Priya says any number's cube is always bigger than its square, since cubing involves one extra…

Why

Priya's reasoning holds up for most numbers you'd think to check — 2³=8 is indeed bigger than 2²=4, and 5³=125 dwarfs 5²=25, so 'one extra multiplication should make it bigger' seems to check out every time she tries a normal-sized number. Test the very smallest whole numbers, though, and the rule quietly breaks: for n=1, the cube is 1×1×1=1, and the square is 1×1=1 — exactly EQUAL, not bigger. For n=0, both the cube and the square are 0 — equal again, not bigger. Cubing does add an extra multiplication, but multiplying by 1 (or by 0) doesn't grow a number the way multiplying by 2, 5, or 10 does, which is exactly why the extra step makes no difference right at these two special values. Test: line up n, n², and n³ for n=0,1,2,3 side by side — 0,0,0 then 1,1,1 then 2,4,8 then 3,9,27 — and watch the cube only pull ahead of the square starting from n=2 onward. No, this is not true for every number: it holds for every whole number 2 and above, but fails right at 0 and 1, where cube and square come out exactly equal.