1. **Calculer:** 1) $\sqrt{2} \sqrt{64} = \sqrt{2 \times 64} = \sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}$. 2) $\frac{\sqrt{36+64}}{49} = \frac{\sqrt{100}}{49} = \frac{10}{49}$. 3) $\sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$. 4) $\sqrt{144} - 2\sqrt{(-4)^2} = 12 - 2\sqrt{16} = 12 - 2 \times 4 = 12 - 8 = 4$. 5) $\sqrt{6} \sqrt{25} + 51 = \sqrt{6 \times 25} + 51 = \sqrt{150} + 51 = 5\sqrt{6} + 51$. 6) $3 \sqrt{7} \times 2 \sqrt{\frac{14}{78}} = 6 \sqrt{7} \times \sqrt{\frac{14}{78}} = 6 \sqrt{\frac{7 \times 14}{78}} = 6 \sqrt{\frac{98}{78}} = 6 \sqrt{\frac{49}{39}} = 6 \times \frac{7}{\sqrt{39}} = \frac{42}{\sqrt{39}}$. 7) $\frac{\sqrt{45}}{2 \sqrt{20}} = \frac{\sqrt{9 \times 5}}{2 \sqrt{4 \times 5}} = \frac{3\sqrt{5}}{2 \times 2 \sqrt{5}} = \frac{3 \sqrt{5}}{4 \sqrt{5}} = \frac{3}{4}$. 8) $\sqrt{28} - \sqrt{7} + \sqrt{10} - \sqrt{36} = \sqrt{4 \times 7} - \sqrt{7} + \sqrt{10} - 6 = 2 \sqrt{7} - \sqrt{7} + \sqrt{10} - 6 = \sqrt{7} + \sqrt{10} - 6$. 2. **Simplifier:** A) $7\sqrt{18} - 5\sqrt{98} - 2\sqrt{50} = 7 \times 3 \sqrt{2} - 5 \times 7 \sqrt{2} - 2 \times 5 \sqrt{2} = 21\sqrt{2} - 35\sqrt{2} - 10 \sqrt{2} = (21 - 35 - 10) \sqrt{2} = -24 \sqrt{2}$. B) $\frac{\sqrt{27 + \sqrt{3}}}{\sqrt{48}}$ cannot be simplified directly without approximations; leave it as is. C) $\frac{\sqrt{50 + \sqrt{2}}}{\sqrt{8} + \sqrt{2}}$ also does not simplify nicely; rationalizing the denominator if needed. 3. **Rendre les dénominateurs rationnels :** - $\frac{2\sqrt{3}}{3\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{15}}{15}$. - $\frac{1 + \sqrt{10}}{2\sqrt{5} - \sqrt{7}} \times \frac{2\sqrt{5} + \sqrt{7}}{2\sqrt{5} + \sqrt{7}} = \frac{(1 + \sqrt{10})(2\sqrt{5} + \sqrt{7})}{(2\sqrt{5})^2 - (\sqrt{7})^2} = \frac{(1 + \sqrt{10})(2\sqrt{5} + \sqrt{7})}{20 - 7} = \frac{(1 + \sqrt{10})(2\sqrt{5} + \sqrt{7})}{13}$. - $\frac{-15}{4 - \sqrt{11}} \times \frac{4 + \sqrt{11}}{4 + \sqrt{11}} = \frac{-15 (4 + \sqrt{11})}{16 - 11} = \frac{-15 (4 + \sqrt{11})}{5} = -3 (4 + \sqrt{11}) = -12 - 3 \sqrt{11}$. 4. **Développer et réduire :** A) $(5x - 2)(3x - 7) = 15x^2 - 35x - 6x + 14 = 15x^2 - 41x + 14$. B) $-2\sqrt{3}[5\sqrt{3} + 2] + 7(3\sqrt{3} - 5) = -2\sqrt{3} \times 5\sqrt{3} - 2\sqrt{3} \times 2 + 7 \times 3 \sqrt{3} - 7 \times 5 = -2 \times 5 \times 3 - 4 \sqrt{3} + 21 \sqrt{3} - 35 = -30 + 17 \sqrt{3} - 35 = -65 + 17 \sqrt{3}$. C) $(2\sqrt{7} - 3)^2 - (5 + 3\sqrt{7})^2 = [(2\sqrt{7})^2 - 2 \times 2\sqrt{7} \times 3 + 3^2] - [5^2 + 2 \times 5 \times 3\sqrt{7} + (3\sqrt{7})^2] = (4 \times 7 - 12\sqrt{7} + 9) - (25 + 30 \sqrt{7} + 9 \times 7) = (28 - 12 \sqrt{7} + 9) - (25 + 30 \sqrt{7} + 63) = (37 - 12\sqrt{7}) - (88 + 30\sqrt{7}) = 37 - 12 \sqrt{7} - 88 - 30 \sqrt{7} = -51 - 42 \sqrt{7}$. D) $(\frac{\sqrt{2}}{3} + 7x)(\frac{\sqrt{2}}{3} - 7x) - x^2 = (\frac{\sqrt{2}}{3})^2 - (7x)^2 - x^2 = \frac{2}{9} - 49x^2 - x^2 = \frac{2}{9} - 50x^2$. 