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

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

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 $\mathrm{\Delta }x=8.66-8=0.66$.

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 $\boxed{{\displaystyle 2.055}}$.