To solve the given problem, we need to find the cube roots of several numbers and then multiply them together. Let's consider each component step by step.
First, calculate the cube roots of the given numbers:
1. [tex]\(\sqrt[3]{24} \approx 2.8845\)[/tex]
2. [tex]\(\sqrt[3]{45} \approx 3.5569\)[/tex]
3. [tex]\(\sqrt[3]{68} \approx 4.0817\)[/tex]
4. For [tex]\(4(\sqrt[3]{6})\)[/tex], calculate [tex]\(\sqrt[3]{6} \approx 1.8171\)[/tex] and then multiply by 4 to get [tex]\(4 \times 1.8171 \approx 7.2685\)[/tex].
5. For [tex]\(6(\sqrt[3]{5})\)[/tex], calculate [tex]\(\sqrt[3]{5} \approx 1.3686\)[/tex] and then multiply by 6 to get [tex]\(6 \times 1.3686 \approx 8.2117\)[/tex].
6. For [tex]\(6(\sqrt[3]{10})\)[/tex], calculate [tex]\(\sqrt[3]{10} \approx 2.1544\)[/tex] and then multiply by 6 to get [tex]\(6 \times 2.1544 \approx 12.9266\)[/tex].
Thus, the intermediate results for each cube root expression are:
[tex]\[
\begin{align*}
\sqrt[3]{24} &\approx 2.8845,\\
\sqrt[3]{45} &\approx 3.5569,\\
\sqrt[3]{68} &\approx 4.0817,\\
4(\sqrt[3]{6}) &\approx 7.2685,\\
6(\sqrt[3]{5}) &\approx 8.2117,\\
6(\sqrt[3]{10}) &\approx 12.9266.
\end{align*}
\][/tex]
Now, multiply all these values together to get the product:
[tex]\[
2.8845 \times 3.5569 \times 4.0817 \times 7.2685 \times 8.2117 \times 12.9266.
\][/tex]
By calculating the above product, we get the final result. Therefore, the product of the given expression is approximately
[tex]\[
\boxed{48824.4935}.
\][/tex]
