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. Para racionalizar una fracción con raíz cuadrada en el denominador, multiplicamos numerador y denominador por la raíz cuadrada del denominador para eliminar la raíz del 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: **1)** $$\frac{3}{\sqrt{2}} = \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{\cancel{\sqrt{2}}\cancel{\sqrt{2}}} = \frac{3\sqrt{2}}{2}.$$ **2)** $$\frac{10}{\sqrt{8}} = \frac{10}{\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} = \frac{10\sqrt{8}}{8}.$$ Simplificamos $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$, entonces: $$\frac{10 \times 2\sqrt{2}}{8} = \frac{20\sqrt{2}}{8} = \frac{\cancel{20}\sqrt{2}}{\cancel{8}} = \frac{5\sqrt{2}}{2}.$$ **3)** $$\frac{7}{\sqrt{4}} = \frac{7}{2}$$ porque $\sqrt{4} = 2$. No es necesario racionalizar más. **4)** $$\frac{6}{\sqrt{16}} = \frac{6}{4} = \frac{3}{2}$$ porque $\sqrt{16} = 4$. **5)** $$\frac{9}{\sqrt{32}} = \frac{9}{\sqrt{32}} \times \frac{\sqrt{32}}{\sqrt{32}} = \frac{9\sqrt{32}}{32}.$$ Simplificamos $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$, entonces: $$\frac{9 \times 4\sqrt{2}}{32} = \frac{36\sqrt{2}}{32} = \frac{\cancel{36}\sqrt{2}}{\cancel{32}} = \frac{9\sqrt{2}}{8}.$$ **6)** $$\frac{4}{\sqrt{64}} = \frac{4}{8} = \frac{1}{2}$$ porque $\sqrt{64} = 8$. **7)** $$\frac{12}{\sqrt{8}} = \frac{12}{\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} = \frac{12\sqrt{8}}{8}.$$ Como antes, $\sqrt{8} = 2\sqrt{2}$, entonces: $$\frac{12 \times 2\sqrt{2}}{8} = \frac{24\sqrt{2}}{8} = 3\sqrt{2}.$$ **8)** $$\frac{14}{\sqrt{128}} = \frac{14}{\sqrt{128}} \times \frac{\sqrt{128}}{\sqrt{128}} = \frac{14\sqrt{128}}{128}.$$ Simplificamos $\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}$, entonces: $$\frac{14 \times 8\sqrt{2}}{128} = \frac{112\sqrt{2}}{128} = \frac{\cancel{112}\sqrt{2}}{\cancel{128}} = \frac{7\sqrt{2}}{8}.$$ 4. Resumen final: $$\frac{3\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}, \frac{7}{2}, \frac{3}{2}, \frac{9\sqrt{2}}{8}, \frac{1}{2}, 3\sqrt{2}, \frac{7\sqrt{2}}{8}.$$