Quadratwurzeln
L¨ osungen+ (L+)
(a) √
144 = 12 (b) √
324 = 18 (c) √
529 = 23 (d) √
576 = 24
(e) √
225 = 15 (f) √
400 = 20 (g) √
256 = 16 (h) √
625 = 25 Aufgabe 1.2
(a) √
119250736929: 6 Vor- und 0 Nachkommastellen (b) √
36692168704: 6 Vor- und 0 Nachkommastellen (c) √
279016.26275601: 3 Vor- und 4 Nachkommastellen (d) √
0.0000184041: 0 Vor- und 5 Nachkommastellen Aufgabe 1.3
22 09 = (40 + 7)2 16 42
6 09
6 09 (2·40 + 7)·7
0
√2209 = 47
Aufgabe 1.4 30 25 = (50 + 5)2 25 52
5 25
5 25 (2·50 + 5)·5
0
√3025 = 55
Aufgabe 1.5
5 33 61 = (200 + 30 + 1)2
4 22
1 33
1 29 (2·20 + 3)·3
0 04 61
04 61 (2·200 + 2·30 + 1)·1
0
√53361 = 231
Aufgabe 1.6
32 03 56 = (500 + 60 + 6)2
25 52
7 03
6 36 (2·50 + 6)·6
0 67 56
67 56 (2·500 + 2·60 + 6)·6
0
√320356 = 566
Aufgabe 1.7 x2 = 1
2
x1+ a x1
= 1 2
3 4 +5
6
= 19 24
|x2−x1|=
19 24− 3
4
= 1 24 < 1
10 N¨aherung:
r5 8 ≈ 19
24 Aufgabe 1.8
x2 = 1 2
x1+ a x1
= 1 2
2 + 3
2
= 7 4
|x2−x1|= 7 4−2
= 1 4 ≥ 1
10 x2 →x1
x2 = 1 2
x1+ a x1
= 1 2
7 4 +12
7
= 97 56
|x2−x1|=
97 56− 7
4
= 1 56 < 1
10 N¨aherung: √
3≈ 97 56 Aufgabe 1.9 x2 = 1
2
x1+ a x1
= 1 2
1 + 1
2
= 3 4
|x2−x1|= 3 4−1
= 1 4 ≥ 1
10 x2 →x1
x2 = 1 2
x1+ a x1
= 1 2
3 4 +2
3
= 17 24
|x2−x1|=
17 24− 3
4
= 1 24 < 1
10
N¨aherung:
2 ≈ 24 Aufgabe 2.1
(a) √
6 400 000 000 = 80 000 (b) √
36 100 000 000 = 190 000
(c) √
19 600 000 000 = 140 000 (d) √
5 760 000 = 2 400 Aufgabe 2.2
(a) √
0.0049 = 0.07 (b) √
0.0000000256 = 0.00016 (c) √
0.00000324 = 0.0018 (d) √
0.000529 = 0.023 Aufgabe 2.3
(a) √
4·1026= 2·1013 (b) √
2.25·1026= 1.5·1013
(c) √
6.4·1053= 8·1026 (d) √
5.76·1068= 2.4·1034 Aufgabe 2.4
(a) √
9 + 16 =√ 9 +√
√ 16
25 = 3 + 4 5 = 7 falsch (b) √
25 + 144 =√
25 +√
√ 144
169 = 5 + 12 13 = 17 falsch (c) √
49 + 576 =√
49 +√
√ 576
625 = 7 + 24 25 = 31 falsch
Radizieren und Addieren d¨urfen nicht vertauscht werden.
