### Instruction: Given that $a+b=3$ and $a^3+b^3=81$, find $ab$.### Response: I notice that the expressions $a+b$ and $a^3+b^3$ are related by a cubic identity: $(a+b)^3=a^3+b^3+3ab(a+b)$.
If I plug in the given values, I get $3^3=81+3ab(3)$, which simplifies to $27=81+9ab$.
Subtracting $81$ from both sides, I get $-54=9ab$.
Dividing by $9$, I get $-6=ab$.
Therefore, $ab=-6$.