Context: In $\dfrac{5}{9}(5x + 9)(x + \sqrt{5k + 9})(x - \sqrt{5k + 9}) = 0$ the constant $k$ is positive, and the solutions multiply to $81$. We need $k$.

Answer: 36/5

Why: A product equals zero exactly when one of its variable factors is zero; the constant $\dfrac{5}{9}$ contributes no solution. Setting $5x + 9 = 0$ gives $x = -\dfrac{9}{5}$, and the other two factors give $x = -\sqrt{5k + 9}$ and $x = \sqrt{5k + 9}$. Their product is $\left(-\dfrac{9}{5}\right)\left(-\sqrt{5k + 9}\right)\left(\sqrt{5k + 9}\right) = \dfrac{9}{5}(5k + 9)$. Setting $\dfrac{9}{5}(5k + 9) = 81$ gives $5k + 9 = 45$, so $5k = 36$ and $k = \dfrac{36}{5}$.

Common slips: Treating $\dfrac{5}{9}$ as a solution or as a factor of the product; or losing one negative sign, which turns the product into $-\dfrac{9}{5}(5k + 9)$ and forces a negative $k$, contradicting the stated condition.