Aufgabe 2.5 (a)
q 51925 =
q144 25 = 125 (b) q
3014 =q
121 4 = 112 (c)
q 21425 =
q64 25 = 85
Aufgabe 2.6 (a) √
64·121 =√ 64·√
121 = 8·11 = 88 (b) √
81·484 =√ 81·√
484 = 9·22 = 198 (c) √
36·324 =√ 36·√
324 = 6·18 = 108 Aufgabe 2.7
(a) √ 2·√
8 =√
2·8 =√
2·2·2 = 4 (b) √
2·√
18 =√
2·18 =√
2·2·3 = 6 (c) √
5·√
45 =√
5·45 =√
5·5·3 = 15 Aufgabe 2.8
(a)
r100 529 =
√100
√529 = 10 23 (b)
r100 169 =
√100
√169 = 10 13 (c)
r100 289 =
√100
√289 = 10 17 Aufgabe 2.9
(a)
√150
√54 =
r150 54 =
r6·25 6·9 =
r25 9 =
√25
√9 = 5 3 (b)
√567
√343 =
r567 343 =
r7·81 7·49 =
r81 49 =
√81
√49 = 9 7 (c)
√75
√108 = r 75
108 =
r3·25 3·36 =
r25 36 =
√25
√36 = 5 6 Aufgabe 2.10
(a) p
(−116)2 = 116 (b) q
19 60 ·q
19 60 = 1960 (c)
q
1−10112
= q
−(1011−1)2
= 1011−1
(a) p√
81 =√ 9 = √
3 (b) p
64 +√
289 =√
64 + 17 = √
81 =√ 9
(c) r
72 + q
68 +√ 169 =
q
72 +√
68 + 13 = q
72 +√ 81
=√
72 + 9 =√ 81 = 9 Aufgabe 2.12
(a) √
293·293 = 293 (b) √
293: 293 =√ 1 = 1 (c) √
293−293 =√ 0 = 0 (d) √
293+ 293=√
2·293 =√
294 = 247 Aufgabe 2.13
(a) √
0.0045·√
450000 =√ 45·√
45 = 45 (b) √
0.0003·√
30000 =√ 3·√
3 = 3 (c) √
0.092·√
920 =√ 9.2·√
9.2 = 9.2 (d) √
11.5·√
1150 =√
115·√
115 = 115 Aufgabe 2.14
(a) √ 10·√
4.9 = √
10·4.9 =√ 49 = 7 (b)
√361000
√1000 =
r361000 1000 =√
361 = 19 (c) √
1000·√
0.121 =√
1000·0.121 =√
121 = 11 (d)
√5290
√10 =
r5290 10 =√
529 = 23
Aufgabe 2.15 (a) √
7·√ 7·√
7·√ 7 = 49 (b) √
6·√ 6·√
6·√ 6 = 36
Aufgabe 2.16 (a) √
3·√ 15·√
8·√
10 =√ 3·√
3·√ 5·√
2·√ 2·√
2·√ 2·√
5 = 3600 (b) √
6·√ 5·√
8·√
15 =√ 2·√
3·√ 5·√
2·√ 2·√
2·√ 3·√
5 = 3600 Aufgabe 2.17
(a) √
0.5·√
50 =√ 0.5·√
10·√ 5 =√
5·√ 5 = 5 (b) √
38 000·√
3.8 =√
380·√
100·√
3.8 = √
380·√
380 = 380 (c) √
0.123·√
0.00123 =√
0.0123·√ 10·√
0.00123
=√
0.0123·√
0.0123 = 0.0123 Aufgabe 2.18
(a) r 7
65 r13
35 = r 7
65· 13 35 =
