CALCUL NUMERIQUE
Correction des exercices **

Exercice 1

\( \displaystyle A=\frac{1-\frac{1}{3}}{\frac{3}{2}}\times \frac{3}{4}-\frac{3}{5}\)
\( \displaystyle \quad =\frac{\displaystyle\frac{2}{3}}{\displaystyle\frac{3}{2}}\times \frac{3}{4}-\frac{3}{5}\)
\( \displaystyle \quad =\frac{2}{3}\times \frac{2}{3}\times \frac{3}{4}-\frac{3}{5}\)
\( \displaystyle \quad =\frac{2\times 2\times 3}{3\times 3\times 4}-\frac{3}{5}\)
\( \displaystyle \quad =\frac{\color{red}2\color{red}\times \color{red}2 \color{red}\times \color{red}3}{3\times \color{red}3\color{red}\times\color{red} 4}-\frac{3}{5}\)
\( \displaystyle \quad =\frac{1}{3}-\frac{3}{5}\)
\( \displaystyle \quad =\frac{1\times 5}{3 \times 5}-\frac{3 \times 3}{5 \times 3}\)
\( \displaystyle \quad =\frac{5}{15}-\frac{9}{15}\)
\( \displaystyle \quad =\frac{5-9}{15}\)
\( \displaystyle \quad =-\frac{4}{15}\)


\( \displaystyle B=\frac{2}{5}+\frac{5}{12}-\frac{1}{15}\)
\( \displaystyle \quad =\frac{2}{5}-\frac{1}{15}+\frac{5}{12}\)
\( \displaystyle \quad =\frac{2\times 3}{5\times 3}-\frac{1}{15}+\frac{5}{12}\)
\( \displaystyle \quad =\frac{6}{15}-\frac{1}{15}+\frac{5}{12}\)
\( \displaystyle \quad =\frac{6-1}{15}+\frac{5}{12}\)
\( \displaystyle \quad =\frac{5}{15}+\frac{5}{12}\)
\( \displaystyle \quad =\frac{{\color{red}5}\times 1}{{\color{red}5}\times 3}+\frac{5}{12}\)
\( \displaystyle \quad =\frac{1}{3}+\frac{5}{12}\)
\( \displaystyle \quad =\frac{1\times 4}{3\times 4}+\frac{5}{12}\)
\( \displaystyle \quad =\frac{4}{12}+\frac{5}{12}\)
\( \displaystyle \quad =\frac{4+5}{12}\)
\( \displaystyle \quad =\frac{9}{12}\)
\( \displaystyle \quad =\frac{3}{4}\)

\( \displaystyle C=\frac{5}{6}-\frac{11}{6}\times \frac{3}{8}\)
\( \displaystyle \quad =\frac{5}{6}-\frac{11\times 3}{6\times 8}\)
\( \displaystyle \quad =\frac{5}{6}-\frac{11{\color{red}\times \color{red}3}}{{\color{red}3}\times 2\times 8}\)
\( \displaystyle \quad =\frac{5}{6}-\frac{11}{16}\)
\( \displaystyle \quad =\frac{5\times 8}{6\times 8}-\frac{11\times 3}{16\times 3}\)
\( \displaystyle \quad =\frac{40}{48}-\frac{33}{48}\)
\( \displaystyle \quad =\frac{40-33}{48}\)
\( \displaystyle \quad =\frac{7}{48}\)


\( \displaystyle D=\frac{7}{3}+\frac{2}{3}\times \frac{5}{6}\)
\( \displaystyle \quad =\frac{7}{3}+\frac{2\times 5}{3\times 6}\)
\( \displaystyle \quad =\frac{7}{3}+\frac{{\color{red}2} \times 5}{3\times 3 {\color{red} \times \color{red}2}}\)
\( \displaystyle \quad =\frac{7}{3}+\frac{5}{9}\)
\( \displaystyle \quad =\frac{7\times 3}{3\times 3}+\frac{5}{9}\)
\( \displaystyle \quad =\frac{21}{9}+\frac{5}{9}\)
\( \displaystyle \quad =\frac{21+5}{9}\)
\( \displaystyle \quad =\frac{26}{9}\)

