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, and the ratio between their surface areas.

Given :
Side (a) of cube = 12 cm

We know that,

Volume of 8 cubes
= \({a}^3\)
= \({12}^3\) \({cm}^3\)
= 1728 \({cm}^3\).

Now, Let the side of the smaller cube be \(\left(a_{1}\right)\).

Thus, volume of 1 smaller cube
= \(\frac{1728}{8}\) \({cm}^3\)
\(\Rightarrow \) \({\left(a_{1}\right)}^3\) = 216 \({cm}^3\)
\(\Rightarrow \) \(\left(a_{1}\right)= 6 cm \)

Therefore, the side of the smaller cubes will be 6 cm.

Ratio between surface areas of cube
= \(\frac{Surface area of bigger cube}{Surface area of smaller cube}\)
= \([\frac{6 × \left(a_{2}\right)^2}{6 × \left(a_{1}\right)^2}]\)
= \((\frac{12}{6})^2\)
= \((\frac{2}{1})^2\)
= \(\frac{4}{1}\)

Therefore, the ratio between the surface areas of these cubes is 4:1.