A 4 cm cube is cut into 1 cm cubes. The total surface area of all the small cubes is

### Concept & Multiplied Surface Area When a solid is cut into smaller pieces, its total volume remains constant, but the total surface area increases because new faces are exposed. $$ \text{Total Surface Area} = \text{Number of Cubes} \times \text{Surface Area of One Cube} $$ ### Step-by-Step Solution * Determine the total number of small cubes: Volume of large cube = $4^3 = 64\text{ cm}^3$. Volume of one small cube = $1^3 = 1\text{ cm}^3$. Total cubes = $\frac{64}{1} = 64$. * Calculate the surface area of one small cube: A cube has 6 faces. Surface area = $6 \times (side)^2 = 6 \times 1^2 = 6\text{ cm}^2$. * Calculate the total surface area of all small cubes combined: $64 \times 6\text{ cm}^2 = 384\text{ cm}^2$. ### Exam Strategy & Shortcut If a cube of side $n \times a$ is cut into cubes of side $a$, the number of cubes is $n^3$. The surface area of each is proportional to $\frac{1}{n^2}$. The total surface area increases by a factor of $n$. Here, $n = 4$. Original surface area = $6 \times 4^2 = 96$. Total new surface area = $96 \times 4 = 384$. ### Common Pitfall Calculating the surface area of the original $4\text{ cm}$ cube ($96\text{ cm}^2$) and selecting that as the answer, forgetting that cutting the cube exposes new surfaces. ### Final Answer Therefore, the correct answer is **384 cm²**.