5. **Factoriser :** E) $5\sqrt{18} - 10\sqrt{8} x + (12 - 5)(3 - 2x) = 15\sqrt{2} - 20 \sqrt{2} x + 7(3 - 2x) = 15 \sqrt{2} - 20 \sqrt{2} x + 21 - 14x$ (expression; factorization depends on grouping, no common factor). F) $4x^2 + 4 \sqrt{5} x + 5$ is a perfect square trinomial: $(2x + \sqrt{5})^2$. G) $3x^2 - 4 + (5\sqrt{3} x + 8)(3x + 2) = 3x^2 - 4 + 15 \sqrt{3} x^2 + 10 \sqrt{3} x + 24 x + 16 = (3x^2 + 15 \sqrt{3} x^2) + (10 \sqrt{3} x + 24 x) + ( -4 + 16 ) = x^2(3 + 15\sqrt{3}) + x(10\sqrt{3} + 24) + 12$. H) $(4x + 5)^2 - 49 = (4x + 5 - 7)(4x + 5 + 7) = (4x - 2)(4x + 12)$. 6. **Calculer :** 1) $(\sqrt{5})^{-3} + (-\frac{2}{5})^2 = \frac{1}{(\sqrt{5})^3} + \frac{4}{25} = \frac{1}{5 \sqrt{5}} + \frac{4}{25}$. 2) $[(\frac{1}{2})^{-3} \times 5^7]^{-4} = [8 \times 78125]^{-4} = (625000)^{-4} = \frac{1}{625000^4}$. 3) $(11^2 - 12^2)^{8025} = (-23)^{8025}$ noting $11^2=121$, $12^2=144$, so negative base to odd power remains negative. 7. **Ecrire sous forme puissance :** 1) $(\sqrt{\frac{2}{3}})^7 \times \left(\frac{3}{12}^{-5}\right)^2 = \left(\frac{2}{3}\right)^{\frac{7}{2}} \times \left(\frac{1}{4}^{-5}\right)^2 = \left(\frac{2}{3}\right)^{\frac{7}{2}} \times 4^{10}$. 2) $\left(-\frac{\sqrt{3}}{2}\right)^6 \times \left(-\frac{5 \sqrt{3}}{4}\right)^6 = \left(\frac{\sqrt{3}}{2}\right)^6 \times \left(\frac{5 \sqrt{3}}{4}\right)^6$ (since power is even, negatives vanish) = $\left(\frac{3^{1/2}}{2}\right)^6 \times \left(\frac{5 \times 3^{1/2}}{4}\right)^6 = \frac{3^3}{2^6} \times \frac{5^6 3^3}{4^6} = \frac{3^3 5^6 3^3}{2^6 4^6} = \frac{3^{6} 5^{6}}{2^6 (2^2)^6} = \frac{3^6 5^6}{2^{6+12}} = \frac{3^6 5^6}{2^{18}}$. 3) $\frac{4 \times 10000 \times 5^2}{10^{-6} \times 2^3 \times 125} = \frac{4 \times 10^4 \times 25}{10^{-6} \times 8 \times 125} = \frac{4 \times 10^4 \times 25}{10^{-6} \times 1000} = \frac{10^6}{10^{-6} \times 1000} = 10^6 \times 10^{6} / 1000 = 10^{12} / 10^3 = 10^9$. **Final answers summarized:** 1.1 $8\sqrt{2}$ 1.2 $\frac{10}{49}$ 1.3 $4$ 1.4 $4$ 1.5 $5\sqrt{6} + 51$ 1.6 $\frac{42}{\sqrt{39}}$ 1.7 $\frac{3}{4}$ 1.8 $\sqrt{7} + \sqrt{10} - 6$ 2.A $-24 \sqrt{2}$ 3.1 $\frac{2\sqrt{15}}{15}$ 3.2 $\frac{(1 + \sqrt{10})(2\sqrt{5} + \sqrt{7})}{13}$ 3.3 $-12 - 3 \sqrt{11}$ 4.A $15x^2 - 41x + 14$ 4.B $-65 + 17\sqrt{3}$ 4.C $-51 - 42\sqrt{7}$ 4.D $\frac{2}{9} - 50x^2$ 5.F $(2x + \sqrt{5})^2$ 5.H $(4x - 2)(4x + 12)$ 6.1 $\frac{1}{5 \sqrt{5}} + \frac{4}{25}$ 6.2 $\frac{1}{625000^4}$ 6.3 $(-23)^{8025}$ 7.1 $\left(\frac{2}{3}\right)^{\frac{7}{2}} \times 4^{10}$ 7.2 $\frac{3^{6} 5^{6}}{2^{18}}$ 7.3 $10^{9}$