To determine which expression is equivalent to [tex]\( 8\sqrt{6} \)[/tex], we will compare it to each of the given options one by one. We aim to match the value of [tex]\( 8\sqrt{6} \)[/tex] with one of the square roots presented.
First, let's note the given expression and compute its value:
[tex]\[ 8\sqrt{6} \][/tex]
Now, let's examine each of the different options to see which one will be equivalent to [tex]\( 8\sqrt{6} \)[/tex].
Option A: [tex]\( \sqrt{384} \)[/tex]
To check if this is equivalent, we need to find out if:
[tex]\[ 8\sqrt{6} = \sqrt{384} \][/tex]
Recall that [tex]\( \sqrt{a} = b \implies b^2 = a \)[/tex]. So we square both sides for [tex]\( 8\sqrt{6} \)[/tex]:
[tex]\[ (8\sqrt{6})^2 = 64 \times 6 = 384 \][/tex]
Thus, [tex]\( \sqrt{384} \)[/tex] equals [tex]\( 8\sqrt{6} \)[/tex], confirming that:
[tex]\[ \sqrt{384} = 8\sqrt{6} \][/tex]
So option A is correct.
Option A:
[tex]\[ 8\sqrt{6} = \sqrt{384} \][/tex]
Next, for completeness, let's briefly check the other options to ensure none of them match.
Option B: [tex]\( \sqrt{576} \)[/tex]
Squaring [tex]\( 8\sqrt{6} \)[/tex] again to check:
[tex]\[ (8\sqrt{6})^2 = 384 \neq 576 \][/tex]
Thus, [tex]\( \sqrt{576} \)[/tex] does not equal [tex]\( 8\sqrt{6} \)[/tex].
Option C: [tex]\( \sqrt{96} \)[/tex]
Squaring [tex]\( 8\sqrt{6} \)[/tex]:
[tex]\[ (8\sqrt{6})^2 = 384 \neq 96 \][/tex]
Thus, [tex]\( \sqrt{96} \)[/tex] does not equal [tex]\( 8\sqrt{6} \)[/tex].
Option D: [tex]\( \sqrt{48} \)[/tex]
Squaring [tex]\( 8\sqrt{6} \)[/tex]:
[tex]\[ (8\sqrt{6})^2 = 384 \neq 48 \][/tex]
Thus, [tex]\( \sqrt{48} \)[/tex] does not equal [tex]\( 8\sqrt{6} \)[/tex].
Given these evaluations, we identify that the correct answer is:
[tex]\[ \boxed{ \sqrt{384} } \][/tex]
Thus, the correct option is:
Answer: A. [tex]\( \sqrt{384} \)[/tex]
