A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

Hint: Calculate the volume of the bigger cube and divide it by 8 to get the volume of the smaller cube. Then, use the volume of the cube formula to determine the side. Then, use the formula for surface area to find the ratio.

Complete step-by-step answer:
The solid cube of side 12 cm is cut into eight cubes of equal volumes.
We know the formula for the volume of a cube of side a is \[{a^3}\].
\[V = {a^3}...........(1)\]
The volume of the bigger cube of side 12 cm is given as follows:
\[{V_B} = {(12)^3}\]
We know that the value of the cube of 12 is 1728.
\[{V_B} = 1728c{m^3}\]
The volume of each smaller cube is the volume of the bigger cube divided by 8.
\[{V_S} = \dfrac{{{V_B}}}{8}\]
\[{V_S} = \dfrac{{1728}}{8}\]
\[{V_S} = 216c{m^3}\]
Using equation (1), we get the side of the smaller cube to be as follows:
\[{a_S}^3 = 216\]
The cube root of 216 is 6.
\[{a_S} = 6cm\]
Hence, the side of the smaller cube is 6 cm.
The formula for surface area of the cube is given as follows:
\[S = 6{a^2}...........(2)\]
Using equation (2) to find the ratio of the surface areas of the cubes, we have:
\[\dfrac{{{S_B}}}{{{S_S}}} = \dfrac{{6{a_B}^2}}{{6{a_S}^2}}\]
Substituting the value of sides of the bigger and smaller cube, we have:
\[\dfrac{{{S_B}}}{{{S_S}}} = \dfrac{{{{(12)}^2}}}{{{{(6)}^2}}}\]
\[\dfrac{{{S_B}}}{{{S_S}}} = {2^2}\]
\[\dfrac{{{S_B}}}{{{S_S}}} = 4\]
Hence, the ratio of surface area of the bigger cube to the smaller cube is 4.

Note: When one cube is divided into eight small cubes of equal volume. Then the volume of the smaller cube is \[\dfrac{1}{8}\] times the volume of the bigger cube.