The sides of the large cake are 2 times the length of the sides of the small one.
-- The amount of the frosting depends on the area of the cake.
-- The area grows as fast as the square of the sides grows.
-- The ratio of the areas is the square of the ratio of the sides.
-- If you make the sides 2 times as long, the area becomes (2²) = 4 times as much.
-- If you made the sides 7 times as long, you'd need (7²) = 49 times as much frosting.
-- If you made the sides 10 times as long, you'd need 10² = 100 times as much.