ⋅
− 4 2
9 ⋅ a 3b⋅ f)
− 1
= 27a
b ⋅
5
a
−3
b ⋅ ⋅
2 4
a ⋅ a
3 2
a ⋅ a
c)
6
a
= = a 5
a
f)
c)
f)
c)
12a ⋅ b
3
2a b
=
6 5
3 4
6a b
2
3 ( (3x) )
6 6 6
= (3x)
3 x=
8 4
x y
(xy)
3
8 4
x y 5
= x y=
3 3
x y
9 ⋅
10
9
=
10
− 4
4

a
x b −
⋅ ⋅
3 3 5
d) a³ ⋅ x³
= 5
b
11. Schreibe ohne Bruch.
2
a
3 a) x
= a² a
− 3
x ⋅
d) 10000
5
a 10 −
= ⋅
12. Schreibe ohne Bruch.
1
a) 2 x = x
d)
x
2
−
3
=
3
1
x²
13. Vereinfache.
a)
d)
2a ⋅ 4,5a
= 9a² 3a
=
27
3
= 9 3 =
e)
b)
( 5b)
− 3
1
= 3
(5b)
1
3 4
a ⋅ b
=
− 3
a
9
b
4
⋅
−
5
e) a b
2
= 9
− 5
a⋅
2
b
−
⋅
1
b) 2 3 a ⋅ b = a ⋅
1
3
e) 5 5
b)
5 = 5³
=
6
a
a³
5
27
e) 13,5
Potenzrechnung – Vermischte Aufgaben 3
5
5
= 2 3=
2
3 b
f)
3x
y
=
− 2 5 −
3
x²
y
5
7
3
c) 10
= 7 ⋅
f)
5
5x
4
y
3
10 −
5 − 4
= 5x y ⋅
2
c) 3 3
x = x²
1 1
3 2 13 ⋅ 5
f)
=
3
13
⋅ 5
c)
f)
3 x
=
6
x
0,4
5 ⎛ ⎞
8
⎜ x ⎟
⎝ ⎠
=
1
4 x =
4
x
1. Vereinfache.
a) 4a³ + 3x² - 5z 4 + 2a³ + z 4 – 2x³ b) (15a 4 – 12a 3+n + 9a 1-n ) : 3a²
= 6a³ + 3x² - 4z 4 – 2x³ = 5a² – 4a 1+n + 3a -1-n
c) (8x³ - 28x² - 12x + 2) : ( 4x + 2) d) (x 4 – 1) : (x – 1)
= 2x² - 8x + 1 = x³ + x² + x + 1
e) (6a²b 4 c³ + 9a 5 c³ + 9a 5 b²c 6 ) 2
= 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
f) 3x²(4x³ – 5x 4 )
= 12x 5 – 15x 6
2. Vereinfache
3 7
a ⋅ b
a) 2 4
a ⋅ b
= a b³
b)
z ⋅ z
m
z
= 1
n m− n
c)
4z ⋅ 8y
6 3
2y ⋅ z
5 7
=
16yz²

5 6
5 5
3 2
2
2
3
x+ b
⎛ x ⋅ y ⎛ ⎞x
y ⋅ ⎞ ⎛ 2x ⋅ y x⎞ 2y ⎛ ⋅ 250a ⎞
⎜ :
2 3 ⎟3
5
d) a ⋅ b
⎜
a ⋅ b
⎟ ⎜ :
2 3 ⎟2 2⎜
x b⎟
⎝ ⎝ ⎠ ⎠e)
3a ⋅ 2b
2a 3b ⋅ 75a ⋅ a
⎝ ⎠ ⎝ ⎠ f)
20 25 5 10
10
= x y a b
= y ⋅ 3a²
=
3
3. Vereinfache.
a)
d)
4 3 6
6x 9y 0,5z 3x
=
⋅ ⋅ ⋅
3 2
1,5y ⋅ 18z
4
3x
⋅
xz
4
n+ 2
x +
n 1
2x
n
x
n+
x −
= x² + 2x
1 −
4. Vereinfache.
a)
x + x
x + x
= x²
6 5
4 3
5n− 1 1 5n + 5+
n
a ⋅ b a b⋅
⋅
d)
5n 6+
6n
= a b
5. Vereinfache
b)
e)
2
(3x +
2
6x )
4 2
x ⋅ y
3
x
5
⋅y ⋅
=
9xy³
5
1− x 1
+ 7 2
x x
=
1
7
x
(a ⋅ x )
2 − 3 4
(a ⋅ x )
c)
x²
=
a²
n 1
a +
n 2
a
f) n
a +
n + 1
a
= a
b)
x y
5 6 4 5
22x y − 121x y
6 7
77x y+
c) y x
3 4
11x
y
= x²
y
= 2x²y²
11xy− 7x³y³ +
(s − s ) ⋅ s
e)
n+ 2 n + 1
= s s −
6 5 n − 4
f)
− 3 5 2 −
+ +
2n+ 1 3n 1 +
3n 2n −1
4
(x²y³ − xy )²
4 6 7 8
= x y 2x³y
+ x²y +
5 5 8 7
4 5 3 4 5 3 2
1,2xy z ⋅ (0,5x²yz 0,8xy²z − 1,2xyz ) + (x y − x y + x y ) : ( xy)
a)
b)
6 6 7 9 6 8
= 0,6x³y z 0,96x²y − z 1,44x² y z + = x²y³ x y² − x³y +
c)
4 5
x ⋅ x 5
: 3
8 2x b³
=
b³x
20
12
4
⎛ 2 5
⎜ 7
e) ⎝
⎞
⎟
⎠ f)
=
400
49
2 2
⎛
d) ⎜
⎝
5r − 7s
7c + 2d
⋅
21c ⎞
5r
⎟
⎠
6d ⎛
7s
⎜
⎝
+
−
⎞
⎟
⎠
= 9
2x ⋅
5x 5x ⋅ 4x
: 9 8
4y 8y
=
4 6 2 3
5
x
y