To determine which choices are equivalent to the expression [tex]\(4 \sqrt{6}\)[/tex], we will evaluate each given option step by step.
Choice A: [tex]\( \sqrt{16} \cdot \sqrt{6} \)[/tex]
First, let's simplify this expression:
[tex]\[
\sqrt{16} = 4
\][/tex]
So,
[tex]\[
\sqrt{16} \cdot \sqrt{6} = 4 \cdot \sqrt{6} = 4 \sqrt{6}
\][/tex]
This is equivalent to [tex]\(4 \sqrt{6}\)[/tex].
Choice B: 96
This option is simply a number.
[tex]\[
96 \neq 4 \sqrt{6}
\][/tex]
Therefore, this is not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
Choice C: [tex]\( \sqrt{32} \cdot \sqrt{3} \)[/tex]
Next, we simplify:
[tex]\[
\sqrt{32} \cdot \sqrt{3} = \sqrt{32 \cdot 3} = \sqrt{96}
\][/tex]
So we need to check:
[tex]\[
\sqrt{96} \neq 4 \sqrt{6}
\][/tex]
Therefore, this is not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
Choice D: [tex]\( \sqrt{96} \)[/tex]
Here, we have:
[tex]\[
\sqrt{96}
\][/tex]
We need to compare this with [tex]\(4 \sqrt{6} = 4 \cdot 2.45 \approx 9.8\)[/tex].
[tex]\[
\sqrt{96} \approx 9.8
\][/tex]
Since the two values are roughly equal, this one is equivalent to [tex]\(4 \sqrt{6}\)[/tex].
Choice E: [tex]\( \sqrt{4} \cdot \sqrt{36} \)[/tex]
Simplifying this, we find:
[tex]\[
\sqrt{4} = 2 \ \text{and} \ \sqrt{36} = 6
\][/tex]
So,
[tex]\[
\sqrt{4} \cdot \sqrt{36} = 2 \cdot 6 = 12
\][/tex]
Since [tex]\(12 \neq 4 \sqrt{6}\)[/tex], this option is not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
Choice F: [tex]\( \sqrt{24} \)[/tex]
Lastly, we evaluate:
[tex]\[
\sqrt{24} \approx 4.9
\][/tex]
Since [tex]\(4.9 \neq 4 \cdot 2.45\)[/tex], this option is also not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
Therefore, the choices that are equivalent to [tex]\(4 \sqrt{6}\)[/tex] are:
A and D.
