Use $\Delta y \approx f^{\prime}(x) \Delta x$ to find a decimal approximation of the radical expression. \[ \sqrt[3]{8.66} \] What is the value found by using $\Delta y \approx f^{\prime}(x) \Delta x$ ? \[ \sqrt[3]{8.66} \approx \square \text { (Round to three decimal places as needed.) } \]

Step 1 ：The problem is asking for a decimal approximation of the cube root of 8.66 using the linear approximation formula \(\Delta y \approx f^{\prime}(x) \Delta x\).

Step 2 ：The function we are dealing with is \(f(x) = \sqrt[3]{x}\), and we want to find an approximation for \(f(8.66)\).

Step 3 ：We can choose a point close to 8.66 where we know the exact value of the function. A good choice is \(x = 8\), because \(\sqrt[3]{8} = 2\).

Step 4 ：The derivative of \(f(x)\) is \(f^{\prime}(x) = \frac{1}{3\sqrt[3]{x^2}}\).

Step 5 ：We can use the linear approximation formula with \(x = 8\), \(f(x) = 2\), \(f^{\prime}(x) = \frac{1}{3\sqrt[3]{64}} = \frac{1}{12}\), and \(\Delta x = 8.66 - 8 = 0.66\).

Step 6 ：Using these values in the linear approximation formula, we get an approximation of 2.055 for \(\sqrt[3]{8.66}\).

Step 7 ：Final Answer: The decimal approximation of the radical expression \(\sqrt[3]{8.66}\) is \(\boxed{2.055}\).