Given the expression [tex]\(4 \sqrt{3}\)[/tex], we need to identify which choices are equivalent to this expression.
Let's examine each choice one by one:
Choice A: [tex]\(\sqrt{48}\)[/tex]
To determine if this is equivalent to [tex]\(4 \sqrt{3}\)[/tex]:
[tex]\[
\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4 \sqrt{3}
\][/tex]
Thus, [tex]\(\sqrt{48}\)[/tex] is indeed equivalent to [tex]\(4 \sqrt{3}\)[/tex].
Choice B: [tex]\(\sqrt{24} \bullet \sqrt{2}\)[/tex]
To determine if this is equivalent to [tex]\(4 \sqrt{3}\)[/tex]:
[tex]\[
\sqrt{24} \times \sqrt{2} = \sqrt{24 \times 2} = \sqrt{48}
\][/tex]
We have already determined that [tex]\(\sqrt{48} = 4 \sqrt{3}\)[/tex].
Thus, [tex]\(\sqrt{24} \bullet \sqrt{2}\)[/tex] is indeed equivalent to [tex]\(4 \sqrt{3}\)[/tex].
Choice C: [tex]\(3 \sqrt{16}\)[/tex]
To determine if this is equivalent to [tex]\(4 \sqrt{3}\)[/tex]:
[tex]\[
3 \sqrt{16} = 3 \times 4 = 12
\][/tex]
Clearly, [tex]\(12\)[/tex] is not equivalent to [tex]\(4 \sqrt{3}\)[/tex].
Thus, [tex]\(3 \sqrt{16}\)[/tex] is not equivalent to [tex]\(4 \sqrt{3}\)[/tex].
Choice D: [tex]\(\sqrt{4} \bullet \sqrt{3}\)[/tex]
To determine if this is equivalent to [tex]\(4 \sqrt{3}\)[/tex]:
[tex]\[
\sqrt{4} \times \sqrt{3} = \sqrt{4 \times 3} = \sqrt{12}
\][/tex]
[tex]\(\sqrt{12}\)[/tex] does not simplify to [tex]\(4 \sqrt{3}\)[/tex], but rather [tex]\(2 \sqrt{3}\)[/tex].
Thus, [tex]\(\sqrt{4} \bullet \sqrt{3}\)[/tex] is not equivalent to [tex]\(4 \sqrt{3}\)[/tex].
Choice E: [tex]\(\sqrt{12} \bullet \sqrt{4}\)[/tex]
To determine if this is equivalent to [tex]\(4 \sqrt{3}\)[/tex]:
[tex]\[
\sqrt{12} \times \sqrt{4} = \sqrt{12 \times 4} = \sqrt{48}
\][/tex]
We have already determined that [tex]\(\sqrt{48} = 4 \sqrt{3}\)[/tex].
Thus, [tex]\(\sqrt{12} \bullet \sqrt{4}\)[/tex] is indeed equivalent to [tex]\(4 \sqrt{3}\)[/tex].
Choice F: 48
48 is a numerical value and not in a radical form related to [tex]\(4 \sqrt{3}\)[/tex]. It does not make mathematical sense in this context.
Thus, 48 is not equivalent to [tex]\(4 \sqrt{3}\)[/tex].
In summary:
The equivalent choices to [tex]\(4 \sqrt{3}\)[/tex] are:
- A. [tex]\(\sqrt{48}\)[/tex]
- B. [tex]\(\sqrt{24} \bullet \sqrt{2}\)[/tex]
- E. [tex]\(\sqrt{12} \bullet \sqrt{4}\)[/tex]
Therefore, the correct answers are A, B, and E.
