Grundoperationen mit ganzen Zahlen
4 +5⋅6 + 2 =
4 ⋅ 5 + 6(3 + 2) =
(2 ⋅ 2
3
)
(
)
− 24 ⋅ 3 − 2 ⋅ 42 − 5 : 3 =
2 ⋅ (7 ⋅ 3 − 6 ) ⋅ 3 + 72 : 6 =
(
)
3 ⋅ 2 2 + 2 ⋅ 2 3 − 8 ⋅ 2 + 52 : 4 =
(3 ⋅ 2
2
)
(
)
+ 2 ⋅ 23 − 4 ⋅ 4 2 + 5 : 4 =
(− 1) + (+ 3 ) − (− 2) =
(− 2) − (− 3) + (− 1) =
(− 10 ) − (+ 4 ) + (+ 13 ) =
(− 2) − (+ 2) + (− 2) =
(+ 2) − (− 2) + (− 2) − (+ 2) =
(− (− 2)) − (+ (− 2)) − (− (− 2)) =
− 4 − (+ 2) + (+ 5 ) =
(+ 7 ) − (− (− 10 ) − 10 ) =
(− 2) − ((− 6) − − 4 ) =
(+ 8 ) − ((− 21) − (− 4 ) − 1) =
− (− 9 ) − ((+ 3 ) + (+ 7 ) − − 2 ) =
(+ 8 ) − ((− 3) + (− 2) − − 1) =
1
(− 15 )(− 2)(+ 5 ) =
− 10 ⋅ (− 10 )(+ 10 ) =
(− 2)(+ 3 )(− 7 ) =
(− 7)((+ 4) − (− 5 )) =
(+ 10 )((− 6 ) − 8) =
((− 3 ) − (− 2)) ⋅ − 4
(− 1)2 − (− 1)3 =
(− 2)3 − (− 2) =
(− 5)2 + (− 5)2 =
(+ 4)(− 2)4 − 2(− 5)2 + (− 3)2 (− 2)2 =
2
2
2
3
7(− 3 ) − (− 7 ) + (− 3 ) (− 2) =
(+ 8) − (− 3 )2 (− 7 ) + (− 1)4 (− 3) =
[(− 18) : (− 3 ) − (+ 2)]⋅ [(− 1) − (− 2)(+ 5)] =
[(− 49 ) : (+ 7) − (+ 3)]⋅ [(+ 2) + (+ 7)(− 6 )] =
[(− 4 )(− 7 ) − (− 3 )]⋅ [(+ 3 ) − (+ 2)(− 4)] =
(+ 10 )(− 1)6 + (+ 7 )(− 2)2 + (+ 2)(+ 3 )2 =
(+ 5 )(− 2)2 − [(− 2)2 (+ 3) + (+ 6)(− 3)2 ] =
(+ 5 )(− 2)2 − (− 2)([ + 3) + (+ 2)(− 3)2 + (+ 6)(− 1)4 ] =
2
Bruchrechnen mit Zahlen
1 3
+
7 7
=
3 4
3
+ −
5 11 55
=
 − 2  5
−
−  =
 3  4
(− 5) ⋅
4 30
−
− 9 18
=
5
−2 3 −3
+ 2⋅
+ +
2
 3 2  18
=




 3 + 1  ⋅ 2 − 21 =
1
4
20
3+ 

3

4 2   2+5
 −  : −
 =
9 
 9 15  
1
−1
5
2
(
6)
1−
20
=




4
1 −
: 3
4 4

 1− 
5

3
1+
=
2
1
(− 2)
(− 1)3 + 2
2
(− 5 )
+
1
3
2
(− 3) +
=
3
2 5
+
a a
=
1 1
+
=
a b
2 b
+
a c
=
−2 5 7
+ +
x
x x
=
3 5 1
− +
=
a b b
a (− 5 ) a
+
−
−3
b
b
1
=
1
1+
a
a+
1
a
=
2
−a
3
a
1
+
(− 2) 3
(− 1)
2+
a
1+
=
a
b
=
=
2
+b
3
1
=
1
1+
1
1+
1+
1
a
1
2+
2+
2
3+
3
1
4
=
4
Umformen von Termen mit Variablen
a+a+a+b+b =
p+p+p−q−q−q−q+r −p−p−p−p =
4a + 3a + 9a =
7a + 5a + 13b + 9b + 2b =
31x + 12y + 7 x − 9 x − 10 y − 2y =
25f − 19g − 5 − 18f − 7 − g =
7 x + 9 x + (− 10 x ) + (− 20 x ) =
8f + 5g + 7h + (− 3f ) + (− 5g) + h =
(17a − 66 − 2c ) + (5c − 20a − b) =
a − (b − c − d) =
2k − (a + k + m + n) =
9a − (4b − 3a ) =
(7x + 3 y ) − (7x + 3 y + z ) =
(− 19m − 5n + 1) − (4m − 2n − 3) =
16 x − y − (3 x − y ) + (4 x − 1) − (x + y )
20a − (16a − (2a + b )) =
=
12p − 4q − (6p − 4q) − ((3p + 4q) − (2p + 9q)) =
15t − ((3t − 6u) − v ) − (5u − (20t + 4v )) =
7m − 5n − (5m − (3n − n) − (2m + n) − 5n) =
(7a + 5b − (3a + b)) − (5a + 3b − (2a − b))
=
a2 ⋅ a =
a5 ⋅ a2 =
r ⋅ rs =
5p ⋅ 4q =
8ab ⋅ 4ac =
2
3
cd ⋅ ce =
3
4
6 xy ⋅ 2xy =
5ax ⋅ 3ax ⋅ ax =
5
(5ab )2 =
(2acd)2 =
(3a )2 ⋅ 5a 3
=
5 3 2
1 2 2
 a ⋅ ac ⋅ a bc  =
3
4
5

