To determine which expression is equivalent to [tex]\(8 \sqrt{6}\)[/tex], let's follow these steps:
1. Express the terms inside a single square root:
- We start with [tex]\(8 \sqrt{6}\)[/tex].
- We can think of [tex]\(8\)[/tex] as [tex]\(8 = \sqrt{8^2}\)[/tex].
2. Combine the terms under one square root:
- Using the property of square roots, [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex], we can combine [tex]\(\sqrt{8^2}\)[/tex] and [tex]\(\sqrt{6}\)[/tex]:
[tex]\[
8 \sqrt{6} = \sqrt{8^2} \cdot \sqrt{6} = \sqrt{8^2 \cdot 6}
\][/tex]
3. Calculate the value under the square root:
- [tex]\(8^2 = 64\)[/tex], so the term becomes:
[tex]\[
\sqrt{64 \cdot 6}
\][/tex]
- Now calculate inside the square root:
[tex]\(64 \cdot 6 = 384\)[/tex].
4. Final expression:
- Thus, the expression simplifies to:
[tex]\[
\sqrt{384}
\][/tex]
So, the expression equivalent to [tex]\(8 \sqrt{6}\)[/tex] is [tex]\(\sqrt{384}\)[/tex].
The correct answer is:
[tex]\[ \boxed{\sqrt{384}} \][/tex]
Thus, the correct answer is option D: [tex]\(\sqrt{384}\)[/tex].
