Quadratwurzeln L¨osungen+ Pr¨ufungsvorbereitung
Aufgabe 1.1
6 Vorkommastellen und 5 Nachkommastellen Aufgabe 1.2
10 56 25 = (300 + 20 + 5)2
9 32
1 56
1 24 (2·30 + 2)·2
32 25
32 25 (2·300 + 2·20 + 5)·5
0
√105625 = 325
Aufgabe 1.3 x1 = 1
2
x0+ a x0
= 1 2
11
6 +11/12 11/6
= 1 2
11 6 + 11
12· 6 11
= 1 2
11 6 +1
2
= 1 2·14
6 = 7 6 x2 = 1
2
x1+ a x1
= 1 2
7
6 +11/12 7/6
= 1 2
7 6 +11
12· 6 7
= 1 2
7 6+ 11
14
= 1 2· 82
42 = 41 42 N¨aherung:
r11 12 ≈ 41
42 Aufgabe 1.4
(a) √
16<√
19<√
25 ⇒ 4<√ 19<5 (b) √
100 <√
101 <√
121 ⇒ 10<√
101 <11 (c) √
289 <√
300 <√
324 ⇒ 17<√
300 <18 Aufgabe 2.1
(a) √
0.000324 = 0.018 (b) √
3.61 = 1.9 (c) √
16.9·109 = 1.3·104 = 130 000 Aufgabe 2.2
(a) √ 6√
15√ 5√
2 = √ 2√
3√ 3√
5√ 5√
2 = 2·3·5 = 30 (b) √
2252 = 225
(c) 17 = 17 = 289 (d) p
(−35)2 = 35 (e) √
8.1·109 =√
81·108 = 9·104 (f) √
0.144·107 =√
1.44·106 = 1.2·103 (g) √
0.00043·√
0.043 =√
0.0043·√
0.0043 = 0.0043 (h) √
0.0156·√
15 600 =√
15.6·√
15.6 = 15.6 Aufgabe 2.3
(a)
√28
√7 = r28
7 =√ 4 = 2
(b)
√3240
√10 =
r3240 10 =√
324 = 18
(c) r7
3 r5
6 r7
5 = r7
3· 5 12· 7
5 = r49
36 = 7 6 Aufgabe 2.4
(a) 2√ 3 =√
4√ 3 = √
12 (b) 3√
2 =√ 9√
2 = √ 18 (c) 5√
5 =√ 25√
5 =√ 125 (d) 10√
7 =√ 100√
7 = √ 700 Aufgabe 2.5
(a) √
441 000 000 =√
441·√
1 000 000 = 21·1000 = 21 000 (b) √
0.000064 = 0.008 (c) √
1118= 119
(d) q
614 = r25
4 =
√25
√4 = 5 2 (e)
qp√
256 =p√
16 =√ 4 = 2
q p √ p √ √
Aufgabe 3.1 (a) √
12 =√
4·3 = √ 4√
3 = 2√ 3 (b) √
9000 =√
900·10 = 30√ 10 (c) √
75 =√
25·3 = 5√ 3 (d) √
723=√
722·7 = √ 722·√
7 = 711·√ 7 Aufgabe 3.2
(a) √
50 +√
32−√
75 +√
150 = 5√
2 + 4√
2−5√
3 + 5√
6 = 9√
2−5√
3 + 5√ 6 (b) 7 +√
3
7−√ 3
= 49−3 = 46 (c) 4 +√
22
= 16 + 2·4·√
2 + 2 = 18 + 8√ 2 (d) √
3 +√ 52
= 3 + 2√ 3√
5 + 5 = 8 + 2√ 15 Aufgabe 3.3
q 3−√
2 + q
3 +√ 2
2
= 3−√ 2
+ 2 q
3−√ 2
q 3 +√
2 + 3 +√ 2
= 6 + 2 q
3−√ 2
3 +√ 2
= 6 + 2√
9−2 = 6 + 2√ 7 Aufgabe 3.4
(a) 2
√8 = 2 2√
2 = 1
√2 = 1·√
√ 2 2·√
2 = 1 2
√ 2
(b)
√3
√3−√ 2 =
√3· √ 3 +√
2
√3−√ 2
· √ 3 +√
2
= 3 +√ 6
3−2 = 3 +√ 6
