Matemáticas 2 · Tema 1
Operaciones con fracciones
Video
Mini apunte
| Suma y resta | $\dfrac{a}{b} \pm \dfrac{c}{d} = \dfrac{ad \pm cb}{bd}$ |
| Multiplicación | $\dfrac{a}{b}\left(\dfrac{c}{d}\right) = \dfrac{ac}{bd}$ |
| División | $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{ad}{bc}$ ó $\dfrac{\dfrac{a}{b}}{\dfrac{c}{d}} = \dfrac{ad}{bc}$ |
| Potencia | $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ |
| Raíces | $\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$ |
Todas las fracciones se deben simplificar al menos: mitades, terceras y quintas.
Ejemplos
Suma y resta
$$-\dfrac{3}{4} - \dfrac{8}{6} = \dfrac{-18-32}{24} = \dfrac{-50}{24} = \mathbf{-\dfrac{25}{12}}$$
Multiplicación
$$-\dfrac{5}{8}\left(\dfrac{1}{2}\right)\left(\dfrac{4}{10}\right) = \dfrac{-20}{160} = \dfrac{-10}{80} = \mathbf{-\dfrac{1}{8}}$$
División
$$\dfrac{3}{4} \div \dfrac{8}{12} = \dfrac{36}{32} = \dfrac{18}{16} = \mathbf{\dfrac{9}{8}} \qquad \text{ó} \qquad \dfrac{\dfrac{3}{4}}{\dfrac{8}{12}} = \dfrac{36}{32} = \mathbf{\dfrac{9}{8}}$$
Potencia
$$\left(\dfrac{5}{-3}\right)^3 = \dfrac{(5)^3}{(-3)^3} = \mathbf{\dfrac{125}{-27}}$$
Raíces
$$\sqrt{\dfrac{36}{25}} = \dfrac{\sqrt{36}}{\sqrt{25}} = \mathbf{\dfrac{6}{5}}$$
Ejercicios
Bloque 1
$-\dfrac{4}{8} + \dfrac{1}{4} =$
$\dfrac{4}{3}\left(\dfrac{5}{15}\right) =$
$\dfrac{3}{4} \div \dfrac{-2}{6} =$
$\left(\dfrac{2}{5}\right)^2 =$
$\sqrt{\dfrac{25}{36}} =$
$-\dfrac{2}{4} - \dfrac{5}{3} =$
$\dfrac{2}{4}\left(\dfrac{7}{14}\right)\left(\dfrac{-1}{3}\right) =$
$\dfrac{-\frac{1}{2}}{-\frac{4}{6}} =$
$\left(\dfrac{1}{-5}\right)^3 =$
$\sqrt[3]{\dfrac{-8}{1000}} =$
Bloque 2
$\dfrac{3}{6} - \dfrac{2}{5} =$
$-\dfrac{1}{3}\left(-\dfrac{2}{4}\right) =$
$-\dfrac{1}{2} \div \dfrac{5}{3} =$
$\left(\dfrac{-1}{4}\right)^2 =$
$\sqrt{\dfrac{49}{4}} =$
$-\dfrac{3}{5} - \dfrac{2}{10} =$
$-\dfrac{3}{9}\left(\dfrac{4}{5}\right)\left(\dfrac{1}{3}\right) =$
$\dfrac{-\frac{2}{5}}{-\frac{4}{3}} =$
$\left(-\dfrac{2}{5}\right)^3 =$
$\sqrt[3]{\dfrac{-64}{125}} =$
Bloque 3
$-\dfrac{3}{5} - \dfrac{2}{4} =$
$-\dfrac{2}{5}\left(-\dfrac{4}{6}\right) =$
$-\dfrac{2}{3} \div \dfrac{1}{4} =$
$\left(-\dfrac{1}{3}\right)^2 =$
$\sqrt{\dfrac{81}{49}} =$
$\dfrac{7}{10} - \dfrac{6}{10} =$
$-\dfrac{2}{3}\left(-\dfrac{9}{6}\right)\left(\dfrac{-1}{3}\right) =$
$\dfrac{-\frac{1}{3}}{\frac{2}{6}} =$
$\left(\dfrac{4}{6}\right)^3 =$
$\sqrt[3]{\dfrac{-1}{8}} =$
Bloque 4
$-\dfrac{2}{8} + \dfrac{6}{9} =$
$-\dfrac{6}{3}\left(\dfrac{5}{20}\right) =$
$\dfrac{-6}{4} \div \dfrac{3}{-5} =$
$\left(-\dfrac{2}{4}\right)^2 =$
$\sqrt{\dfrac{4}{49}} =$
$-\dfrac{4}{5} - \dfrac{6}{9} =$
$\dfrac{4}{8}\left(\dfrac{1}{4}\right)\left(\dfrac{-3}{2}\right) =$
