Let the side lengths of the two cubes be $a_1$ and $a_2$, respectively. The volume ($V$) of a cube is given by $V = a^3$, and the total surface area ($A$) is given by $A = 6a^2$.

Cube Volume Ratio Calculation

The ratio of the volumes of the two cubes is given as $V_1 : V_2 = 64 : 1331$.

Using the volume formula, we can write the ratio as:

$ \frac{V_1}{V_2} = \frac{a_1^3}{a_2^3} = \left(\frac{a_1}{a_2}\right)^3 $

We are given:

$ \left(\frac{a_1}{a_2}\right)^3 = \frac{64}{1331} $

To find the ratio of the side lengths ($\frac{a_1}{a_2}$), we take the cube root of both sides:

$ \frac{a_1}{a_2} = \sqrt[3]{\frac{64}{1331}} = \frac{\sqrt[3]{64}}{\sqrt[3]{1331}} = \frac{4}{11} $

So, the ratio of the side lengths of the two cubes is $4 : 11$.

Cube Surface Area Ratio Calculation

Now, we need to find the ratio of their total surface areas ($A_1 : A_2$).

The ratio of the total surface areas is:

$ \frac{A_1}{A_2} = \frac{6a_1^2}{6a_2^2} = \left(\frac{a_1}{a_2}\right)^2 $

Substitute the ratio of the side lengths we found ($ \frac{a_1}{a_2} = \frac{4}{11} $):

$ \frac{A_1}{A_2} = \left(\frac{4}{11}\right)^2 = \frac{4^2}{11^2} = \frac{16}{121} $

Therefore, the ratio of the total surface areas of the two cubes is $16 : 121$.