A cube has volume 512 cm^3. Find its total surface area.

Introduction / Context:For a cube, volume and surface area are functions of the side length a. If the volume is known, find a as the cube root, then compute surface area as 6a^2. This is a straightforward application of cube geometry formulas.

Given Data / Assumptions:

• Volume V = a^3 = 512 cm^3.
• Total surface area S = 6a^2.

Concept / Approach:Take the cube root of 512 to find a; then square a and multiply by 6 to obtain S.

Step-by-Step Solution:a = ∛512 = 8 cmS = 6a^2 = 6 * 8^2 = 6 * 64 = 384 cm^2

Verification / Alternative check:Compute volume from a = 8: 8^3 = 512, confirming side length.

Why Other Options Are Wrong:64 cm^2 is one face area (a^2) times 1; 256 cm^2 is 4a^2; 512 cm^2 equals 8a^2; only 384 cm^2 equals 6a^2.

Common Pitfalls:Accidentally using 6a instead of 6a^2; mixing units (keeping “cm” vs “cm^2”).

Final Answer:384 cm2