Use the product rule to simplify. Assume that all variables represent non-negative values.
\[
\sqrt{24 x^{3}}
\]
$\sqrt{24 x^{3}}=\square$ (Simplify your answer. Type an exact answer, using radicals as needed.)

Step 1 ：The given expression is a square root of a product. We can simplify this by taking the square root of each factor separately. The square root of a product is the product of the square roots, so we can simplify \(\sqrt{24 x^{3}}\) as \(\sqrt{24} * \sqrt{x^{3}}\).

Step 2 ：The square root of 24 can be simplified by finding the prime factorization of 24, which is \(2^3 * 3\). We can then pair up the factors and take out one of each pair from under the square root.

Step 3 ：The square root of \(x^{3}\) can be simplified by using the rule \(\sqrt{x^n} = x^{n/2}\).

Step 4 ：Let's calculate these in the next step.

Step 5 ：\(x = x\)

Step 6 ：\(\sqrt{24} = 2\sqrt{6}\)

Step 7 ：\(\sqrt{x^{3}} = x^{1.5}\)

Step 8 ：\(\sqrt{24 x^{3}} = 2\sqrt{6}x^{1.5}\)

Step 9 ：The simplified form of \(\sqrt{24 x^{3}}\) is \(2\sqrt{6}x^{1.5}\). This is the final answer.

Step 10 ：Final Answer: \(\boxed{2\sqrt{6}x^{1.5}}\)