1) Calculer : 1. Calcul de A = $(-\sqrt{25})^2$\: \rightarrow -\sqrt{25} = -5$ donc $A = (-5)^2 = 25$. 2. Calcul de B = $\sqrt{10^6}$\: comme $10^6 = (10^3)^2$, $B = 10^3 = 1000$. 3. Calcul de C = $\sqrt{3} \times \sqrt{12}$\: $\sqrt{3 \times 12} = \sqrt{36} = 6$. 4. Calcul de D = $\frac{\sqrt{50}}{\sqrt{98}} = \sqrt{\frac{50}{98}} = \sqrt{\frac{25}{49}} = \frac{5}{7}$. 5. Calcul de E = $\sqrt{6}\sqrt{100} + 4 - \sqrt{3}\sqrt{16} - 8 = \sqrt{600} + 4 - \sqrt{48} - 8$\. Simplifions $\sqrt{600} = \sqrt{100 \times 6} = 10\sqrt{6}$ et $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$. Ainsi $E = 10\sqrt{6} + 4 - 4\sqrt{3} - 8 = 10\sqrt{6} - 4\sqrt{3} - 4$. 2) Simplifier les expressions : A = $-\sqrt{54} + 4\sqrt{24} + 3\sqrt{6}$\. Simplifions chaque racine : $\sqrt{54} = 3\sqrt{6}$, $\sqrt{24} = 2\sqrt{6}$. Donc A = $-3\sqrt{6} + 4 \times 2\sqrt{6} + 3\sqrt{6} = -3\sqrt{6} + 8\sqrt{6} + 3\sqrt{6} = 8\sqrt{6}$. B = $7\sqrt{10} - 5\sqrt{90} + 2\sqrt{640}$. $\sqrt{90} = 3\sqrt{10}$, $\sqrt{640} = 8\sqrt{10}$ donc B = $7\sqrt{10} - 5 \times 3\sqrt{10} + 2 \times 8\sqrt{10} = 7\sqrt{10} - 15\sqrt{10} + 16\sqrt{10} = 8\sqrt{10}$. C = $\sqrt{98} + \sqrt{72} - 5\sqrt{18}$. $\sqrt{98} = 7\sqrt{2}$, $\sqrt{72} = 6\sqrt{2}$, $\sqrt{18} = 3\sqrt{2}$. Ainsi C = $7\sqrt{2} + 6\sqrt{2} - 5 \times 3\sqrt{2} = 13\sqrt{2} - 15\sqrt{2} = -2\sqrt{2}$. D = $2\sqrt{63} - 4\sqrt{28} - \sqrt{175}$. $\sqrt{63} = 3\sqrt{7}$, $\sqrt{28} = 2\sqrt{7}$, $\sqrt{175} = 5\sqrt{7}$. Donc D = $2 \times 3\sqrt{7} - 4 \times 2\sqrt{7} - 5\sqrt{7} = 6\sqrt{7} - 8\sqrt{7} - 5\sqrt{7} = -7\sqrt{7}$. Exercice 2 : 1) Rationaliser les dénominateurs : A = $\frac{12}{3\sqrt{5}} = \frac{12}{3\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{12\sqrt{5}}{3 \times 5} = \frac{12\sqrt{5}}{15} = \frac{4\sqrt{5}}{5}$. B = $\frac{-4\sqrt{6}}{8\sqrt{2}} = \frac{-4\sqrt{6}}{8\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{-4\sqrt{12}}{8 \times 2} = \frac{-4 \times 2\sqrt{3}}{16} = \frac{-8\sqrt{3}}{16} = -\frac{\sqrt{3}}{2}$. C = $\frac{3 - \sqrt{10}}{7\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{(3 - \sqrt{10})\sqrt{3}}{7 \times 3} = \frac{(3-\sqrt{10})\sqrt{3}}{21}$. D = $\frac{2\sqrt{7}}{5 - \sqrt{15}} \times \frac{5 + \sqrt{15}}{5 + \sqrt{15}} = \frac{2\sqrt{7}(5 + \sqrt{15})}{25 - 15} = \frac{2\sqrt{7}(5 + \sqrt{15})}{10} = \frac{\sqrt{7}}{5}(5 + \sqrt{15})$. E = $\frac{4\sqrt{3} + \sqrt{11}}{6\sqrt{2} + 5\sqrt{3}} \times \frac{6\sqrt{2} - 5\sqrt{3}}{6\sqrt{2} - 5\sqrt{3}} = \frac{(4\sqrt{3} + \sqrt{11})(6\sqrt{2} - 5\sqrt{3})}{(6\sqrt{2})^2 - (5\sqrt{3})^2} = \frac{(4\sqrt{3})(6\sqrt{2}) - (4\sqrt{3})(5\sqrt{3}) + (\sqrt{11})(6\sqrt{2}) - (\sqrt{11})(5\sqrt{3})}{72 - 75}$. Numérateur : $24 \sqrt{6} - 20 \sqrt{9} + 6 \sqrt{22} - 5 \sqrt{33} = 24\sqrt{6} - 60 + 6\sqrt{22} - 5\sqrt{33}$. Dénominateur : $72 - 75 = -3$. Donc E = $\frac{24\sqrt{6} - 60 + 6\sqrt{22} - 5\sqrt{33}}{-3} = -8\sqrt{6} + 20 - 2\sqrt{22} + \frac{5}{3}\sqrt{33}$. 2) Résoudre les équations : * $-5x^2 + 3 = 0$ \rightarrow $-5x^2 = -3$ \rightarrow $x^2 = \frac{3}{5}$ \rightarrow $x = \pm \sqrt{\frac{3}{5}}$. * $4x^2 + 2 = 1$ \rightarrow $4x^2 = -1$ pas de solution réelle. * $2x^2 - 7 = 3x^2$ \rightarrow $2x^2 - 3x^2 = 7$ \rightarrow $-x^2 = 7$ pas de solution réelle. * $-10x^2 - 8 = -8$ \rightarrow $-10x^2 = 0$ \rightarrow $x^2 = 0$ \rightarrow $x = 0$. Exercice 3 : 1) Développer puis réduire : A = $(3\sqrt{5}x - 4\sqrt{2})^2 = (3\sqrt{5}x)^2 - 2\times 3\sqrt{5}x \times 4\sqrt{2} + (4\sqrt{2})^2 = 9 \times 5 x^2 - 24 \sqrt{10} x + 16 \times 2 = 45x^2 - 24\sqrt{10} x + 32$. B = $(2x - 3)(-x + 1) - (\sqrt{6}x + 2)^2$. Calculons $(2x - 3)(-x + 1) = -2x^2 + 2x + 3x - 3 = -2x^2 + 5x - 3$. $(\sqrt{6}x + 2)^2 = 6x^2 + 4\sqrt{6} x + 4$. Donc B = $(-2x^2 + 5x -3) - (6x^2 + 4\sqrt{6} x + 4) = -8x^2 + (5x - 4\sqrt{6} x) - 7$. C = $(-4 - 2x)^2 = (-4)^2 + 2 \times (-4)(-2x) + (-2x)^2 = 16 + 16x + 4x^2 = 4x^2 + 16x + 16$. D = $2\sqrt{7} x (3\sqrt{3} x + 5\sqrt{2} x)(3\sqrt{3} x - 5\sqrt{2} x)$. Le produit $(a+b)(a-b) = a^2 - b^2$ donc $(3\sqrt{3} x)^2 - (5\sqrt{2} x)^2 = 9 \times 3 x^2 - 25 \times 2 x^2 = 27x^2 - 50x^2 = -23x^2$. Donc D = $2\sqrt{7} x (-23 x^2) = -46 \sqrt{7} x^3$. E = $\sqrt{7} - 2\sqrt{6} \times \sqrt{7} + 2\sqrt{6}$. Attention à l'ordre; $- 2\sqrt{6} \times \sqrt{7} = -2\sqrt{42}$. Donc E = $\sqrt{7} - 2\sqrt{42} + 2\sqrt{6}$. 2) Factoriser puis réduire : A = $3 \sqrt{18} x^2 - 2 \sqrt{24} x$. $\sqrt{18} = 3\sqrt{2}$, $\sqrt{24} = 2\sqrt{6}$. Donc A = $3 \times 3\sqrt{2} x^2 - 2 \times 2\sqrt{6} x = 9\sqrt{2} x^2 - 4\sqrt{6} x$. Facteur commun : $x \sqrt{2}$. $9 \sqrt{2} x^2 - 4 \sqrt{6} x = x \sqrt{2} (9 x - 4 \sqrt{3})$. B = $21 ab^5 c^3 - 35 a^4 b c^2 + 14 a^3 b^2 c$. PGCD: $7 a b c$. $7 a b c (3 b^4 c^2 - 5 a^3 + 2 a^2 b)$. C = $17 