10. Simplificar: $$ E=\sqrt[n]{\frac{92^{n^{2}}+138^{n^{2}}}{69^{n^{2}}+46^{n^{2}}}} $$

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Para simplificar la expresión $$ E=\sqrt[n]{\frac{92^{n^{2}}+138^{n^{2}}}{69^{n^{2}}+46^{n^{2}}}} $$, sigamos estos pasos:

1. **Descomponer los números en factores primos:**
   $$
   \begin{align*}
   92 &= 4 \times 23 = 2^2 \times 23 \
   138 &= 6 \times 23 = 2 \times 3 \times 23 \
   69 &= 3 \times 23 \
   46 &= 2 \times 23
   \end{align*}
   $$

2. **Reescribir la expresión original usando los factores descompuestos:**
   $$
   E = \sqrt[n]{\frac{(2^2 \times 23)^{n^2} + (2 \times 3 \times 23)^{n^2}}{(3 \times 23)^{n^2} + (2 \times 23)^{n^2}}}
   $$

3. **Factorizar $$23^{n^2}$$ en el numerador y el denominador:**
   $$
   E = \sqrt[n]{\frac{2^{2n^2} \times 23^{n^2} + 2^{n^2} \times 3^{n^2} \times 23^{n^2}}{3^{n^2} \times 23^{n^2} + 2^{n^2} \times 23^{n^2}}}
   $$

4. **Simplificar cancelando $$23^{n^2}$$:**
   $$
   E = \sqrt[n]{\frac{2^{2n^2} + 2^{n^2} \times 3^{n^2}}{3^{n^2} + 2^{n^2}}}
   $$

5. **Factorizar $$2^{n^2}$$ en el numerador:**
   $$
   E = \sqrt[n]{\frac{2^{n^2}(2^{n^2} + 3^{n^2})}{2^{n^2} + 3^{n^2}}} = \sqrt[n]{2^{n^2}} = 2^{\frac{n^2}{n}} = 2^{n}
   $$

**Respuesta final:**
$$
E = 2^{n}
$$
$$ E = 2^{n} $$

Simplificar:

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