1. The problem gives us two equations: $$(a+b)^2=81$$ and $$(a-b)^2=1$$. 2. Expand both equations using the identity $$(x+y)^2 = x^2 + 2xy + y^2$$: $$(a+b)^2 = a^2 + 2ab + b^2 = 81$$ $$(a-b)^2 = a^2 - 2ab + b^2 = 1$$ 3. Let’s denote: $$S = a^2 + b^2$$ and $$P = ab$$. Then the equations become: $$S + 2P = 81$$ $$S - 2P = 1$$ 4. Subtract the second equation from the first: $$(S + 2P) - (S - 2P) = 81 - 1$$ $$4P = 80$$ 5. Solve for $$P$$: $$P = \frac{80}{4} = 20$$ 6. Therefore, the value of $$ab$$ is $$20$$.