Select the correct answer.

Which expression is equivalent to [tex]$8 \sqrt{6}$[/tex]?
A. [tex]$\sqrt{14}$[/tex]
B. [tex][tex]$\sqrt{48}$[/tex][/tex]
C. [tex]$\sqrt{96}$[/tex]
D. [tex]$\sqrt{384}$[/tex]



Answer :

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].