First, let's interpret the original mathematical expression:
[tex]\[
4 \sqrt{6}
\][/tex]
Now, let's evaluate each choice one by one to determine which, if any, are equivalent to the expression [tex]\(4 \sqrt{6}\)[/tex].
### Choice A: [tex]\(\sqrt{32} \cdot \sqrt{3}\)[/tex]
To simplify, we use the property [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]:
[tex]\[
\sqrt{32} \cdot \sqrt{3} = \sqrt{32 \cdot 3} = \sqrt{96}
\][/tex]
The numerical value of [tex]\(\sqrt{96}\)[/tex] is approximately:
[tex]\[
\sqrt{96} \approx 9.797958971132713
\][/tex]
### Choice B: [tex]\(\sqrt{4} \cdot \sqrt{36}\)[/tex]
Similarly, we apply the property [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]:
[tex]\[
\sqrt{4} \cdot \sqrt{36} = \sqrt{4 \cdot 36} = \sqrt{144}
\][/tex]
The numerical value of [tex]\(\sqrt{144}\)[/tex] is:
[tex]\[
\sqrt{144} = 12
\][/tex]
### Choice C: [tex]\(\sqrt{16} \cdot \sqrt{5}\)[/tex]
Using the same property:
[tex]\[
\sqrt{16} \cdot \sqrt{5} = \sqrt{16 \cdot 5} = \sqrt{80}
\][/tex]
The numerical value of [tex]\(\sqrt{80}\)[/tex] is approximately:
[tex]\[
\sqrt{80} \approx 8.94427190999916
\][/tex]
### Choice D: [tex]\(\sqrt{96}\)[/tex]
This is left as is:
[tex]\[
\sqrt{96}
\][/tex]
The numerical value of [tex]\(\sqrt{96}\)[/tex] is approximately:
[tex]\[
\sqrt{96} \approx 9.797958971132712
\][/tex]
### Choice E: [tex]\(\sqrt{24}\)[/tex]
This is left as is:
[tex]\[
\sqrt{24}
\][/tex]
The numerical value of [tex]\(\sqrt{24}\)[/tex] is approximately:
[tex]\[
\sqrt{24} \approx 4.898979485566356
\][/tex]
### Choice F: 96
This is a simple number without any operations involved, so it remains:
[tex]\[
96
\][/tex]
### Comparison with the Original Expression
We compare each choice with [tex]\(4 \sqrt{6}\)[/tex]:
[tex]\[
4 \sqrt{6} \approx 9.797958971132712
\][/tex]
- Choice A: [tex]\(\sqrt{32} \cdot \sqrt{3} \approx 9.797958971132713\)[/tex]
- Choice B: [tex]\(\sqrt{4} \cdot \sqrt{36} = 12\)[/tex]
- Choice C: [tex]\(\sqrt{16} \cdot \sqrt{5} \approx 8.94427190999916\)[/tex]
- Choice D: [tex]\(\sqrt{96} \approx 9.797958971132712\)[/tex]
- Choice E: [tex]\(\sqrt{24} \approx 4.898979485566356\)[/tex]
- Choice F: 96
Among these values, only Choice A and Choice D are approximately equal to [tex]\(4 \sqrt{6}\)[/tex].
Thus, the choices that are equivalent to [tex]\(4 \sqrt{6}\)[/tex] are:
- Choice D: [tex]\(\sqrt{96}\)[/tex]
Therefore, the correct answer is:
[tex]\[
\text{The choices that are equivalent to } 4 \sqrt{6} \text{ are } \boxed{\text{D}}
\][/tex]