r 7·13 5·13·5·7 =
r 1 5·5 = 1
5 (b)
r1 2
r2 3
r3 4 =
r1 2· 2
3· 3 4 =
r1 4 = 1
2 Aufgabe 2.19
(a) 3√ 22
= 9·2 = 18 (b) 2√
52
= 4·5 = 20 (c) 7√
32
= 49·3 = 147 Aufgabe 2.20
(a) √ 3· 8
3·√ 3 = √
3·√ 3· 8
3 = 3· 8 3 = 8 (b)
√12 3 ·
√12 4 =
√12·√ 12 3·4 = 12
12 = 1 (c)
√2 2
!2
=
√22 22 = 2
4 = 1 2
(a) p√
7·p 7√
7 =p√
7·7·√ 7 = √
7·7 = 7 (b)
q 5√
3· q
15√ 3 =
q 5√
3·15·√ 3 = √
5·15·3
=√
15·15 = 15 (c) p√
3√ 12p√
2√
18 =p√
36·√
36 =√ 36 = 6 Achtung: falsche Aufgabenstellung in (c) Aufgabe 2.22
(a) √ 3 √
3 + 1
=√ 3·√
3 +√
3·1 = 3 +√ 3 (b) √
2 √ 8−√
2
=√ 2·√
8−√ 2·√
2 = 4−2 = 2 (c) √
10q
1 5 +
q1 2
=√ 10·
r1 5 +√
10· r1
2 =√ 2 +√
5
Aufgabe 2.23 (a) 3 +√
2
3−√ 2
= 9−2 = 7 (b) 4−√
3
4 +√ 3
= 16−3 = 13 (c) √
7−√ 5 √
7 +√ 5
= 7−5 = 2 Aufgabe 2.24
(a) p
3 + 2√ 2p
3−2√ 2 =√
32−22 ·2 = √ 1 = 1 (b) p
7 + 2√ 6p
7−2√ 6 =√
72−22 ·6 = √ 25 = 5 Aufgabe 2.25
(a) q
7 + 3√ 5 +
q
7−3√ 5
2
= 7 + 3√ 5
+ 2 q
7 + 3√ 5
q
7−3√
5 + 7−3√ 5
= 14 + 2 q
7 + 3√ 5
7−3√ 5
= 14 + 2√
49−45 = 14 + 2√
4 = 18
(b) q
8 + 2√ 7−
q
8−2√ 7
2
= 8 + 2√ 7
−2 q
8 + 2√ 7
q
8−2√
7 + 8−2√ 7
= 16−2 q
8 + 2√ 7
8−2√ 7
= 16−2√
64−28 = 16−2√
36 = 16−12 =4 Aufgabe 2.26
(a) 2√ 7 =√
4√ 7 = √
28 (b) 2q
1 2 =√
4q
1 2 =√
2 (c) 15√
5 = √ 125√
5 = q1
5
(d) 10√
0.31 = √
100·√
0.31 =√ 31 (e) 23
q3 2 =
r4 9
r3 2 =
r2 3 Aufgabe 3.1
(a) √
28 = 2√ 7 (b) √
48 = 4√ 3 (c) √
92 = 2√ 23 (d) √
98 = 7√ 2 Aufgabe 3.2
(a) √
162 = 9√ 2 (b) √
604 = 2√ 151 (c) √
882 = 21√ 2 (d) √
925 = 5√ 37 Aufgabe 3.3
(a) 1
√3 =
√3
√3·√ 3 =
√3 3 = 1
3
√3
(b) 1
√5 =
√5
√5·√ 5 =
√5 5 = 1
5
√5
(c) 1
√8 = 1 2√
2 =
√2 2√
2·√ 2 =
√2 4 = 1
4
√2
(a) r1
7 = 1
√7 =
√7
√7·√ 7 =
√7 7 = 1
7
√7
(b) r 2
11 =
√2
√11 =
√2·√
√ 11 11·√
11 =
√22 11 = 1
11
√22
(c) r 1
18 = 1 3√
2 =
√2 3√
2·√ 2 =
√2 6 = 1
6
√2
Aufgabe 3.5
(a)
√12
√20 = 2√ 3 2√
5 =
√3·√
