The volumes of two spheres are in the ratio of $512:3375$. The ratio of their surface areas is:

Let the radii of the two spheres be $r_1$ and $r_2$.
Step 1: Use the volume ratio to find the ratio of their radii
The formula for the volume of a sphere is $V=\frac{4}{3}\pi r^3$.

$$\frac{V_1}{V_2}=\frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3}={\left(\frac{r_1}{r_2}\right)}^3=\frac{512}{3375}$$

Taking the cube root on both sides:

$$\frac{r_1}{r_2}=\frac{\sqrt[3]{512}}{\sqrt[3]{3375}}=\frac{8}{15}$$

Step 2: Find the ratio of their surface areas
The formula for the surface area of a sphere is $A=4\pi r^2$.

$$\frac{A_1}{A_2}=\frac{4\pi r_1^2}{4\pi r_2^2}={\left(\frac{r_1}{r_2}\right)}^2$$

$$\frac{A_1}{A_2}={\left(\frac{8}{15}\right)}^2=\frac{64}{225}$$

Therefore, the ratio of their surface areas is $64:225$.