(c)
√2 +√
√ 3 2 +√
5 =
√2 +√ 3 √
2−√ 5
√2 +√ 5
(√ 2−√
5
= 2−√
10 +√ 6−√
15 2−5
=−2 3 − 1
3
√ 6 + 1
3
√ 10 + 1
3
√ 15
(a) √
b2 =|b|
(b) √
a2+a2 +a2+a2 =√
4a2 = 2|a|
(c) √
9b4c2 = 3b2|c|
(d) p
4x2−12xy+ 9y2 =p
(2x−3y)2 =|2x−3y|
(e) p
(x−3)2+ 6x=√
x2−6x+ 9−12x=√
x2+ 6x+ 9 =p
(x+ 3)2 =|x+ 3|
(f) √ x32
=x3 Aufgabe 4.2
(a) s
x4 y2 =
√ x4 py2 = x2
|y|
(b) r1
t
√ t3·
q t3√
t = r1
t
√
t3·t3√ t =
q t2
√ t4
=√
t2·t2 =t2 Aufgabe 5.1
√
3x=x+ 1
√
3x−x= 1 x √
3−1
= 1
x= 1
√3−1
x= 1·(√ 3 + 1) (√
3−1)(√ 3 + 1) x= 1 +√
3 3−1 x= 1
2+ 1 2
√ 3
Aufgabe 5.2 3x−√
7 =√ 5x+ 4 3x−√
5x= 4 +√ 7 x(3−√
5) = 4 +√ 7 x= 4 +√
7 3−√
5 x= (4 +√
7)(3 +√ 5) (3−√
5)(3 +√ 5) x= 12 + 4√
5 + 3√ 7 +√
35 9−5
= 3 +√ 5 + 3
4
√ 7 + 1
4
√ 35
Aufgabe 6.1 3x2 = 0.48
x2 = 0.16 x=±0.4 L={±0.4}
Aufgabe 6.2 (x+ 9)2 = 4x2 x+ 9 = 2x
9 =x x= 9
x+ 9 =−2x 3x=−9
x=−3 L={9,−3}
Aufgabe 6.3
(4x+ 3)2 = (3x−10)2 4x+ 3 = +(3x−10) 4x+ 3 = 3x−10
x=−13
4x+ 3 = −(3x−10) 4x+ 3 = −3x+ 10
7x= 7 x= 1 L={−13,1}
√2x+ 4 =√
8−x ||2 2x+ 4 = 8−x
3x= 4 x= 43 Probe:
L:
q
2· 43 + 4 = q8
3 +123 = q20
3
R:
q
8− 43 = q24
3 − 43 = q20
3 (stimmt) L={43}
Aufgabe 7.2
√x+ 4 = 7√
x || −√ x 4 = 6√
x || : 2 2 = 3√
x ||2 4 = 9x || : 9 x= 49
Probe:
L:
q4
9 + 4 = 23 +123 = 143 R: 7
q4
9 = 723 = 143 (stimmt) L={49}
Aufgabe 7.3
√x2−2 = x−1 ||2
x2−2 = x2−2x+ 1 || −x2−1
−3 = −2x || : (−2) x= 32
Probe:
L:
q
3 2
2
−2 = q9
4 −84 = q1
4 = 12 R: 32 −1 = 12 (stimmt)
L={32}
Aufgabe 7.4
√x+√ 3 =√
x+ 15 (√
x+√
3)2 =√
x+ 152 (Doppelprodukt!) x+ 2√
x√
3 + 3 =x+ 15 || −x−3 2√
3x= 12
√
3x= 6 3x= 36
x= 12 Probe:
L: √
12 +√
3 = 2√ 3 +√
3 = 3√ 3 R: √
12 + 15 =√
27 = 3√ 3 L={12}
Aufgabe 7.5
√x+ 5 +√ x=√
5x+ 5 ||2 (binomische Formel) x+ 5 + 2√
x+ 5√
x+x= 5x+ 5 (Produkt mit Wurzeln isolieren) 2√
x+ 5√
x= 3x ||2 4(x+ 5)x= 9x2 4x2+ 20x= 9x2
0 = 5x2 −20x faktorisieren 0 = 5x(x−4) L¨osungen ablesen x1 = 0
x2 = 4 Probe f¨ur x= 0
L: √
0 + 5 +√ 0 =√
5 R: √
5·0 + 5 =√ 5 (ok) Probe f¨ur x= 4
L: √
4 + 5 +√ 4 = 5 R: √
5·4 + 5 = 5 (ok) L={0,4}
