1. Calculer : $10^4$ et $10^{-2}$. - $10^4 = 10000$ car $10^4 = 10 \times 10 \times 10 \times 10$. - $10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01$. 2. Écrire sous forme d'une puissance : - $(\frac{2}{9})^3 \times (\frac{1}{2})^{10} = \frac{2^3}{9^3} \times \frac{1}{2^{10}} = \frac{2^3}{9^3} \times 2^{-10} = \frac{2^{3-10}}{9^3} = \frac{2^{-7}}{9^3}$. - $(\frac{1}{2})^7 \times (\frac{3}{4})^{-3} = 2^{-7} \times \left(\frac{3}{4}\right)^{-3} = 2^{-7} \times \left(\frac{4}{3}\right)^3 = 2^{-7} \times \frac{4^3}{3^3} = 2^{-7} \times \frac{2^{6}}{3^3} = \frac{2^{-7+6}}{3^3} = \frac{2^{-1}}{3^3}$. - $(\frac{4}{5})^5 \div (\frac{1}{3})^{-1} = (\frac{4}{5})^5 \times (\frac{1}{3})^{1} = \frac{4^5}{5^5} \times \frac{1}{3} = \frac{4^5}{5^5 \times 3}$. - $(\frac{2}{3})^6 \div (1)^{-42} = (\frac{2}{3})^6 \times 1^{42} = (\frac{2}{3})^6$. - $(-\frac{9}{11})^2 = \frac{81}{121}$. - $(\frac{7}{3})^{24} \times (\frac{7}{3})^5 = (\frac{7}{3})^{24+5} = (\frac{7}{3})^{29}$. - $\frac{(\frac{11}{2})^5}{11^2} = \frac{11^5}{2^5} \times \frac{1}{11^2} = \frac{11^{5-2}}{2^5} = \frac{11^3}{2^5}$. 3. Écrire sous forme de $10^n$ : - $A = 100000 = 10^5$. - $B = 0.0001 = 10^{-4}$. - $C = 100 \times 0.1 \times 100000 = 10^2 \times 10^{-1} \times 10^5 = 10^{2-1+5} = 10^6$. - $D = 10^5 \times (10^2)^3 \div 0.001 \times 10 = 10^5 \times 10^{6} \times 10^{3} \times 10^{1} = 10^{5+6+3+1} = 10^{15}$ (car $0.001 = 10^{-3}$ donc division par $0.001$ = multiplication par $10^3$). - $E = 1000 \times 0.01 \div 10^2 = 10^3 \times 10^{-2} \times 10^{-2} = 10^{3-2-2} = 10^{-1}$. 4. Calculer les expressions : - $A = 5 + (\frac{4}{3})^3 = 5 + \frac{64}{27} = \frac{135}{27} + \frac{64}{27} = \frac{199}{27} \approx 7.37$. - $B = (\frac{9}{5})^{-1} \times \frac{(\frac{4}{5} + 1)}{16} = \frac{5}{9} \times \frac{(\frac{4}{5} + \frac{5}{5})}{16} = \frac{5}{9} \times \frac{\frac{9}{5}}{16} = \frac{5}{9} \times \frac{9}{5 \times 16} = \frac{5}{9} \times \frac{9}{80} = \frac{45}{720} = \frac{1}{16}$. 5. Donner l'écriture scientifique : - $A = 1562 = 1.562 \times 10^3$. - $B = 437.18 \times 10^4 = 4.3718 \times 10^2 \times 10^4 = 4.3718 \times 10^{6}$. - $C = 0.000623 \times 10^{-7} = 6.23 \times 10^{-4} \times 10^{-7} = 6.23 \times 10^{-11}$. - $D = -0.0037 \times 254.8 \times 0.0006 = -3.7 \times 10^{-3} \times 2.548 \times 10^{2} \times 6 \times 10^{-4} = -3.7 \times 2.548 \times 6 \times 10^{-3+2-4} = -56.56 \times 10^{-5} = -5.656 \times 10^{-4}$. 6. Simplifier l'expression : $$E = \frac{a^4 \times b^9 \times b^{10}}{(a^3)^2 \times b^{11}} = \frac{a^4 \times b^{19}}{a^{6} \times b^{11}} = a^{4-6} \times b^{19-11} = a^{-2} \times b^{8} = \frac{b^{8}}{a^{2}}.$$