To determine which expression is equivalent to [tex]\( 8 \sqrt{6} \)[/tex], we will compare it to the square roots provided in the options. Below is a detailed step-by-step calculation and comparison.
1. Calculate [tex]\( 8 \sqrt{6} \)[/tex]:
[tex]\[ 8 \sqrt{6} \][/tex]
First, find the value of [tex]\(\sqrt{6}\)[/tex]. The square root of 6 is approximately 2.449.
[tex]\[ 8 \times 2.449 \approx 19.595 \][/tex]
2. Compare the value of [tex]\( 8 \sqrt{6} \)[/tex] to the given options:
- Option A: [tex]\(\sqrt{576}\)[/tex]
[tex]\[ \sqrt{576} = 24 \][/tex]
Comparing this to our target value [tex]\( 19.595 \)[/tex], we see:
[tex]\[ 24 \neq 19.595 \][/tex]
- Option B: [tex]\(\sqrt{48}\)[/tex]
[tex]\[ \sqrt{48} \approx 6.928 \][/tex]
Comparing this to our target value [tex]\( 19.595 \)[/tex], we see:
[tex]\[ 6.928 \neq 19.595 \][/tex]
- Option C: [tex]\(\sqrt{384}\)[/tex]
[tex]\[ \sqrt{384} \approx 19.595 \][/tex]
Comparing this to our target value [tex]\( 19.595 \)[/tex], we see:
[tex]\[ 19.595 \approx 19.595 \][/tex]
This matches our target value.
- Option D: [tex]\(\sqrt{96}\)[/tex]
[tex]\[ \sqrt{96} \approx 9.798 \][/tex]
Comparing this to our target value [tex]\( 19.595 \)[/tex], we see:
[tex]\[ 9.798 \neq 19.595 \][/tex]
From these comparisons, we find that the expression [tex]\(\sqrt{384}\)[/tex] (Option C) is approximately equal to [tex]\( 8 \sqrt{6} \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
