To determine which calculation correctly simplifies [tex]\(\sqrt{48}\)[/tex] using prime factorization, we will examine each provided option step by step.
Option A: [tex]\(\sqrt{48}=\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}=2 \sqrt{12}\)[/tex]
- Breaking 48 into its prime factors, [tex]\[
48 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3
\][/tex]
- According to this option, [tex]\(\sqrt{48}\)[/tex] is simplified as follows:
[tex]\[
\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 2\sqrt{12}
\][/tex]
- However, [tex]\(\sqrt{12}\)[/tex] can be further simplified:
[tex]\[
\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}
\][/tex]
So, [tex]\[
2\sqrt{12} = 2 \cdot 2 \sqrt{3} = 4 \sqrt{3}
\][/tex]
- Therefore, this answer, initially given as [tex]\(2\sqrt{12}\)[/tex], doesn't show the simplest form directly.
Option B: [tex]\(\sqrt{48}=\sqrt{4 \cdot 12}=2 \sqrt{12}\)[/tex]
- 48 is factored into 4 and 12:
[tex]\[
\sqrt{48} = \sqrt{4 \cdot 12}
\][/tex]
Given that:
[tex]\[
\sqrt{4 \cdot 12} = 2 \sqrt{12}
\][/tex]
- Again acknowledging [tex]\(\sqrt{12}\)[/tex] simplifies further:
[tex]\[
\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}
\][/tex]
And hence,
[tex]\[
2\sqrt{12} = 2 \cdot 2 \sqrt{3} = 4 \sqrt{3}
\][/tex]
- So, this option also does not immediately provide the simplest form.
Option C: [tex]\(\sqrt{48}=\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}=4 \sqrt{3}\)[/tex]
- Considering the same prime factorization:
[tex]\[
48 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3
\][/tex]
Let's simplify this:
[tex]\[
\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}
\][/tex]
Grouping the pairs of 2’s:
[tex]\[
\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = \sqrt{4 \cdot 4 \cdot 3} = \sqrt{16 \cdot 3}
\][/tex]
Simplifying further:
[tex]\[
\sqrt{16 \cdot 3} = 4 \sqrt{3}
\][/tex]
- This form [tex]\(\sqrt{48} = 4 \sqrt{3}\)[/tex] is indeed the simplest form.
Option D: [tex]\(\sqrt{48}=\sqrt{16 \cdot 3}=4 \sqrt{3}\)[/tex]
- Similar to option C:
[tex]\[
\sqrt{48} = \sqrt{16 \cdot 3}
\][/tex]
Since [tex]\(16 = 4^2\)[/tex],
[tex]\[
\sqrt{16 \cdot 3} = 4 \sqrt{3}
\][/tex]
- This reaffirms the simplest form directly.
After verifying all options, we see that options C and D correctly simplify [tex]\(\sqrt{48}\)[/tex] into its simplest form, [tex]\(4 \sqrt{3}\)[/tex].
Therefore, the correct answer is 3.
