To answer this question, we need to simplify the expression [tex]\(\sqrt{48}\)[/tex] using the method of prime factorization and find which option matches our simplified form.
Let's analyze each given option:
Option A: [tex]\[
\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 2 \sqrt{12}
\][/tex] This statement is incorrect because it stops the simplification too early.
Option B: [tex]\[
\sqrt{48} = \sqrt{4 \cdot 12} = 2 \sqrt{12}
\][/tex] This statement is incorrect for the same reason as Option A.
Option C: [tex]\[
\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 4 \sqrt{3}
\][/tex] Let's check the simplification here: - The prime factorization of 48 is [tex]\(2 \cdot 2 \cdot 2 \cdot 2 \cdot 3\)[/tex]. - This can be written as [tex]\(\sqrt{(2 \cdot 2) \cdot (2 \cdot 2) \cdot 3}\)[/tex]. - Simplifying further, [tex]\(\sqrt{(2^2) \cdot (2^2) \cdot 3} = \sqrt{4 \cdot 4 \cdot 3} = 4 \sqrt{3}\)[/tex].
This statement is correct.
Option D: [tex]\[
\sqrt{48} = \sqrt{16 \cdot 3} = 4 \sqrt{3}
\][/tex] This statement is also correct because: - [tex]\(16\)[/tex] is the square of [tex]\(4\)[/tex], so it can be pulled out of the square root. - [tex]\(\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3}\)[/tex].
Thus, the correct options are C and D, but if we are to choose only one based on the simplification step shown explicitly, the correct simplified representation in simplest form is [tex]\(\boxed{D}\)[/tex].