Exercice 2

\( \displaystyle A=\frac{1}{1+\displaystyle\frac{1}{1+\displaystyle\frac{1}{3}}}\)
\( \displaystyle \quad =\frac{1}{1+\frac{1}{\frac{3}{3}+\frac{1}{3}}}\)
\( \displaystyle \quad =\frac{1}{1+\displaystyle\frac{1}{\displaystyle\frac{4}{3}}}\)
\( \displaystyle \quad =\frac{1}{1+\displaystyle\frac{3}{4}}\)
\( \displaystyle \quad =\frac{1}{\displaystyle\frac{4}{4}+\displaystyle\frac{3}{4}}\)
\( \displaystyle \quad =\frac{1}{\displaystyle\frac{7}{4}}\)
\( \displaystyle \quad =\frac{4}{7}\)

\( \displaystyle B=\frac{2}{2-\displaystyle\frac{1}{2-\displaystyle\frac{1}{2-\displaystyle \frac{1}{2}}}}\)
\( \displaystyle \quad =\frac{2}{2-\displaystyle\frac{1}{2-\displaystyle\frac{1}{\displaystyle \frac{4}{2}-\displaystyle \frac{1}{2}}}}\)
\( \displaystyle \quad =\frac{2}{2-\displaystyle\frac{1}{2-\displaystyle\frac{1}{\displaystyle \frac{3}{2}}}}\)
\( \displaystyle \quad =\frac{2}{2-\displaystyle\frac{1}{2-\displaystyle\frac{2}{3}}}\)
\( \displaystyle \quad =\frac{2}{2-\displaystyle\frac{1}{\displaystyle\frac{6}{3}-\displaystyle\frac{2}{3}}}\)
\( \displaystyle \quad =\frac{2}{2-\displaystyle\frac{1}{\displaystyle\frac{4}{3}}}\)
\( \displaystyle \quad =\frac{2}{2-\displaystyle\frac{3}{4}}\)
\( \displaystyle \quad =\frac{2}{\displaystyle\frac{8}{4}-\displaystyle\frac{3}{4}}\)
\( \displaystyle \quad =\frac{2}{\displaystyle\frac{5}{4}}\)
\( \displaystyle \quad =2\times \frac{4}{5}\)
\( \displaystyle \quad =\frac{8}{5}\)

Exercice 3

\( \displaystyle A=\frac{7\times 10^{35}}{56\times 10^{34}}\)
\(\quad \displaystyle =\frac{7}{56}\times \frac{10^{35}}{10^{34}}\)
\(\quad =0.125\times 10^{35-34}\)
\(\quad =0.125\times 10^{1}\)
\( \quad =1.25\times 10^{-1}\times 10^{1}\)
\( \quad =1.25\times 10^{-1+1}\)
\( \quad =1.25\times 10^{0}\)


\( \displaystyle B=\frac{35\times 10^{22}\times 3\times \left(10^{-2}\right)^{6}}{42\times 10^{10}}\)
\( \displaystyle \quad =\frac{35\times 3}{42}\times \frac{10^{22}\times \left(10^{-2}\right)^{6}}{10^{10}}\)
\( \displaystyle \quad =2.5\times \frac{10^{22}\times 10^{-2\times 6}}{10^{10}}\)
\( \displaystyle \quad =2.5\times \frac{10^{22}\times 10^{-12}}{10^{10}}\)
\( \quad =2.5\times 10^{22+(-12)-10}\)
\( \quad =2.5\times 10^{0}\)


\( \displaystyle C=\frac{6\times 10^{8}\times 1.6\times 10^{13}}{0.4\times 10^{14}}\)
\( \displaystyle \quad =\frac{6\times 1.6}{0.4}\times \frac{10^{8}\times 10^{13}}{10^{14}}\)
\( \quad =24\times 10^{8+13-14}\)
\( \quad =24\times 10^{7}\)
\( \quad =2.4\times 10^{1}\times 10^{7}\)
\( \quad =2.4\times 10^{1+7}\)
\( \quad =2.4\times 10^{8}\)


\( \displaystyle D=\frac{7\times \left(10^{5}\right)^{2}\times 10^{-3}}{35\times 10^{3}}\)
\( \displaystyle \quad =\frac{7}{35}\times \frac{\left(10^{5}\right)^{2}\times 10^{-3}}{10^{3}}\)
\( \displaystyle \quad =0.2\times \frac{10^{5\times 2}\times 10^{-3}}{10^{3}}\)
\( \displaystyle \quad =0.2\times \frac{10^{10}\times 10^{-3}}{10^{3}}\)
\( \quad =0.2\times 10^{10+(-3)-3}\)
\( \quad =0.2\times 10^{4}\)
\( \quad =2\times 10^{-1}\times 10^{4}\)
\( \quad =2\times 10^{-1+4}\)
\( \quad =2\times 10^{3}\)

