19:I[44578,["/_next/static/immutable/chunks/2gtpnylaz50zo.js","/_next/static/immutable/chunks/22kg_k_j3c_ao.js","/_next/static/immutable/chunks/3s-qhi50ildb6.js"],"default"] 1a:T1cc5,{ "consigne": "Calcule d'abord avec les puissances de 10, puis convertis. Tu peux écrire 1,2 × 10⁴, 1,2x10^4 ou 1,2e4.", "donnees": [ { "cle": "mult", "tex": "Multiplier : on additionne les exposants" }, { "cle": "div", "tex": "Diviser : on soustrait l'exposant du bas ; retirer un nombre négatif, c'est ajouter" }, { "cle": "carre", "tex": "Élever au carré : on multiplie l'exposant par 2" }, { "cle": "nombres", "tex": "Avec des nombres devant : on calcule les nombres entre eux, puis les puissances de 10 entre elles" }, { "cle": "long", "tex": "1\\ \\text{mm} = 10^{-3}\\ \\text{m}\\ ;\\quad 1\\ \\mu\\text{m} = 10^{-6}\\ \\text{m}\\ ;\\quad 1\\ \\text{cm} = 10^{-2}\\ \\text{m}" }, { "cle": "surf", "tex": "1\\ \\text{cm}^{2} = 10^{-4}\\ \\text{m}^{2}\\ ;\\quad 1\\ \\text{mm}^{2} = 10^{-6}\\ \\text{m}^{2}" }, { "cle": "vol", "tex": "1\\ \\text{L} = 10^{-3}\\ \\text{m}^{3}\\ ;\\quad 1\\ \\text{mL} = 10^{-6}\\ \\text{m}^{3}" }, { "cle": "disque", "tex": "S = \\dfrac{\\pi \\times D^{2}}{4}" }, { "cle": "bar", "tex": "1\\ \\text{bar} = 10^{5}\\ \\text{Pa}" } ], "questions": [ { "enonce": "Calcule 10⁵ × 10³.", "donnees": ["mult"], "reponse": 100000000, "tolerancePct": 1, "explication": "10⁵ × 10³ = 10⁵⁺³ = 10⁸." }, { "enonce": "Calcule 10⁴ × 10⁻⁶.", "donnees": ["mult"], "reponse": 0.01, "tolerancePct": 1, "explication": "On additionne les exposants : 4 + (−6) = −2, donc 10⁻²." }, { "enonce": "Calcule 10⁻⁴ × 10⁻².", "donnees": ["mult"], "reponse": 1e-06, "tolerancePct": 1, "explication": "−4 + (−2) = −6, donc 10⁻⁶." }, { "enonce": "Calcule 10³ ÷ 10⁻⁴.", "donnees": ["div"], "reponse": 10000000, "tolerancePct": 1, "aide": "Calcule l'exposant : 3 − (−4). Retirer un nombre négatif, c'est ajouter.", "explication": "3 − (−4) = 7, donc 10⁷. L'erreur classique est de calculer 3 − 4 et de trouver 10⁻¹." }, { "enonce": "Calcule 10⁻⁶ ÷ 10⁻⁴.", "donnees": ["div"], "reponse": 0.01, "tolerancePct": 1, "explication": "−6 − (−4) = −2, donc 10⁻²." }, { "enonce": "Calcule (10⁻²)².", "donnees": ["carre"], "reponse": 0.0001, "tolerancePct": 1, "explication": "On multiplie l'exposant par 2 : 10⁻⁴." }, { "enonce": "Calcule 1 ÷ 10⁻³.", "donnees": ["div"], "reponse": 1000, "tolerancePct": 1, "explication": "1 = 10⁰, et 0 − (−3) = 3, donc 10³." }, { "enonce": "Calcule (4 × 10⁵) × (3 × 10⁻²).", "donnees": ["mult", "nombres"], "reponse": 12000, "tolerancePct": 1, "explication": "4 × 3 = 12 et 10⁵ × 10⁻² = 10³ : 12 × 10³ = 1,2 × 10⁴." }, { "enonce": "Calcule (1,5 × 10⁵) × (2 × 10⁻⁴).", "donnees": ["mult", "nombres"], "reponse": 30, "tolerancePct": 1, "explication": "1,5 × 2 = 3,0 et 10⁵ × 10⁻⁴ = 10¹ : 3,0 × 10¹ = 30." }, { "enonce": "Calcule (6 × 10⁻³) ÷ (3 × 10²).", "donnees": ["div", "nombres"], "reponse": 2e-05, "tolerancePct": 1, "explication": "6 ÷ 3 = 2 et 10⁻³ ÷ 10² = 10⁻⁵ : 2 × 10⁻⁵." }, { "enonce": "Calcule (3 × 10⁴) ÷ (6 × 10⁻²).", "donnees": ["div", "nombres"], "reponse": 500000, "tolerancePct": 1, "explication": "3 ÷ 6 = 0,5 et 10⁴ ÷ 10⁻² = 10⁶ : 0,5 × 10⁶ = 5 × 10⁵." }, { "enonce": "Calcule 16 320 ÷ (3,2 × 10⁻²).", "donnees": ["div", "nombres"], "reponse": 510000, "tolerancePct": 1, "aide": "Écris