x^6 - 5$, pas factorisable avec les informations données. D = $2 x^2 + 5 - 2 \sqrt{10} x$, aucun facteur commun évident ni forme remarquable. E = $( -6 x + 14)(3 x - 5) - (7 - 3 x)(8 x - 9) + 3 x - 7$. Développons : $(-6x)(3x) + (-6x)(-5) + 14(3x) + 14(-5) = -18 x^2 + 30 x + 42 x - 70 = -18 x^2 + 72 x - 70$. $(7)(8x) + (7)(-9) + (-3x)(8x) + (-3x)(-9) = 56 x - 63 - 24 x^2 + 27 x = -24 x^2 + 83 x - 63$. Donc E = $(-18 x^2 + 72 x - 70) - (-24 x^2 + 83 x - 63) + 3 x - 7 = -18 x^2 + 72 x - 70 + 24 x^2 - 83 x + 63 + 3 x - 7$. Simplifions : $(-18 x^2 + 24 x^2) + (72 x - 83 x + 3 x) + (-70 + 63 - 7) = 6 x^2 - 8 x - 14$. F = $(3x + 4)(-5x - 2) - 9 x^2 + 16 - (x + 3)(3x + 4)$. Développons: $(3x)(-5x) + (3x)(-2) + 4(-5x) + 4(-2) = -15 x^2 - 6 x - 20 x - 8 = -15 x^2 - 26 x -8$. Puis $-9 x^2 + 16$ restent. $(x)(3x) + (x)(4) + 3(3x) + 3(4) = 3 x^2 + 4 x + 9 x + 12 = 3 x^2 + 13 x + 12$. Donc F = $(-15 x^2 - 26 x - 8) - 9 x^2 + 16 - (3 x^2 + 13 x + 12)$ = $-15 x^2 - 26 x - 8 - 9 x^2 + 16 - 3 x^2 - 13 x - 12$ = $(-15 - 9 - 3) x^2 + (-26 - 13) x + (-8 + 16 - 12) = -27 x^2 - 39 x - 4$. 3) Calculer : A = $5^{-3} \times (-10)^2 \times (-2)^{-3}$. $5^{-3} = \frac{1}{125}$, $(-10)^2 = 100$, $(-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8}$. Donc A = $\frac{1}{125} \times 100 \times -\frac{1}{8} = -\frac{100}{1000} = -\frac{1}{10}$. B = $\frac{4^3 \times (4^{-2})^3 \times 4}{(4^5)^{-3}}$. $4^3 = 64$, $(4^{-2})^3 = 4^{-6}$, produit $64 \times 4^{-6} \times 4 = 64 \times 4^{-6} \times 4^1 = 64 \times 4^{-5} = 64 \times \frac{1}{4^{5}} = \frac{64}{1024} = \frac{1}{16}$. Denominateur $(4^5)^{-3} = 4^{-15}$. Donc B = $\frac{1}{16} \div 4^{-15} = \frac{1}{16} \times 4^{15} = 4^{15} / 16 = 4^{15} / 4^2 = 4^{13} = 67108864$. C = $\frac{2^{-4} \times 6^9}{6^3 \times 2^{-10}} = 2^{-4} \times 6^{9-3} \times 2^{10} = 2^{-4 + 10} \times 6^6 = 2^{6} \times 6^6 = (2 \times 6)^6 = 12^6 = 2985984$. D = $\frac{7^{2020} + 7^{2019}}{7^{2020} - 7^{2019}} = \frac{7^{2019}(7 + 1)}{7^{2019}(7 - 1)} = \frac{8}{6} = \frac{4}{3}$. 4) Écriture scientifique: A = $0.008 \times 10^{-4} = 8 \times 10^{-3} \times 10^{-4} = 8 \times 10^{-7}$. B = $-451.32 \times (10^2)^{-3} = -451.32 \times 10^{-6} = -4.5132 \times 10^{-4}$. C = $33 \times 10^{-5} \times 12 = 396 \times 10^{-5} = 3.96 \times 10^{-3}$. D = $\frac{495 \times 100}{0.0001 \times 9} = \frac{49500}{0.0009} = 5.5 \times 10^{7}$.