$\dfrac{\frac{2}{3}}{-\frac{5}{4}} =$
$\left(-\dfrac{3}{5}\right)^3 =$
$\sqrt[3]{\dfrac{-125}{1000}} =$
Respuestas
▼Bloque 1
$$-\dfrac{4}{8} + \dfrac{1}{4} = \dfrac{-16+8}{32} = \dfrac{-8}{32} = \mathbf{-\dfrac{1}{4}}$$
$$\dfrac{4}{3}\left(\dfrac{5}{15}\right) = \dfrac{20}{45} = \mathbf{\dfrac{4}{9}}$$
$$\dfrac{3}{4} \div \dfrac{-2}{6} = \dfrac{18}{-8} = \mathbf{-\dfrac{9}{4}}$$
$$\left(\dfrac{2}{5}\right)^2 = \dfrac{(2)^2}{(5)^2} = \mathbf{\dfrac{4}{25}}$$
$$\sqrt{\dfrac{25}{36}} = \dfrac{\sqrt{25}}{\sqrt{36}} = \mathbf{\dfrac{5}{6}}$$
$$-\dfrac{2}{4} - \dfrac{5}{3} = \dfrac{-6-20}{12} = \dfrac{-26}{12} = \mathbf{-\dfrac{13}{6}}$$
$$\dfrac{2}{4}\left(\dfrac{7}{14}\right)\left(\dfrac{-1}{3}\right) = \dfrac{-14}{168} = \mathbf{-\dfrac{1}{12}}$$
$$\dfrac{-\frac{1}{2}}{-\frac{4}{6}} = \dfrac{-6}{-8} = \mathbf{\dfrac{3}{4}}$$
$$\left(\dfrac{1}{-5}\right)^3 = \dfrac{(1)^3}{(-5)^3} = \mathbf{-\dfrac{1}{125}}$$
$$\sqrt[3]{\dfrac{-8}{1000}} = \dfrac{\sqrt[3]{-8}}{\sqrt[3]{1000}} = \mathbf{-\dfrac{2}{10}}$$
Bloque 2
$$\dfrac{3}{6} - \dfrac{2}{5} = \dfrac{15-12}{30} = \dfrac{3}{30} = \mathbf{\dfrac{1}{10}}$$
$$-\dfrac{1}{3}\left(-\dfrac{2}{4}\right) = \dfrac{2}{12} = \mathbf{\dfrac{1}{6}}$$
$$-\dfrac{1}{2} \div \dfrac{5}{3} = \mathbf{-\dfrac{3}{10}}$$
$$\left(\dfrac{-1}{4}\right)^2 = \dfrac{(-1)^2}{(4)^2} = \mathbf{\dfrac{1}{16}}$$
$$\sqrt{\dfrac{49}{4}} = \dfrac{\sqrt{49}}{\sqrt{4}} = \mathbf{\dfrac{7}{2}}$$
$$-\dfrac{3}{5} - \dfrac{2}{10} = \dfrac{-30-10}{50} = \dfrac{-40}{50} = \mathbf{-\dfrac{4}{5}}$$
$$-\dfrac{3}{9}\left(\dfrac{4}{5}\right)\left(\dfrac{1}{3}\right) = \dfrac{-12}{135} = \mathbf{-\dfrac{4}{45}}$$
$$\dfrac{-\frac{2}{5}}{-\frac{4}{3}} = \dfrac{-6}{-20} = \mathbf{\dfrac{3}{10}}$$
$$\left(-\dfrac{2}{5}\right)^3 = \dfrac{(-2)^3}{(5)^3} = \mathbf{-\dfrac{8}{125}}$$
$$\sqrt[3]{\dfrac{-64}{125}} = \dfrac{\sqrt[3]{-64}}{\sqrt[3]{125}} = \mathbf{-\dfrac{4}{5}}$$
Bloque 3
$$-\dfrac{3}{5} - \dfrac{2}{4} = \dfrac{-12-10}{20} = \dfrac{-22}{20} = \mathbf{-\dfrac{11}{10}}$$
$$-\dfrac{2}{5}\left(-\dfrac{4}{6}\right) = \dfrac{8}{30} = \mathbf{\dfrac{4}{15}}$$
$$-\dfrac{2}{3} \div \dfrac{1}{4} = \mathbf{-\dfrac{8}{3}}$$
$$\left(-\dfrac{1}{3}\right)^2 = \dfrac{(-1)^2}{(3)^2} = \mathbf{\dfrac{1}{9}}$$
$$\sqrt{\dfrac{81}{49}} = \dfrac{\sqrt{81}}{\sqrt{49}} = \mathbf{\dfrac{9}{7}}$$
$$\dfrac{7}{10} - \dfrac{6}{10} = \dfrac{70-60}{100} = \dfrac{10}{100} = \mathbf{\dfrac{1}{10}}$$
$$-\dfrac{2}{3}\left(-\dfrac{9}{6}\right)\left(\dfrac{-1}{3}\right) = \dfrac{-18}{54} = \mathbf{-\dfrac{1}{3}}$$
$$\dfrac{-\frac{1}{3}}{\frac{2}{6}} = \dfrac{-6}{6} = \mathbf{-1}$$