d'abord 16 320 = 1,632 × 10⁴.", "explication": "(1,632 ÷ 3,2) × 10⁴ ÷ 10⁻² = 0,51 × 10⁶ = 5,1 × 10⁵. C'est la pression du transformateur sur ses patins, en pascals." }, { "enonce": "Calcule (2,5 × 10⁵) × (4,0 × 10⁻³).", "donnees": ["mult", "nombres"], "reponse": 1000, "tolerancePct": 1, "explication": "2,5 × 4,0 = 10 et 10⁵ × 10⁻³ = 10² : 10 × 10² = 1,0 × 10³." }, { "enonce": "Convertis 45 mm en mètres.", "donnees": ["long"], "reponse": 0.045, "unite": "m", "tolerancePct": 1, "explication": "45 × 10⁻³ m = 4,5 × 10⁻² m." }, { "enonce": "Convertis 350 µm en mètres.", "donnees": ["long"], "reponse": 0.00035, "unite": "m", "tolerancePct": 1, "explication": "350 × 10⁻⁶ m = 3,5 × 10⁻⁴ m." }, { "enonce": "Convertis 31 cm² en mètres carrés.", "donnees": ["surf"], "reponse": 0.0031, "unite": "m²|m2", "tolerancePct": 1, "explication": "31 × 10⁻⁴ m² = 3,1 × 10⁻³ m². C'est la section d'un piston de 63 mm de diamètre." }, { "enonce": "Convertis 250 mm² en mètres carrés.", "donnees": ["surf"], "reponse": 0.00025, "unite": "m²|m2", "tolerancePct": 1, "aide": "Pour une surface, le facteur de conversion est au carré : 1 mm² = (10⁻³ m)² = 10⁻⁶ m².", "explication": "250 × 10⁻⁶ m² = 2,5 × 10⁻⁴ m². Avec 10⁻³, la surface serait mille fois trop grande. La pression calculée serait alors mille fois trop faible." }, { "enonce": "Convertis 0,5 m² en centimètres carrés.", "donnees": ["surf"], "reponse": 5000, "unite": "cm²|cm2", "tolerancePct": 1, "explication": "1 m² = 10⁴ cm², donc 0,5 m² = 5 × 10³ cm²." }, { "enonce": "Convertis 250 mL en mètres cubes.", "donnees": ["vol"], "reponse": 0.00025, "unite": "m³|m3", "tolerancePct": 1, "explication": "250 × 10⁻⁶ m³ = 2,5 × 10⁻⁴ m³." }, { "enonce": "Convertis 60 mL en mètres cubes.", "donnees": ["vol"], "reponse": 6e-05, "unite": "m³|m3", "tolerancePct": 1, "explication": "60 × 10⁻⁶ m³ = 6,0 × 10⁻⁵ m³." }, { "enonce": "Calcule la surface d'un disque de 12 mm de diamètre, en mètres carrés.", "donnees": ["disque", "long"], "reponse": 0.000113, "unite": "m²|m2", "tolerancePct": 2, "aide": "Convertis le diamètre en mètres avant de l'élever au carré.", "explication": "S = π × 0,012² ÷ 4 = 1,13 × 10⁻⁴ m²." }, { "enonce": "Calcule la surface d'un disque de 80 mm de diamètre, en mètres carrés.", "donnees": ["disque", "long"], "reponse": 0.00503, "unite": "m²|m2", "tolerancePct": 2, "explication": "S = π × 0,080² ÷ 4 = 5,03 × 10⁻³ m²." }, { "enonce": "Une force de 8,16 × 10³ N s'exerce sur 1 257 mm². Quelle est la pression ? Réponds en bars.", "donnees": ["surf", "bar"], "reponse": 64.9, "unite": "bar", "tolerancePct": 2, "aide": "Convertis d'abord la surface en m². Calcule ensuite p = F ÷ S en pascals, puis convertis en bars.", "explication": "S = 1 257 × 10⁻⁶ = 1,257 × 10⁻³ m², donc p = 8,16 × 10³ ÷ (1,257 × 10⁻³) ≈ 6,49 × 10⁶ Pa, soit 64,9 bar." } ] } 1b:Tfee,

1\ \text{mm} = 10^{-3}\ \text{m}\ ;\quad 1\ \mu\text{m} = 10^{-6}\ \text{m}\ ;\quad 1\ \text{cm} = 10^{-2}\ \text{m}

1c:T115d,

1\ \text{cm}^{2} = 10^{-4}\ \text{m}^{2}\ ;\quad 1\ \text{mm}^{2} = 10^{-6}\ \text{m}^{2}

1d:Tdfa,

1\ \text{L} = 10^{-3}\ \text{m}^{3}\ ;\quad 1\ \text{mL} = 10^{-6}\ \text{m}^{3}

1e:T8fa,

S = \dfrac{\pi \times D^{2}}{4}

1f:T537,

1\ \text{bar} = 10^{5}\ \text{Pa}

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Gamme — Puissances de 10 et conversions