√ 5 5·√
5 =
√15 5 = 1
5
√15
(b)
√16
√50 = 4 5√
2 = 4√ 2 5√
2·√
2 = 4√ 2 10 = 2
5
√2
(c)
√33
√44 =
√11·√ 3 2√
11 =
√3 2 = 1
2
√3
Aufgabe 3.6 (a) √
50 +√
75 +√
32 = 5√
2 + 5√
3 + 4√
2 = 9√
2 + 5√ 3 (b) √
45−√
180 +√
125 +√
16 = 3√
5−5√
6 + 5√ 5 + 4
= 4 + 8√
5−5√ 6 (c) √
72 +√
80 +√
245−√
18 = 6√
2 + 4√
5 + 7√
5−3√ 2
= 3√
2 + 11√ 5 Aufgabe 3.7
(a) 4
√28+ 3
√150 − 5
√7 = 4 2√
7 + 3 5√
6 − 5
√7
= 2
√7+ 3 5√
6 − 5
√7 = 3 5√
6 − 3
√7
= 3√ 6 5√
6√
6 − 3√
√ 7 7√
7 = 3√ 6 30 − 3√
7 7 = 1
10
√6− 3 7
√7
(b) 1
√27− 6
√24+ 3
√54+ 10
√25 = 1 3√
3 − 6 2√
6 + 3 3√
6+ 2
= 2 + 1 3√
3− 3
√6 + 1
√6 = 2 + 1 3√
3 − 2
√6
= 2 +
√3 9 −2√
6
6 = 2 + 1 9
√3− 1 3
√6
Aufgabe 3.8 (a) √
3 +√ 62
= 3 + 2√ 3√
6 + 6 = 9 + 2√
18 = 9 + 6√ 3 (b) 2√
3−√ 22
= 12−4√
6 + 2 = 14−4√ 6 Aufgabe 3.9
(a) 3 +√ 502
= 9 + 2·3·5√
2 + 50 = 59 + 30√ 2 (b) √
6 +√ 242
= √
6 + 2√ 62
= 3√ 62
= 9·6 = 54 Aufgabe 3.10
(a) 1 +√ 2 +√
52
= 1 + 2 + 5 + 2√
2 + 2√
5 + 2√ 10
= 8 + 2√
2 + 2√
5 + 2√ 10 (b) 1 +√
2−√ 5
1 +√ 2 +√
53BF
= 1 +√ 22
−5
= 1 + 2√
2 + 2−5
=−2 + 2√ 2 Aufgabe 3.11
(a) 3 +√
√ 3
3 = 3 +√ 3
·√
√ 3 3·√
3 = 3√ 3 + 3
3 = 1 +√ 3
(b)
√2 +√
√ 10
5 =
√2 +√ 10
·√
√ 5 5·√
5 =
√10 +√ 50 5
=
√10 + 5√ 2
5 =√
2 + 1 5
√ 10
(a) 1 2 +√
3 = 2−√
3 2 +√
3
2−√
3 = 2−√ 3
4−3 = 2−√ 3
(b) 6
1−2√
2 = 6 1 + 2√ 2 1−2√
2
1 + 2√
2 = 6 + 12√ 2 1−8
=−6 7 − 12
7
√ 2 Aufgabe 3.13
(a) 8−√ 7 3−√
5 = 8−√ 7
3 +√ 5 3−√
5
3 +√
5 = 24 + 8√
5−3√ 7−√
35 4
= 6 + 2√ 5− 3
4
√7− 1 4
√35
(b)
√5−2√
√ 2
5 + 2√ 2 =
√5−2√ 2 √
5−2√ 2
√5 + 2√ 2 √
5−2√
2 = 5−4√ 10 + 8 5−8
= 13−4√ 10
−3 =−13 3 +4
3
√ 10 Aufgabe 3.14
√5
5 − 2 1−√
2 +4 +√
√ 3 3
= 5·√
√ 5 5·√
5 − 2·(1 +√ 2) (1−√
2)·(1 +√
2)+ (4 +√ 3)·√
√ 3 3·√
3
= 5√ 5
5 − 2 + 2√ 2 1−2 +4√
3 + 3 3
=√
5 + 2 + 2√ 2 + 4
3
√ 3 + 1
= 3 + 2√ 2 + 4
3
√3 +√ 5 Aufgabe 4.1
(a) p
p2 =|p|
(b) √
