To determine which of the given expressions is equivalent to [tex]\( 8 \sqrt{6} \)[/tex], we need to square each option and then compare it to the square of [tex]\( 8 \sqrt{6} \)[/tex].
First, let's compute the square of the given expression [tex]\( 8 \sqrt{6} \)[/tex]:
[tex]\[
(8 \sqrt{6})^2 = 8^2 \cdot (\sqrt{6})^2 = 64 \cdot 6 = 384
\][/tex]
Now, let's examine the given options by squaring each one:
- Option A: [tex]\( \sqrt{48} \)[/tex]
[tex]\[
(\sqrt{48})^2 = 48
\][/tex]
- Option B: [tex]\( \sqrt{96} \)[/tex]
[tex]\[
(\sqrt{96})^2 = 96
\][/tex]
- Option C: [tex]\( \sqrt{384} \)[/tex]
[tex]\[
(\sqrt{384})^2 = 384
\][/tex]
- Option D: [tex]\( \sqrt{576} \)[/tex]
[tex]\[
(\sqrt{576})^2 = 576
\][/tex]
We see that the value we obtained for [tex]\( (8 \sqrt{6})^2 \)[/tex], which is 384, matches with [tex]\( \sqrt{384} \)[/tex].
Hence, the correct answer is:
C. [tex]\( \sqrt{384} \)[/tex]
