2006 AMC 10A Problem 2

The rule $x \otimes y = x^3 - y$ means: cube the first number, then subtract the second number.

1. Start with the inside part, because the outside part needs its value. With both inputs equal to $h$:
$$h \otimes h = h^3 - h.$$
2. Now the outside operation has first input $h$ and second input $h^3 - h$. Cube the first input and subtract the whole second input, in parentheses:
$$h \otimes (h^3 - h) = h^3 - (h^3 - h).$$
3. Remove the parentheses. The minus sign flips both signs inside:
$$h^3 - h^3 + h = h.$$

The answer is $\boxed{\textbf{(C)}\ h}$.

A made-up operation is just a rule with two slots. Here the two slots are treated differently (only the first one gets cubed), so it matters which number goes where. Always work from the innermost parentheses outward, and when you plug in an expression, keep it in parentheses so the minus sign reaches every term.

Work from the inside out. The rule says "cube the first input and subtract the second," so
$$h \otimes h = h^3 - h.$$
Now the outer operation has first input $h$ and second input $h^3 - h$:
$$h \otimes (h^3 - h) = h^3 - (h^3 - h) = h^3 - h^3 + h = h.$$

The answer is $\boxed{\textbf{(C)}\ h}$.