Select the correct answer.

Which calculation correctly uses prime factorization to write [tex]\sqrt{48}[/tex] in simplest form?

A. [tex]\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 2 \sqrt{12}[/tex]

B. [tex]\sqrt{48} = \sqrt{4 \cdot 12} = 2 \sqrt{12}[/tex]

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]



Answer :

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.