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Sagot :
Para completar la tabla, vamos a realizar los siguientes cálculos con los valores dados:
1. Para los valores [tex]\( a = \frac{2}{3} \)[/tex], [tex]\( b = -\frac{1}{4} \)[/tex] y [tex]\( c = \frac{5}{6} \)[/tex]:
[tex]\[ a \times c = \frac{2}{3} \times \frac{5}{6} = \frac{10}{18} = 0.5555555555555556 \][/tex]
[tex]\[ b \times c = -\frac{1}{4} \times \frac{5}{6} = -\frac{5}{24} = -0.20833333333333334 \][/tex]
[tex]\[ a(b+c) = \frac{2}{3} \times \left(-\frac{1}{4} + \frac{5}{6}\right) = \frac{2}{3} \times \frac{7}{12} = \frac{14}{36} = 0.3888888888888889 \][/tex]
[tex]\[ (c+b) \times a = \left(\frac{5}{6} - \frac{1}{4}\right) \times \frac{2}{3} = \frac{7}{12} \times \frac{2}{3} = \frac{14}{36} = 0.3888888888888889 \][/tex]
2. Para los valores [tex]\( a = \frac{15}{6} \)[/tex], [tex]\( b = -\frac{18}{10} \)[/tex] y [tex]\( c = -\frac{20}{12} \)[/tex]:
[tex]\[ a \times c = \frac{15}{6} \times -\frac{20}{12} = -\frac{300}{72} = -4.166666666666667 \][/tex]
[tex]\[ b \times c = -\frac{18}{10} \times -\frac{20}{12} = \frac{360}{120} = 3.0 \][/tex]
[tex]\[ a(b+c) = \frac{15}{6} \times \left(-\frac{18}{10} - \frac{20}{12}\right) = \frac{15}{6} \times -\frac{74}{30} = -\frac{1110}{180} = -8.666666666666668 \][/tex]
[tex]\[ (c+b) \times a = \left(-\frac{20}{12} - \frac{18}{10}\right) \times \frac{15}{6} = -\frac{74}{30} \times \frac{15}{6} = -8.666666666666668 \][/tex]
3. Para los valores [tex]\( a = -\frac{3}{7} \)[/tex], [tex]\( b = \frac{18}{21} \)[/tex] y [tex]\( c = -\frac{14}{9} \)[/tex]:
[tex]\[ a \times c = -\frac{3}{7} \times -\frac{14}{9} = \frac{42}{63} = 0.6666666666666666 \][/tex]
[tex]\[ b \times c = \frac{18}{21} \times -\frac{14}{9} = -\frac{252}{189} = -1.3333333333333333 \][/tex]
[tex]\[ a(b+c) = -\frac{3}{7} \times \left(\frac{18}{21} - \frac{14}{9}\right) = -\frac{3}{7} \times -\frac{7}{63} = \frac{21}{63 \times 3} = 0.29931972789115646 \][/tex]
[tex]\[ (c+b) \times a = \left(-\frac{14}{9} + \frac{18}{21}\right) \times -\frac{3}{7} = -\frac{7}{63} \times -\frac{3}{7} = 0.29931972789115646 \][/tex]
Ahora, completamos la tabla con los resultados obtenidos:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline a & b & c & a \times c & b \times c & a(b+c) & (c+b) \times a \\ \hline \frac{2}{3} & -\frac{1}{4} & \frac{5}{6} & 0.5555555555555556 & -0.20833333333333334 & 0.3888888888888889 & 0.3888888888888889 \\ \hline \frac{15}{6} & -\frac{18}{10} & -\frac{20}{12} & -4.166666666666667 & 3.0 & -8.666666666666668 & -8.666666666666668 \\ \hline -\frac{3}{7} & \frac{18}{21} & -\frac{14}{9} & 0.6666666666666666 & -1.3333333333333333 & 0.29931972789115646 & 0.29931972789115646 \\ \hline \end{array} \][/tex]