9u4 = 3u2 (kein Betrag n¨otig) (c) √
k7 =√
k6·k =√ k6√
k=|k3|√ k (d) √
32z13=√
16z12·2z=√
16z12√
2z = 4z6√ 2z
Aufgabe 4.2 (a) √
a2b6 =|ab3| (b) √
5s3t4 =√
5s·s2·t4 =|s|t2√ 5s (c) √
e16m20 =e8m10 (d) p
x7y9 =p
x6y8 ·xy=|x|3y4√ xy
Aufgabe 4.3
(a) ra2
b2 = |a|
|b|
(b)
r100m4
n2 = 10m2
|n|
(c) s
p4 q8 = p2
q4
(d)
r18v11 v3 =√
18v8 = 3v4√ 2 Aufgabe 4.4
(a) √
x2+ 4x+ 4 =p
(x+ 2)2 =|x+ 2|
(b) √
a2−22a+ 121 =p
(a−11)2 =|a−11|
(c) √
20m2+ 40m+ 20 =p
20(m2+ 2m+ 1)
=p
4·5(m+ 1)2 = 2|m+ 1|√ 5 Aufgabe 4.5
(a) p
(xy)2+ (xy)2+ (xy)2 =p
3(xy)2 =|xy|√ 3 (b) √
k7+k6 =p
k6(k+ 1) =|k3|√ k+ 1 (c) p
(r+s)2−4rs=√
r2+ 2rs+s2−4rs=√
r2−2rs+s2
=p
(r−s)2 =|r−s|
(a) √
x−2; D={x∈R: x≥2}
(b) √
x+ 7; D={x∈R: x≥ −7}
(c) √
3−x; D={x∈R: x≤3}
(d) √
2x−3;D ={x∈R: x≥1.5}
Aufgabe 4.7 (a) √
x2−9; D={x∈R: 0≤x≤3}
(b) √
x2+ 1; D ={x∈R: −∞< x <∞}=R (c) √
4−x2; D={x∈R: −2≤x≤2}
(d) √
x2−10x+ 25 =p
(x−5)2; D={x∈R: −∞< x <∞}=R Aufgabe 5.1
x√ 3 +√
12 =√
32−x√ 3 x√
3 +x√ 3 =√
32−√ 12 2x√
3 = 4√
2−2√ 3 x√
3 = 2√ 2−√
3 x= 2√
2−√
√ 3 3 x= 2√
2−√ 3
·√
√ 3 3·√
3 x= 2√
6−3 3 x=−1 + 2
3
√ 6 Aufgabe 5.2
x√ 2−√
5 = √ 20 +x x√
2−x=√
5 + 2√ 5 x √
2−1
= 3√ 5 x= 3√
√ 5 2−1 x= 3√
5· √ 2 + 1
√2−1 √ 2 + 1 x= 3√
10 + 3√ 5 2−1 x= 3√
5 + 3√ 10 Aufgabe 5.3
√
2−x2
= (3−x)(4−x) 2−2x√
2 +x2 = 12−7x+x2 2−2x√
2 = 12−7x 7x−2x√
2 = 10 x 7−2√
2
= 10 x= 10
7−2√ 2 x= 10 7 + 2√
2 7−2√
2
7 + 2√
2 = 70 + 20√ 2 49−8 x= 70
41+ 20 41
√2
Aufgabe 6.1 (a) x2 = 289
x1 = 17 x2 =−17 (b) x2 = 49
64 x1 = 7
8 x2 =−7
8
x1 = 2.1 x2 =−2.1 Aufgabe 6.2
(a) 2x2 = 50 x2 = 25 x=±5 (b) 3
2x2 = 8
27 || · 2 3 x2 = 16
81 x=±4
9
(c) 0.0002x2 = 0.0098 x2 = 0.0098
0.0002 = 98 2 = 49 x2 =±7
Aufgabe 6.3
(a) (x+ 4)(x−4) = 9 x2−16 = 9 x2 = 25
x=±5 (b) (3x−5)(3x+ 5) = 24
9x2−25 = 24 9x2 = 49 x2 = 49
9 x=±7
3 Aufgabe 6.4
(a) (x−7)2 = 121 = 112 x−7 = 11
