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Sagot :
To determine which choices are equivalent to the expression [tex]\(4 \sqrt{6}\)[/tex], let's examine each choice carefully.
1. Choice A: [tex]\(\sqrt{24}\)[/tex]
To find out if [tex]\(\sqrt{24}\)[/tex] is equivalent to [tex]\(4 \sqrt{6}\)[/tex], we need to see if:
[tex]\[ \sqrt{24} = 4 \sqrt{6} \][/tex]
Since [tex]\(\sqrt{24}\)[/tex] simplifies to [tex]\(\sqrt{4 \cdot 6} = 2 \sqrt{6}\)[/tex], it is clear that:
[tex]\[ 2 \sqrt{6} \neq 4 \sqrt{6} \][/tex]
Thus, [tex]\(\sqrt{24}\)[/tex] is not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
2. Choice B: 96
Let's compare 96 to [tex]\(4 \sqrt{6}\)[/tex].
We know that [tex]\(4 \sqrt{6} \approx 4 \cdot 2.449 \approx 9.796\)[/tex].
Clearly, 96 is not close to 9.796, so:
[tex]\[ 96 \neq 4 \sqrt{6} \][/tex]
Hence, 96 is not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
3. Choice C: [tex]\(\sqrt{16} \cdot \sqrt{6}\)[/tex]
Let's simplify [tex]\(\sqrt{16} \cdot \sqrt{6}\)[/tex] and see if it matches [tex]\(4 \sqrt{6}\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
So,
[tex]\[ \sqrt{16} \cdot \sqrt{6} = 4 \cdot \sqrt{6} = 4 \sqrt{6} \][/tex]
This is indeed equivalent to [tex]\(4 \sqrt{6}\)[/tex].
4. Choice D: [tex]\(\sqrt{4} \cdot \sqrt{36}\)[/tex]
Let's simplify [tex]\(\sqrt{4} \cdot \sqrt{36}\)[/tex] and match it to [tex]\(4 \sqrt{6}\)[/tex]:
[tex]\[ \sqrt{4} = 2 \quad \text{and} \quad \sqrt{36} = 6 \][/tex]
So,
[tex]\[ \sqrt{4} \cdot \sqrt{36} = 2 \cdot 6 = 12 \][/tex]
Clearly,
[tex]\[ 12 \neq 4 \sqrt{6} \][/tex]
Hence, [tex]\(\sqrt{4} \cdot \sqrt{36}\)[/tex] is not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
5. Choice E: [tex]\(\sqrt{32} \cdot \sqrt{3}\)[/tex]
Let's simplify [tex]\(\sqrt{32} \cdot \sqrt{3}\)[/tex] and check against [tex]\(4 \sqrt{6}\)[/tex]:
[tex]\[ \sqrt{32} \cdot \sqrt{3} = \sqrt{32 \cdot 3} = \sqrt{96} \][/tex]
Notice here:
[tex]\[ \sqrt{96} \][/tex]
We will address this shortly but for now, let's consider it to compare directly.
6. Choice F: [tex]\(\sqrt{96}\)[/tex]
To check equivalence,
[tex]\[ \sqrt{96} = \sqrt{16 \cdot 6} = \sqrt{16} \cdot \sqrt{6} = 4 \sqrt{6} \][/tex]
Hence, [tex]\(\sqrt{96}\)[/tex] is indeed [tex]\(4 \sqrt{6}\)[/tex].
Thus, the correct choices are:
[tex]\[ \text{C.} \quad \text{and } \quad \text{F.} \][/tex]
Given that choice F covers E through its simplification.
Therefore, the correct choices that are equivalent are:
[tex]\[ C, E, \text{ and } F \][/tex]
1. Choice A: [tex]\(\sqrt{24}\)[/tex]
To find out if [tex]\(\sqrt{24}\)[/tex] is equivalent to [tex]\(4 \sqrt{6}\)[/tex], we need to see if:
[tex]\[ \sqrt{24} = 4 \sqrt{6} \][/tex]
Since [tex]\(\sqrt{24}\)[/tex] simplifies to [tex]\(\sqrt{4 \cdot 6} = 2 \sqrt{6}\)[/tex], it is clear that:
[tex]\[ 2 \sqrt{6} \neq 4 \sqrt{6} \][/tex]
Thus, [tex]\(\sqrt{24}\)[/tex] is not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
2. Choice B: 96
Let's compare 96 to [tex]\(4 \sqrt{6}\)[/tex].
We know that [tex]\(4 \sqrt{6} \approx 4 \cdot 2.449 \approx 9.796\)[/tex].
Clearly, 96 is not close to 9.796, so:
[tex]\[ 96 \neq 4 \sqrt{6} \][/tex]
Hence, 96 is not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
3. Choice C: [tex]\(\sqrt{16} \cdot \sqrt{6}\)[/tex]
Let's simplify [tex]\(\sqrt{16} \cdot \sqrt{6}\)[/tex] and see if it matches [tex]\(4 \sqrt{6}\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
So,
[tex]\[ \sqrt{16} \cdot \sqrt{6} = 4 \cdot \sqrt{6} = 4 \sqrt{6} \][/tex]
This is indeed equivalent to [tex]\(4 \sqrt{6}\)[/tex].
4. Choice D: [tex]\(\sqrt{4} \cdot \sqrt{36}\)[/tex]
Let's simplify [tex]\(\sqrt{4} \cdot \sqrt{36}\)[/tex] and match it to [tex]\(4 \sqrt{6}\)[/tex]:
[tex]\[ \sqrt{4} = 2 \quad \text{and} \quad \sqrt{36} = 6 \][/tex]
So,
[tex]\[ \sqrt{4} \cdot \sqrt{36} = 2 \cdot 6 = 12 \][/tex]
Clearly,
[tex]\[ 12 \neq 4 \sqrt{6} \][/tex]
Hence, [tex]\(\sqrt{4} \cdot \sqrt{36}\)[/tex] is not equivalent to [tex]\(4 \sqrt{6}\)[/tex].
5. Choice E: [tex]\(\sqrt{32} \cdot \sqrt{3}\)[/tex]
Let's simplify [tex]\(\sqrt{32} \cdot \sqrt{3}\)[/tex] and check against [tex]\(4 \sqrt{6}\)[/tex]:
[tex]\[ \sqrt{32} \cdot \sqrt{3} = \sqrt{32 \cdot 3} = \sqrt{96} \][/tex]
Notice here:
[tex]\[ \sqrt{96} \][/tex]
We will address this shortly but for now, let's consider it to compare directly.
6. Choice F: [tex]\(\sqrt{96}\)[/tex]
To check equivalence,
[tex]\[ \sqrt{96} = \sqrt{16 \cdot 6} = \sqrt{16} \cdot \sqrt{6} = 4 \sqrt{6} \][/tex]
Hence, [tex]\(\sqrt{96}\)[/tex] is indeed [tex]\(4 \sqrt{6}\)[/tex].
Thus, the correct choices are:
[tex]\[ \text{C.} \quad \text{and } \quad \text{F.} \][/tex]
Given that choice F covers E through its simplification.
Therefore, the correct choices that are equivalent are:
[tex]\[ C, E, \text{ and } F \][/tex]
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