1. El problema es racionalizar las siguientes expresiones: $$\frac{3}{\sqrt{2}}, \frac{10}{\sqrt{8}}, \frac{7}{\sqrt{4}}, \frac{6}{\sqrt{16}}, \frac{9}{\sqrt{32}}, \frac{4}{\sqrt{64}}, \frac{12}{\sqrt{8}}, \frac{14}{\sqrt{128}}.$$ 2. La racionalización consiste en eliminar la raíz del denominador multiplicando numerador y denominador por la raíz que está en el denominador. La fórmula general es: $$\frac{a}{\sqrt{b}} = \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}.$$ 3. Ahora racionalizamos cada expresión paso a paso: - $$\frac{3}{\sqrt{2}} = \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}.$$ - $$\frac{10}{\sqrt{8}} = \frac{10}{\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} = \frac{10\sqrt{8}}{8}.$$ Simplificamos $$\frac{10\sqrt{8}}{8} = \frac{\cancel{10}\times \sqrt{4 \times 2}}{\cancel{8}} = \frac{5 \times 2 \sqrt{2}}{4} = \frac{10\sqrt{2}}{4} = \frac{\cancel{10}\sqrt{2}}{\cancel{4}} = \frac{5\sqrt{2}}{2}.$$ - $$\frac{7}{\sqrt{4}} = \frac{7}{2}$$ porque $$\sqrt{4} = 2.$$ - $$\frac{6}{\sqrt{16}} = \frac{6}{4} = \frac{3}{2}$$ porque $$\sqrt{16} = 4.$$ - $$\frac{9}{\sqrt{32}} = \frac{9}{\sqrt{32}} \times \frac{\sqrt{32}}{\sqrt{32}} = \frac{9\sqrt{32}}{32}.$$ Simplificamos $$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}.$$ Entonces $$\frac{9\sqrt{32}}{32} = \frac{9 \times 4 \sqrt{2}}{32} = \frac{36\sqrt{2}}{32} = \frac{\cancel{36}\sqrt{2}}{\cancel{32}} = \frac{9\sqrt{2}}{8}.$$ - $$\frac{4}{\sqrt{64}} = \frac{4}{8} = \frac{1}{2}$$ porque $$\sqrt{64} = 8.$$ - $$\frac{12}{\sqrt{8}} = \frac{12}{\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} = \frac{12\sqrt{8}}{8}.$$ Simplificamos $$\sqrt{8} = 2\sqrt{2}.$$ Entonces $$\frac{12\sqrt{8}}{8} = \frac{12 \times 2 \sqrt{2}}{8} = \frac{24\sqrt{2}}{8} = \frac{\cancel{24}\sqrt{2}}{\cancel{8}} = 3\sqrt{2}.$$ - $$\frac{14}{\sqrt{128}} = \frac{14}{\sqrt{128}} \times \frac{\sqrt{128}}{\sqrt{128}} = \frac{14\sqrt{128}}{128}.$$ Simplificamos $$\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}.$$ Entonces $$\frac{14\sqrt{128}}{128} = \frac{14 \times 8 \sqrt{2}}{128} = \frac{112\sqrt{2}}{128} = \frac{\cancel{112}\sqrt{2}}{\cancel{128}} = \frac{7\sqrt{2}}{8}.$$ 4. Resumen final: $$\frac{3\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}, \frac{7}{2}, \frac{3}{2}, \frac{9\sqrt{2}}{8}, \frac{1}{2}, 3\sqrt{2}, \frac{7\sqrt{2}}{8}.$$