(− 2a ) ⋅ 3a =
(− bc ) ⋅ b =
(
)
3 x 2 − 2x 3
(
2
ac − 2a c
2
=
)
=
2a(− b )(− c )(− 3d) =
(− 5n )⋅ 2n (− 2n) =
1 1 
−  − a (− 6 x )(− y )
4
2
2
3
3
2


=
2
 3 
3
− 6(− 2a ) − a  (− a )
 2 
=
x 2 y + x 2 y + x 2 y + xy + xy + xy + xy =
5 x 2 y + 3 xy + xy + 2xy 2
(
)
=
(
14ab 2 + − 3a 2b + 17ab 2 + − 47a 2 b
(
)
4 x 2 − 2x 2 − 4 x + 1
)
=
=
7x : x =
4d 3 : d =
4c 3 : 4c 3
=
(
− 27c 3 : − c 2
(14x
2
)
=
)
− 35 x : (− 7 x ) =
1

  1 
 8apq − bpq  :  − pq  =
8

  4 
5
4
26 x − 39 x + 52x 3 − 13 x 2 : − 26 x 2
(
)(
)
=
 32 2 2 20 3 2

x y +
x y − 36 x 4 y 3 − 4 x 5 y  : − 4 x 2 y

7
 3

(
)
=
6
Faktorisieren
m(u + v ) − 3(u + v ) =
2a(a − b ) − b(a − b ) =
x(y − z ) − (z − y ) =
a 3 (2ab − c ) + a 2 (2ab − c ) =
10q(9e − 6 ) − 5(9e − 6 ) =
2r 2 (r − 7 ) − r (− r + 7 ) =
7m(r + s) − 3n(r + s) − 4(r + s ) − n(r + s) + r + s =
− a(x − y ) + 2b(x − y ) − 3c (x − y ) + 4(x − y ) =
3p 2 (u − v ) − 2p(v − u) − 8(u − v ) + u − v =
(3a − 5c )(m + 4) − (a + c )(4 + m) =
(a − b)(5z − 1) + (2a − 2b)(z + 4) =
q(2x − 3 y ) − (q + 1)(− 2x + 3 y ) =
au + av + bu + bv =
12st + 16s − 27t − 36 =
81ab + 72ad + 36bc + 32cd =
mn − m + n − 1 =
20r 2 s + 4rs 2 − 5r − s =
6ab + 3a − 12b − 6 =
7
a 2 + 2ab + b 2 =
4c 2 + 28cd + 49d 2 =
36u 2 + 60uv + 25v 2 =
x 2 − 2xy + y 2 =
m 2 − 2m + 1 =
p 4 − 8p 2 + 16 =
4c 2 − 9d 2 =
36n 2 − 1 =
x4 − y4 =
a 2 + 8a + 15 =
x 2 + 9 x + 20 =
d 2 + 20d + 91 =
a 2 + 2a − 24 =
t 2 − 6t − 7 =
y 2 − y − 30 =
y 2 + 20 y + 400
=
x 2 − 4x + 4
2
u 2 (4u + 4 ) + (4u + 4 )
=
u 2 + 4u + 4
a 4 − 81
=
a 2 − 6a + 9
8
Bruchrechnen
3 u−4
−
=
2
2
u−v u+v
−
=
3
3
3 + 2x x − 4 2x − 7
−
+
=
y
y
y
2m 3
+ =
3k 2 k
1
2
1
+
+ 2 =
2
mn n
m
1
x
+b−
=
a
2y
1−
u+v
=
u−v
u+v −
1
=
u−v
4
+ 1− z =
z+2
3x + y
1
+
=
2x + 2y x + y
m
1
−
=
2
m −1 m −1
1
1
1
−
+ 2
=
2
a + b (a − b )
a − b2
1
s −1
2s
+
−
=
2
1 + s (s + 1)
1− s2
4x − 2
3
1
2
−
+
−
=
2
x − 1 1+ x x − 1 1− x
3
2v
1
1
+ 2
+
−
=
2
2
u+v u−v
u + 2uv + v
u −v
2
9
a(u − v ) ⋅
u2
=
au 2 − auv
9
1 1− t 2
⋅
⋅
=
t − 1 3t + 3 − 3
1  3
m2 − 1
 m
−
⋅
⋅
=


