To solve this problem, we need to simplify the expression [tex]\(\sqrt{48}\)[/tex] using prime factorization and then determine which of the given choices correctly shows this simplification.
First, let's factorize 48 into its prime factors:
[tex]\[ 48 = 2 \times 2 \times 2 \times 2 \times 3 \][/tex]
We can write [tex]\(\sqrt{48}\)[/tex] using these prime factors:
[tex]\[ \sqrt{48} = \sqrt{2 \times 2 \times 2 \times 2 \times 3} \][/tex]
Grouping the prime factors into pairs:
[tex]\[ \sqrt{48} = \sqrt{(2 \times 2) \times (2 \times 2) \times 3} \][/tex]
[tex]\[ \sqrt{48} = \sqrt{4 \times 4 \times 3} \][/tex]
Since [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{48} = \sqrt{4} \times \sqrt{4} \times \sqrt{3} \][/tex]
[tex]\[ \sqrt{48} = 2 \times 2 \times \sqrt{3} \][/tex]
[tex]\[ \sqrt{48} = 4 \sqrt{3} \][/tex]
Now, let's check the given options to see which one matches with our simplified expression:
A. [tex]\(\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 2 \sqrt{12}\)[/tex]
- This option simplifies incorrectly since [tex]\(2 \sqrt{12}\)[/tex] is not equal to [tex]\(\sqrt{48}\)[/tex].
B. [tex]\(\sqrt{48} = \sqrt{4 \cdot 12} = 2 \sqrt{12}\)[/tex]
- This option also simplifies incorrectly since [tex]\(2 \sqrt{12}\)[/tex] is not equal to [tex]\(\sqrt{48}\)[/tex].
C. [tex]\(\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 4 \sqrt{3}\)[/tex]
- This option is correct as it matches exactly with our simplification.
D. [tex]\(\sqrt{48} = \sqrt{16 \cdot 3} = 4 \sqrt{3}\)[/tex]
- This option is also correct as it correctly simplifies [tex]\(\sqrt{48}\)[/tex] using another step of factorization.
So, the correct answers are:
C. [tex]\(\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 4 \sqrt{3}\)[/tex]
D. [tex]\(\sqrt{48} = \sqrt{16 \cdot 3} = 4 \sqrt{3}\)[/tex]
Given that we need to select only one correct answer:
Considering both C and D are correct and simplifying in similar form, we select
The correct answer is: D. [tex]\(\sqrt{48} = \sqrt{16 \cdot 3} = 4 \sqrt{3}\)[/tex]
- Note: If choosing one, always select a more direct factorization if multiple answers are correct. So the emphasis is always on a logically more simplified version.
