Exercices racines carrées
Correction 1
1. a. 75 = 25×3 = 52×3 b. 32 = 16×2 = 42×2 c. 18 = 9×2 = 32×2 d. 72 = 36×2 = 62×2 e. 1 000=100×10=102×10 f. 242=121×2=112×2 2. a. √
75 =√
52×3 =√ 52×√
3 = 5×√ 3 = 5√
3 b. √
32 =√
42×2 =√ 42×√
2 = 4×√ 2 = 4√
2 c. √
18 =√
32×2 =√ 32×√
2 = 3×√ 2 = 3√
2 d. √
72 =√
62×2 =√ 62×√
2 = 6×√ 2 = 6√
2 e. √
1 000 =√
102×10 =√ 102×√
10
= 10×√
10 = 10√ 10 f. √
242 =√
112×2 =√ 112×√
2 = 11×√
2 = 11√ 2 Correction 2
a. √
32×2 =√ 32×√
2 = 3√ 2 b. √
13×42=√ 13×√
42= 4√ 13 c. √
12 =√
22×3 =√ 22×√
3 = 2√ 3 d. √
48 =√
22×22×3 = 4√ 3 e. √
1 600 =√
42×102= 4×10 = 40 f. √
360 =√
36×10 =√ 62×√
10 = 6√ 10
Correction 3 a. √
5×√ 30 =√
5×30 =√
5×5×6 = 5√ 6 b. √
24×√ 6 =√
24×6 =√
6×4×6 = 6×2 = 12 c. 5√
2×2√
2 = 10(√
2)2
= 20 d. 3√
6×4√
3 = 12√
6×3 = 12√
2×32= 36√ 2 e. √
39×2√
13 = 2√
39×13 = 2√
3×132= 26√ 3 f. 2(√
15)2= 2×15 = 30 Correction 4
1. a. √ 6×√
40 =√
6×40 =√
2×3×8×5 =√ 16×15
= √ 16×√
15 = 4√ 15 b. √
3×√ 15 =√
3×√
3×5 =√ 3×√
3×√ 5 = 3√
5 c. √
8×√ 18 =√
8×18 =√
8×2×9 =√ 16×9
= √ 16×√
9 = 4×3 = 12 2. a. √
2(√
18 + 2)
=√ 2×√
18 + 2√ 2
= √
2×18 + 2√ 2 =√
36 + 2√
2 = 6 + 2√ 2 b. √
5(√
5−√ 45)
=√ 5(√
5−√ 45)
= (√
5)2
−(√
5×√ 45)
= 5−√ 5×45
= 5−√
5×5×9 = 5−√
52×32= 5−√ 52×√
32
= 5−5×3 = 5−15 =−10 Correction 5
1. On a :
√7
√3×
√7
√3
=(√ 7)2 (√
3)2
=7 3.
2. a.
√7
√3 est un nombre dont le carré vaut 7 3.
Etant positif, on peut en déduire l'égalité suivante :É 7
3 =
√7
√3 b.
√7
√3
= Ê7
3
3. On a les carrés suivants : (√a
√b )2
=(√ a)2 (√
b)2 =a b (É
a b
)2
=a b
Ainsi, ces deux nombres sont positifs et ont même carré.
Correction 6
Communément, les élèves gardent en tête que l'inverse du nombre rationnel a
b est b
a. Hors quand est-il alors d'un nom- bre d'une autre nature?
La dénition de l'inverse du nombre réel x non-nul est le nombrex′ tel que leur produit soit égal à 1 ; c'est à dire :
x·x′= 1
Nous utiliserons cette dénition pour montrer que chacun des nombres est l'inverse de
√8 3 : a. 3
√8×
√8 3 =3√
8 3√
8 b. 3√
8 8 ×
√8 3 = 3×8
3×8 = 1 c.
√18
√16×
√8 3 =
√144 3√
16
=
√144
√9×√ 16
=
√144
√144
= 1
Correction 7 a. 1
√3 = 1×√ (√ 3
3)2 =
√3 3
b.
√3
√2 =
√3×√ (√ 2
2)2 =
√6 2
c.
√28
√7
=
√22×7
√7
= 2√
√7 7
= 2
Correction 8 a. 2
√2
= 2×√
√ 2 2×√
2
=2√ 2 2 =√
2
b.
√3
√2 =
√3×√
√ 2 2×√
2 =
√6 2
c.
√5
√15 =
√5×√
√ 15 15×√
15 = È
5×(5×3)
15 = 5×√ 3 15 =
√3 3
d.
É2 18 =
É1 9 =
É(1 3
)2
=1 3
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e.
É27 3 =√
9 = 3
Correction 9 1. A=√
2 + 2√ 2 = 3√
2
2. a. On a les simplications suivantes :
√50 =√
25×2 =√
52×2 =√ 52×√
2
= 5×√ 2 = 5√
√ 2 32 =√
16×2 =√
42×2 =√ 42×√
2
= 4×√ 2 = 4√
2
b. On a la simplications suivante : B=√
50 +√ 32 +√
2 = 5√ 2 + 4√
2 +√ 2
= 10√ 2
3. On a les simplications suivantes : C= 2√
27 + 5√
75 = 2√
32×3 + 5√ 52×3
= 2√ 32×√
3 + 5√ 52×√
3 = 2×3×√
3 + 5×5×√ 3
= 6√
3 + 25√
3 = 31√ 3 Correction 10
a. √ 3 + 2√
3 = (1 + 2)×√
3 = 3×√ 3 = 3√
3 b. √
12 +√ 3 =√
4×3 +√ 3 =√
4 ×√ 3 +√
3
= 2×√ 3 +√
3 = 3√ 3 c. √
3×√ 6 +√
2 =√
3×6 +√ 2 =√
18 +√ 2
= √
9×2 +√ 2 = 3√
2 +√ 2 = 4√
2 d. √
8 +√ 2 =√
4×2 +√ 2 = 2√
2 +√ 2 = 3√
2 Correction 11
a. √ 3 +√
3 = 2√ 3 b. 2√
5 + 3√
5 = (2 + 3)√ 5 = 5√
5 c. √
2−4√
2 = (1−4)√
2 =−3√ 2 d. √
8 +√ 2 =√
4×2 +√ 2 =√
22×√ 2 +√
2
= 2√ 2 +√
2 = 3√ 2 e. √
27−8√ 3 =√
9×3−8√ 2 =√
32×√ 3−8√
3
= 3√ 3−8√
3 =−5√ 3 f. √
50−√ 72 =√
25×2−√ 36×2
= √ 52×√
2−√ 62×√
2 = 5√ 2−6√
2 =−√ 2 Correction 12
1. 2√
48 + 7√ 3−√
75 = 2√
16×3 + 7√ 3−√
25×3
= 2×4×√ 3 + 7√
3−5√ 3
= 8√ 3 + 7√
3−5√
3 = 10√ 3 2. √
63−4√ 2 +√
18×√ 2 + 2√
8−3√ 7
= √
9×7−4√ 2 +√
18×2 + 2√
4×2−3√ 7
= 3√ 7−4√
2 +√
36 + 2×2√ 2−3√
7
= −4√
2 + 6 + 4√ 2 = 6
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