3
 m − 1 m + 1 m 3m + 3m
 a + b a − b   a2 + b2 
=
a⋅
−
 ⋅ 1 −
2a 2 
a−b a+b 
a 2 − ab b 2 − ab
:
=
3a 2b
2ab 2
1  s
 s
=
 2 2 −
:
s−t s+t
s −t
m4 − 1 m2 + 1
:
=
mn
2mn
3
m =
1
+1
m
x +1
2 =
2
x −1
x−
p
−1
1+ p
=
p
1+
p +1




a
1 −
⋅ a
a  a −1

 1−

a +1

=
1− a
u
−1
v
=
2
(
u + v)
1−
4uv
10
Potenzieren mit negativen Exponenten
2
 1
−3
 −  − (+ 2)
2


=
(− 2) ⋅ − 2 −1 −1
(
(
))
(− 6 )−1 + (− 2 −2 ) =
−1
 2
−1
 −  : (− 4 )
 3
(2 − 2 )
−1 −2
( )
−1
− − 12
2
 1  2
−  : − 
 2  4
(− 1)−2
2
(
− 1)
(− 2) −
 1
1−  − 
 2
=
−1
=
−1
−1
  1  −1 −3 
 −  −1 
 3 



=
−1
−2
− 3 + (− 3 ) ⋅ (− 1)
(
)
−1
  1  −1 −3 
 −  −1 
 2 



=
−1
−3
− 2 + (− 2) ⋅ (− 1)
(
)
−
(− 1)2 + (− 12 )⋅
−2
2
(− 2 )⋅ (− 1 )
−1
−2
1
2
 1
− 
 3 =
−2
+ (− 2)
(− 1)−2 ⋅ (− (− 2 2 ))−1 =
2
(
− 2) + 1
1−
−1
1 − (− 1)
11
(u
−1
(2x
+ v −1
−1
)
−1
=
)(
)
− 1 ⋅ 2x −1 + 1 =
 1

 −1 + 1
 3x

−1
=
x −1 + y −1
=
x −2 − y −2
 −1 a 
 2a − 
3

−1
=
−1
1

3 +  : k =
k

2

−1
 m +  : (2n) =
n

m − 1

−1
 2m −

2 

−1
=
  t  −1 1 −1

   + t + 2t −1 
 2 

2


−1
1 − a. 1 − a(a + 1)−1

1− a
 ⋅ a(a − 1)−1

=
(
b
−1
)
−1
=
−1
−1
(
1 − b ) − (1 + b )
⋅
=
(1 − b)−1 + (1 + b)−1
1+ a
−1
−1
−1
(
1 − a ) + (1 + a )
+
=
(1 − a )−1 − (1 + a)−1
12
Radizieren
Potenzieren mit gebrochenen Exponenten
8⋅ 2 =
2 27
⋅
=
3
8
a ⋅ ab ⋅ 2a =
(1+ 2 )
2
(
=
)(
)
5 −1 ⋅
5 +1 =
(3 − 3 )⋅ ( 3 − 1) =
(3 ⋅ 2 − 2)⋅ ( 2 + 2) =
(
)(
3+ 2 ⋅
)
3− 2 =
2
 3 + 5 − 3 − 5  =


1.21⋅ 10 3 =
ab 2 − b 3 =
(x + y )3 ⋅ (x − y ) =
4
1
1+ 3
=
1
2a − b
x−y
x− y
=
=
13
(1+ a )⋅ 1− a
=
1− a
2

1 
 =
x ⋅  x −
x

 a
a   a
a  a

−
⋅
+
 2
  2
+ 4 =
2
2

 

v
−v =
v −1
v −1⋅
1
m2 + 1
+
1
⋅
=
m
m2
(x + 1)2 − x =
2
 x +1
x − 1 
x 2 − 1 ⋅ 
−
=
x + 1 
 x −1

2 + 3 
2 ⋅  3 −
=
1 + 3 

(2)
5
−
1
5
⋅ 3 16

8 ⋅  2 + 2

4
8 −1 : 3 16 −1
2 −1 ⋅
3
2
3 :
4
( 4)
3
−1
(3 )
−1
−
27 ⋅ 3 9 −1




1
2
−1
=
=
1
2
=
2 −2 ⋅ 3 3 −1
=
1 3 2
: 3
43
14

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