Answer: A. $\dfrac{9}{7,396}$

Explanation:1. Define edge lengths: Let $e\_X$ be the edge length of cube X. Let $e\_Y$ be the edge length of cube Y. We are given that $e\_Y = \dfrac{3}{86} e\_X$.

2. Formulate surface area of a cube: The surface area (SA) of a cube with edge length $e$ is $SA = 6e^2$.

3. Calculate the surface area of cube X: $SA\_X = 6(e\_X)^2$

4. Calculate the surface area of cube Y in terms of $e\_X$: $SA\_Y = 6(e\_Y)^2$ Substitute $e\_Y = \dfrac{3}{86} e\_X$: $SA\_Y = 6 \left( \dfrac{3}{86} e\_X \right)^2$ $SA\_Y = 6 \left( \left(\dfrac{3}{86}\right)^2 (e\_X)^2 \right)$ $SA\_Y = 6 \left( \dfrac{3^2}{86^2} (e\_X)^2 \right)$ $SA\_Y = 6 \left( \dfrac{9}{7396} (e\_X)^2 \right)$ (Since $86^2 = 7396$) $SA\_Y = \dfrac{9}{7396} \left( 6 (e\_X)^2 \right)$

5. Relate $SA\_Y$ to $SA\_X$: We know that $SA\_X = 6(e\_X)^2$. Substitute this into the equation for $SA\_Y$: $SA\_Y = \dfrac{9}{7396} SA\_X$

6. Identify the value of n: The problem states that the surface area of cube Y is $n$ times the surface area of cube X, so $SA\_Y = n \cdot SA\_X$. Comparing this with our derived equation, we see that $n = \dfrac{9}{7396}$.

7. Conclusion: The value of $n$ is $\dfrac{9}{7,396}$.