To determine which choices are equivalent to the expression [tex]\(4 \sqrt{3}\)[/tex], let's evaluate each option step-by-step:
### Choice A: [tex]\(3 \sqrt{16}\)[/tex]
1. Evaluate [tex]\(\sqrt{16}\)[/tex]:
[tex]\[
\sqrt{16} = 4
\][/tex]
2. Multiply by 3:
[tex]\[
3 \cdot 4 = 12
\][/tex]
Therefore, [tex]\(3 \sqrt{16} = 12\)[/tex]. This does not equal [tex]\(4 \sqrt{3}\)[/tex].
### Choice B: 48
48 is a constant and does not involve the square root of 3 or any similar structure. It is clear that 48 does not equal [tex]\(4 \sqrt{3}\)[/tex].
### Choice C: [tex]\(\sqrt{24} \cdot \sqrt{2}\)[/tex]
1. Use the property of square roots: [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex].
[tex]\[
\sqrt{24} \cdot \sqrt{2} = \sqrt{24 \cdot 2} = \sqrt{48}
\][/tex]
2. Simplify [tex]\(\sqrt{48}\)[/tex]:
[tex]\[
\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \cdot \sqrt{3}
\][/tex]
Therefore, [tex]\(\sqrt{24} \cdot \sqrt{2} = 4 \sqrt{3}\)[/tex]. This is equivalent to the original expression.
### Choice D: [tex]\(\sqrt{12} \cdot \sqrt{4}\)[/tex]
1. Use the property of square roots: [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex].
[tex]\[
\sqrt{12} \cdot \sqrt{4} = \sqrt{12 \cdot 4} = \sqrt{48}
\][/tex]
2. Simplify [tex]\(\sqrt{48}\)[/tex]:
[tex]\[
\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \cdot \sqrt{3}
\][/tex]
Therefore, [tex]\(\sqrt{12} \cdot \sqrt{4} = 4 \sqrt{3}\)[/tex]. This is equivalent to the original expression.
### Choice E: [tex]\(\sqrt{48}\)[/tex]
1. Simplify [tex]\(\sqrt{48}\)[/tex]:
[tex]\[
\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \cdot \sqrt{3}
\][/tex]
Therefore, [tex]\(\sqrt{48} = 4 \sqrt{3}\)[/tex]. This is equivalent to the original expression.
### Choice F: [tex]\(\sqrt{4} \cdot \sqrt{3}\)[/tex]
1. Use the property of square roots: [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex].
[tex]\[
\sqrt{4} \cdot \sqrt{3} = \sqrt{4 \cdot 3} = \sqrt{12}
\][/tex]
2. Simplify [tex]\(\sqrt{12}\)[/tex]:
[tex]\[
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \cdot \sqrt{3}
\][/tex]
Therefore, [tex]\(\sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}\)[/tex]. This does not equal [tex]\(4 \sqrt{3}\)[/tex].
### Conclusion
The choices that are equivalent to [tex]\(4 \sqrt{3}\)[/tex] are:
- C. [tex]\(\sqrt{24} \cdot \sqrt{2}\)[/tex]
- D. [tex]\(\sqrt{12} \cdot \sqrt{4}\)[/tex]
- E. [tex]\(\sqrt{48}\)[/tex]
So, the equivalent choices are C, D, and E.
