Feuille d'exercices 0. Nombres et calcul
Exercice I.
Parmi les ensembles N,Z,Q,R, donner le plus précis auquel appartiennent les nombres suivants : 1. 3; −5; π; 56
7 ; 0.3333; −78 4 ; −√
25; 5 9; √
3; −38.52719 2. −17; 3.141592654; 22
7 ; √
2; −π
3; 1400; 30
5 ; −15
6 ; e; 175−9 2
Exercice II.
Calculer et simplier :
1. a= 4−2×5−3; b= 4−2×(5−3); c= (7 + 2)×(6 + 1) 2. a=−3 + 2×4−6÷2 + 1; b=−5−(−1) + 3×(2−6) + 3−4×(−5)
3. a=−23−(−3)2+ (−2)−2; b=−4×23−5 + 6×(34−7) + 2−1 + 3×(2−7)3 Exercice III.
Calculer et simplier : 1. a= 1 + 2
3 b= 1
3 +1
6 c=− 8
15 + 3
10 d= 5
4 −3×2 5
2. a= 4
3−1 5 +3
2 b= 7
10 +23 30 −31
20 c= 5
6 −2 3× 7
4 d= 2
3 ÷6 5+ 2
5
3. a= 1
4−3 8 ×4
5 −3
2 b=
−4
3 −2×5 4
7 3 −3
c= 7
5 −2 1 +1 2 −1
3
−2
3−9 2
4. a= 3
4× 7
2 −8 3
b= 1
4 −2 3 −11
24
× 8
49 c=
1 2− 2
3 1 2+ 2
3 +1
6 d= 3 +
3 2 +5
4 1 1 2 −1
−1 5
Exercice IV.
Exprimer, si nécessaire, à l'aide de puissances de2, de 3, de 5 ou de10 (ou autres) :
1. a= 47 b= 63 c= 34×3−5
2. a= 23×45 b= 182×124 c= 604×(452)3
3. a= 45×36
27×6 b= 10010
217×522 c= 3−3×2−5 (−6)−7×(32)4 4. a= (−2)3×(53)2×46
(102×3)3×2−4 b= (52×33)2×2−2×(−3)−2
2−4×(24×32)3×5 c= 2502×104×1205 1003×10−2×(302)3
Exercice V.
1. Combien vaut (−1)n en fonction den? Calculer et simplier :
2. a= (−1)2n b= (−1)2n+1 c= (−1)n+ (−1)n+1
3. a= (−1)n−(−1)n−1 b= 3(−1)n−2(−1)1−n c= −(−1)n+ 2(−1)n+1−3(−1)2n (−1)n−4(−1)2n+1−(−1)n−1
1
Exercice VI.
Calculer et simplier :
1. a=√
8×√
2 b=√
24×√
54 c=
√3×√
√ 10 2×√
6
2. a=
√18×√
√ 15
90 b=
√200
√25×√
24 c=
√42×√
√ 22 11×√
60
3. a=√
75 + 4√
12−2√
27 b= 9√
28−4√
63 + 2√ 175
4. a=p
2 +√ 3 +p
2−√ 3
2
b= q
(2 +√ 5)2+
q (2−√
5)2
5. a=
√ 2 1 +√
2 b= 2−√
3 2 +√
3 c= 2√
3−7 3 +√
5
6. a=√
3
2
√ 7−√
3− 2
√ 7 +√
3
b=√
3 2
√ 7−√
32 − 2
√ 7 +√
32
!
Exercice VII.
Développer et réduire : 1. a= (3 +√
5)2; b= (2√
7−4)2; c= (√
3−1)(1 +√
3); d= (2 +√
2)2−(1−√ 8)2 2. a= (1 +√
2)(1 +√
3); b= (√ 10−√
6)(√ 3 +√
5)
3. (a+b)3; (a−b)3; c= (a+b)(a2−ab+b2); d= (a−b)(a2+ab+b2) 4. a= (2 +x)(3 + 2x); b= (x+ 1)2−2(x+ 1)(3−x)
5. a= (x2+√
2x+ 1)(x2−√
2x+ 1); b= (3x2−5x+ 2)(−2x3−5x2+ 7) Exercice VIII.
Factoriser :
1. a=x2+ 3x b= 3x4−5x3+ 2x2 c=x8−4x6
2. a=x2−2x+ 1 b= 4−(x−3)2 c= 2x4−72
3. a=−7(3x−2)−3(3x−2)2 b= (2x+ 5)2−10−4x c= (x−5)(−x+ 3)−4(x−5) 4. a= (2x+ 5)(6x−3)−(1−2x)2 b= 1−x
3 +x2−1
4 c= (x−3)(x+ 5)−2(x−3)2+3−x 5
Exercice IX.
Calculer de tête :
1. a= 7 + 8 b= 14 + 35 c= 3.7 + 2.6 d= 59 + 27 e= 59 + 27 2. a= 85−53 b= 8.2−2.8 c= 74−28 d= 48−91 e= 122−79 3. a= 5−7 + 6 b= 35 + 48−63 c=−147 + 381−269 d= 3.84−2.67−4.38 + 5.12 4. a= 5×7 b= 9×8 c= 6×13 d= 112 e= 3.4×2.7 5. a= 18×12 b= 23×37 c= 56×83 d= 5.7×6.9 e= 285×142 6. a= 28÷4 b= 87÷3 c= 294÷7 d= 5848÷13 e= 543÷9 Exercice X.
Décomposer en produit de facteurs premiers : a= 360; b= 2088; c= 4158000; d= 1001 2