\( \displaystyle E=\frac{1.6\times 10^{-12}}{4\times 10^{-9}}\)
\( \displaystyle \quad =\frac{1.6}{4}\times \frac{10^{-12}}{10^{-9}}\)
\( \quad =0.4\times 10^{-12-(-9)}\)
\( \quad =0.4\times 10^{-12+9}\)
\( \quad =0.4\times 10^{-3}\)
\( \quad =4\times 10^{-1}\times 10^{-3}\)
\( \quad =4\times 10^{-1+(-3)}\)
\( \quad =4\times 10^{-4}\)


\( \displaystyle F=\frac{5\times 10^{8}\times 6\times 10^{3}}{2\times \left(10^{-4}\right)^{3}}\)
\( \displaystyle \quad =\frac{5\times 6}{2}\times \frac{10^{8}\times 10^{3}}{\left(10^{-4}\right)^{3}}\)
\( \quad \displaystyle =15\times \frac{10^{8}\times 10^{3}}{10^{-4\times 3}}\)
\( \quad \displaystyle =15\times \frac{10^{8}\times 10^{3}}{10^{-12}}\)
\( \quad =15\times 10^{8+3-(-12)}\)
\( \quad =15\times 10^{23}\)
\( \quad =1.5\times 10^{1}\times 10^{23}\)
\( \quad =1.5\times 10^{1+23}\)
\( \quad =1.5\times 10^{24}\)


\( \displaystyle G=\frac{2\times 10^{9}\times 7\times 10^{-6}}{35\times 10^{2}}\)
\( \displaystyle \quad =\frac{2\times 7}{35}\times \frac{10^{9}\times 10^{-6}}{10^{2}}\)
\( \quad =0.4\times 10^{9+(-6)-2}\)
\( \quad =0.4\times 10^{1}\)
\( \quad =4\times 10^{-1}\times 10^{1}\)
\( \quad =4\times 10^{-1+1}\)
\( \quad =4\times 10^{0}\)


\( \displaystyle H=\frac{3\times 10^{8}\times 4\times 10^{-5}}{6\times 10^{7}}\)
\( \displaystyle \quad =\frac{3\times 4}{6}\times \frac{10^{8}\times 10^{-5}}{10^{7}}\)
\( \quad =2\times 10^{8+(-5)-7}\)
\( \quad =2\times 10^{-4}\)

Exercice 4

Calculs lorsque les nombres de départ sont respectivement \(5\), \(-2\) et \(\displaystyle \frac{1}{4}\):
\( \displaystyle \left(5+\frac{5}{2}\right)\times \frac{1}{3}-\frac{6}{5}\)
\( \displaystyle \quad =\left(\frac{10}{2}+\frac{5}{2}\right)\times \frac{1}{3}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{15}{2}\times \frac{1}{3}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{15}{6}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{5}{2}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{5\times 5}{2\times 5}-\frac{6\times 2}{5 \times 2}\)
\( \displaystyle \quad =\frac{25}{10}-\frac{12}{10}\)
\( \displaystyle \quad =\frac{25-12}{10}\)
\( \displaystyle \quad =\frac{13}{10}\)

\( \displaystyle \left(-2+\frac{5}{2}\right)\times \frac{1}{3}-\frac{6}{5}\)
\( \displaystyle \quad =\left(-\frac{4}{2}+\frac{5}{2}\right)\times \frac{1}{3}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{1}{2}\times \frac{1}{3}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{1}{6}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{1\times 5}{6\times 5}-\frac{6\times 6}{5 \times 6}\)
\( \displaystyle \quad =\frac{5}{30}-\frac{36}{30}\)
\( \displaystyle \quad =\frac{5-36}{30}\)
\( \displaystyle \quad =-\frac{31}{30}\)

\( \displaystyle \left(\frac{1}{4}+\frac{5}{2}\right)\times \frac{1}{3}-\frac{6}{5}\)
\( \displaystyle \quad =\left(\frac{1}{4}+\frac{10}{4}\right)\times \frac{1}{3}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{11}{4}\times \frac{1}{3}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{11}{12}-\frac{6}{5}\)
\( \displaystyle \quad =\frac{11\times 5}{12\times 5}-\frac{6\times 12}{5 \times 12}\)
\( \displaystyle \quad =\frac{55}{60}-\frac{72}{60}\)
\( \displaystyle \quad =\frac{55-72}{60}\)
\( \displaystyle \quad =-\frac{17}{60}\)

Correction des exercices d'application sur le calcul numérique (révisions) pour la troisième (3ème)
© Planète Maths