A solid metallic sphere of radius R is melted and recast into n identical small solid spheres of radius r. Let S be the surface area of the original large sphere and s be the sum of the surface areas of all n small spheres. Identify the single correct relationship between s and S: 

Volume Conservation:

The total volume remains unchanged during melting. The volume of the large sphere equals n times the volume of one small sphere. Since volume depends on the cube of the radius, the cube of the large radius R equals n multiplied by the cube of the small radius r. Taking the cube root, the large radius R equals the small radius r multiplied by n raised to the power of one-third. Therefore, the ratio of the large radius R to the small radius r is n raised to the power of one-third.

Surface Area Comparison:

Surface area depends on the square of the radius. The surface area S of the large sphere is proportional to the square of R. The surface area of a single small sphere is proportional to the square of r, so the sum of the surface areas s of all n small spheres is n multiplied by the square of r.

Ratio of Surface Areas:

Dividing the total small surface area s by the large surface area S gives n multiplied by the square of the ratio of r to R. Since R divided by r equals n raised to the power of one-third, r divided by R equals one divided by n raised to the power of one-third. Squaring this fraction gives one divided by n raised to the power of two-thirds.

Final Relation:

Multiplying n by one divided by n raised to the power of two-thirds simplifies to n raised to the power of one-third (since one minus two-thirds equals one-third).

Thus, the sum of the surface areas s equals n raised to the power of one-third multiplied by S.