1. Para los valores [tex]\( a = \frac{2}{3} \)[/tex], [tex]\( b = -\frac{1}{4} \)[/tex] y [tex]\( c = \frac{5}{6} \)[/tex]:
[tex]\[ a \times c = \frac{2}{3} \times \frac{5}{6} = \frac{10}{18} = 0.5555555555555556 \][/tex]
[tex]\[ b \times c = -\frac{1}{4} \times \frac{5}{6} = -\frac{5}{24} = -0.20833333333333334 \][/tex]
[tex]\[ a(b+c) = \frac{2}{3} \times \left(-\frac{1}{4} + \frac{5}{6}\right) = \frac{2}{3} \times \frac{7}{12} = \frac{14}{36} = 0.3888888888888889 \][/tex]
[tex]\[ (c+b) \times a = \left(\frac{5}{6} - \frac{1}{4}\right) \times \frac{2}{3} = \frac{7}{12} \times \frac{2}{3} = \frac{14}{36} = 0.3888888888888889 \][/tex]
2. Para los valores [tex]\( a = \frac{15}{6} \)[/tex], [tex]\( b = -\frac{18}{10} \)[/tex] y [tex]\( c = -\frac{20}{12} \)[/tex]:
[tex]\[ a \times c = \frac{15}{6} \times -\frac{20}{12} = -\frac{300}{72} = -4.166666666666667 \][/tex]
[tex]\[ b \times c = -\frac{18}{10} \times -\frac{20}{12} = \frac{360}{120} = 3.0 \][/tex]
[tex]\[ a(b+c) = \frac{15}{6} \times \left(-\frac{18}{10} - \frac{20}{12}\right) = \frac{15}{6} \times -\frac{74}{30} = -\frac{1110}{180} = -8.666666666666668 \][/tex]
[tex]\[ (c+b) \times a = \left(-\frac{20}{12} - \frac{18}{10}\right) \times \frac{15}{6} = -\frac{74}{30} \times \frac{15}{6} = -8.666666666666668 \][/tex]
3. Para los valores [tex]\( a = -\frac{3}{7} \)[/tex], [tex]\( b = \frac{18}{21} \)[/tex] y [tex]\( c = -\frac{14}{9} \)[/tex]:
[tex]\[ a \times c = -\frac{3}{7} \times -\frac{14}{9} = \frac{42}{63} = 0.6666666666666666 \][/tex]
[tex]\[ b \times c = \frac{18}{21} \times -\frac{14}{9} = -\frac{252}{189} = -1.3333333333333333 \][/tex]
[tex]\[ a(b+c) = -\frac{3}{7} \times \left(\frac{18}{21} - \frac{14}{9}\right) = -\frac{3}{7} \times -\frac{7}{63} = \frac{21}{63 \times 3} = 0.29931972789115646 \][/tex]
[tex]\[ (c+b) \times a = \left(-\frac{14}{9} + \frac{18}{21}\right) \times -\frac{3}{7} = -\frac{7}{63} \times -\frac{3}{7} = 0.29931972789115646 \][/tex]
Ahora, completamos la tabla con los resultados obtenidos:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline a & b & c & a \times c & b \times c & a(b+c) & (c+b) \times a \\ \hline \frac{2}{3} & -\frac{1}{4} & \frac{5}{6} & 0.5555555555555556 & -0.20833333333333334 & 0.3888888888888889 & 0.3888888888888889 \\ \hline \frac{15}{6} & -\frac{18}{10} & -\frac{20}{12} & -4.166666666666667 & 3.0 & -8.666666666666668 & -8.666666666666668 \\ \hline -\frac{3}{7} & \frac{18}{21} & -\frac{14}{9} & 0.6666666666666666 & -1.3333333333333333 & 0.29931972789115646 & 0.29931972789115646 \\ \hline \end{array} \][/tex]
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