x1 = 18
∨ x−7 = −11 x2 =−4
(b) (2x+ 3)2 = 529 = 232 2x+ 3 = 23
2x= 20 x1 = 10
∨ 2x+ 3 = −23 2x=−26 x2 =−13 Aufgabe 6.5
(2x−11)2 = (x−10)2 2x−11 =x−10
x= 1
∨ 2x−11 = −(x−10) 2x−11 = −x+ 10
3x= 21 x= 7 Aufgabe 6.6
(2x+ 5)2 = (4x−7)2 2x+ 5 = 4x−7
−2x=−12 x= 6
∨ 2x+ 5 = −(4x−7) 2x+ 5 = −4x+ 7
6x= 2 x= 1 3 Aufgabe 6.7
a =x, b= 2x und c= 5x S = 2(ab+bc+ca) 1.36 = 2(2x2+ 10x2+ 5x) 0.68 = 2x2+ 10x2+ 5x 0.68 = 17x2
x2 = 0.04 x= 0.2 dm
Kantenl¨angen: 2 cm, 4 cm, 10 cm Aufgabe 7.1
√x−1 = 3 ||2 x−1 = 9 ||+ 1
x= 10 L: √
10−1 =√ 9 = 3
L={10}
Aufgabe 7.2
√x+ 6 = 2√ x ||2 x+ 6 = 4x || −x
6 = 3x ||: 2 2 =x
L: √
2 + 6 =√ 8 R: 2√
2 =√ 4·√
2 =√
4·2 =√ 8 L={2}
Aufgabe 7.3
√x+ 6 =√
2x ||2 x+ 6 = 2x || −x
6 =x L: √
6 + 6 =√ 12 R: √
2·6 =√ 12 L={6}
Aufgabe 7.4
√x+ 1 =x−1 ||2
x+ 1 =x2−2x+ 1 || −x−1 0 =x2−3x
0 =x(x−3) x1 = 0
x2 = 3 x= 0; L: √
0 + 1 =√
11 = 1; R: 0−1 =−1 x= 3; rhs: √
3 + 1 =√
4 = 2; lhs: 3−1 = 2 L={3}
Aufgabe 7.5
x+ 2√
x−2 = 1 || −x 2√
x−2 = 1−x ||2 4(x−2) = 1−2x+x2
4x−8 = 1−2x+x2 0 =x2−6x+ 9 0 = (x−3)2 x= 3
L: 3 + 2√
3−2 = 3 + 2√
1 = 3 + 2·1 = 5 R: 1
L={ } Aufgabe 7.6
√
x2−4−x= 1 ||+x
√
x2−4 = x+ 1 ||2
x2−4 = x2+ 2x+ 1 || −x2−1
−5 = 2x ||: 2
−2.5 = x L: √
6.25−4−(−2.5) = √
2.25 + 2.5 = 1.5 + 2.5 = 4 R: 1
L={ } Aufgabe 7.7
√x−7−1 =√
x−8 ||2 (√
x−7−1)2 =x−8 BF2 x−7−2√
x−7 + 1 =x−8 ||+ 2√
x−7−x+ 8 2 = 2√
x−7 ||: 2 1 =√
x−7 ||2 x−7 = 1
x= 8 L: √
8−7−1 = √
1−1 = 1−1 = 0 R: √
8−8 =√ 0 = 0 L={8}
√x+ 1 =√
x+ 11 ||2 x+ 2√
x+ 1 =x+ 11 || −x−1 2√
x= 10 ||: 2
√x= 5 ||2 x= 25 L: √
25 + 1 = 5 + 1 = 6 R: √
25 + 11 =√ 36 = 6 L={25}
Aufgabe 7.9 q√
x+ 4−x= 2 ||2
√x+ 4−x= 4 ||+x
√x+ 4 =x+ 4 ||2 x+ 4 =x2+ 8x+ 16
0 =x2+ 7x+ 12 0 = (x+ 3)(x+ 4) x1 =−3
x2 =−4 x=−3; lhs:
q√
−3 + 4−(−3) = √
1 + 3 = 2; rhs: 2 x=−4; lhs:
q√
−4 + 4−(−4) = √
0 + 4 = 2; rhs: 2 L={−3,−4}