$$\left(\dfrac{4}{6}\right)^3 = \dfrac{(4)^3}{(6)^3} = \dfrac{64}{216} = \mathbf{\dfrac{8}{27}}$$
$$\sqrt[3]{\dfrac{-1}{8}} = \dfrac{\sqrt[3]{-1}}{\sqrt[3]{8}} = \mathbf{-\dfrac{1}{2}}$$
Bloque 4
$$-\dfrac{2}{8} + \dfrac{6}{9} = \dfrac{-18+48}{72} = \dfrac{30}{72} = \mathbf{\dfrac{5}{12}}$$
$$-\dfrac{6}{3}\left(\dfrac{5}{20}\right) = \dfrac{-30}{60} = \mathbf{-\dfrac{1}{2}}$$
$$\dfrac{-6}{4} \div \dfrac{3}{-5} = \dfrac{30}{12} = \mathbf{\dfrac{15}{6}} = \mathbf{\dfrac{5}{2}}$$
$$\left(-\dfrac{2}{4}\right)^2 = \dfrac{(-2)^2}{(4)^2} = \dfrac{4}{16} = \mathbf{\dfrac{1}{4}}$$
$$\sqrt{\dfrac{4}{49}} = \dfrac{\sqrt{4}}{\sqrt{49}} = \mathbf{\dfrac{2}{7}}$$
$$-\dfrac{4}{5} - \dfrac{6}{9} = \dfrac{-36-30}{45} = \dfrac{-66}{45} = \mathbf{-\dfrac{22}{15}}$$
$$\dfrac{4}{8}\left(\dfrac{1}{4}\right)\left(\dfrac{-3}{2}\right) = \dfrac{-12}{64} = \mathbf{-\dfrac{3}{16}}$$
$$\dfrac{\frac{2}{3}}{-\frac{5}{4}} = \mathbf{-\dfrac{8}{15}}$$
$$\left(-\dfrac{3}{5}\right)^3 = \dfrac{(-3)^3}{(5)^3} = \mathbf{-\dfrac{27}{125}}$$
$$\sqrt[3]{\dfrac{-125}{1000}} = \dfrac{\sqrt[3]{-125}}{\sqrt[3]{1000}} = \mathbf{-\dfrac{5}{10}} = \mathbf{-\dfrac{1}{2}}$$
Ejercicios extra
▼Bloque 5
$-\dfrac{7}{14} + \dfrac{3}{7} =$
$-\dfrac{5}{10} - \dfrac{3}{4} =$
$\dfrac{6}{5}\left(\dfrac{7}{25}\right) =$
$\dfrac{4}{9}\left(\dfrac{5}{15}\right)\left(-\dfrac{3}{8}\right) =$
$\dfrac{3}{5} \div -\dfrac{5}{8} =$
$-\dfrac{4}{7} \div -\dfrac{6}{9} =$
$\left(\dfrac{2}{3}\right)^2 =$
$\left(-\dfrac{3}{8}\right)^3 =$
$\sqrt{\dfrac{49}{81}} =$
$\sqrt[3]{-\dfrac{64}{125}} =$
Bloque 6
$-\dfrac{8}{16} + \dfrac{5}{10} =$
$-\dfrac{2}{5} - \dfrac{4}{7} =$
$\dfrac{5}{6}\left(\dfrac{8}{24}\right) =$
$\dfrac{7}{9}\left(\dfrac{3}{12}\right)\left(-\dfrac{4}{5}\right) =$
$\dfrac{3}{4} \div -\dfrac{7}{10} =$
$-\dfrac{5}{6} \div -\dfrac{7}{8} =$
$\left(\dfrac{5}{8}\right)^2 =$
$\left(-\dfrac{4}{9}\right)^3 =$
$\sqrt{\dfrac{16}{49}} =$
$\sqrt[3]{-\dfrac{27}{64}} =$
Bloque 7
$-\dfrac{9}{18} + \dfrac{4}{9} =$
$-\dfrac{7}{14} - \dfrac{2}{5} =$
$\dfrac{3}{7}\left(\dfrac{4}{21}\right) =$
$\dfrac{5}{11}\left(\dfrac{2}{14}\right)\left(-\dfrac{5}{9}\right) =$
$\dfrac{5}{6} \div -\dfrac{3}{4} =$
$-\dfrac{2}{3} \div -\dfrac{5}{8} =$
$\left(\dfrac{4}{9}\right)^2 =$
$\left(-\dfrac{5}{8}\right)^3 =$
$\sqrt{\dfrac{36}{49}} =$
$\sqrt[3]{-\dfrac{125}{216}} =$
Bloque 8
$-\dfrac{5}{12} + \dfrac{3}{8} =$
$-\dfrac{3}{7} - \dfrac{1}{4} =$
$\dfrac{8}{9}\left(\dfrac{5}{27}\right) =$
$\dfrac{4}{7}\left(\dfrac{6}{20}\right)\left(-\dfrac{3}{5}\right) =$
$\dfrac{7}{10} \div -\dfrac{2}{5} =$
$-\dfrac{4}{9} \div -\dfrac{3}{7} =$
$\left(\dfrac{3}{8}\right)^2 =$
$\left(-\dfrac{4}{11}\right)^3 =$
$\sqrt{\dfrac{25}{64}} =$
$\sqrt[3]{-\dfrac{8}{27}} =$