Question

select the correct answer.
which calculation correctly uses prime factorization to write \\(\sqrt{48}\\) in simplest form?
\\(\bigcirc\\) a. \\(\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 2\sqrt{12}\\)
\\(\bigcirc\\) b. \\(\sqrt{48} = \sqrt{4 \cdot 12} = 2\sqrt{12}\\)
\\(\bigcirc\\) c. \\(\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 4\sqrt{3}\\)
\\(\bigcirc\\) d. \\(\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}\\)

Explanation:

Step1: Recall prime factorization of 48

Prime factorization of 48 is \(2\times2\times2\times2\times3\) (since \(2^4\times3 = 16\times3=48\)). Also, \(16\times3\) is a valid factorization where 16 is a perfect square.

Step2: Analyze each option

• Option A: After prime factorization, \(\sqrt{2\times2\times2\times2\times3}\) should simplify to \(4\sqrt{3}\), not \(2\sqrt{12}\) (since \(\sqrt{2\times2\times2\times2\times3}=\sqrt{(2\times2\times2\times2)\times3}=\sqrt{16\times3} = 4\sqrt{3}\)). So A is wrong.
• Option B: \(\sqrt{4\times12}\) is not using prime factorization (12 is not a prime factor), and also \(\sqrt{4\times12}=2\sqrt{12}\) but \(\sqrt{12}\) can be simplified further, so B is wrong.
• Option C: Prime factorization of 48 is \(2\times2\times2\times2\times3\), but the step of simplification: \(\sqrt{2\times2\times2\times2\times3}=\sqrt{(2\times2)\times(2\times2)\times3}=2\times2\times\sqrt{3}=4\sqrt{3}\). However, the option says "uses prime factorization", but let's check D.
• Option D: \(\sqrt{48}=\sqrt{16\times3}\), here 16 is a perfect square (\(16 = 4^2\)) and 3 is prime. Then \(\sqrt{16\times3}=\sqrt{16}\times\sqrt{3}=4\sqrt{3}\). Also, 16 is a factor obtained from prime factorization (\(2^4 = 16\)). This uses a valid factorization (16 and 3, where 16 is a perfect square) to simplify the square root.

Answer:

C. \(\boldsymbol{\sqrt{48}=\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}=4\sqrt{3}}\), D. \(\boldsymbol{\sqrt{48}=\sqrt{16 \cdot 3}=4\sqrt{3}}\)

Wait, actually, let's re - evaluate. The question is "which calculation correctly uses prime factorization".

• Option C: It starts with prime factorization (\(2\times2\times2\times2\times3\)) and then simplifies correctly.
• Option D: It uses \(16\times3\), where 16 is \(2^4\) (from prime factorization). But does "using prime factorization" mean starting with prime factors?

Prime factorization of 48 is \(2^4\times3\).

For Option C: \(\sqrt{48}=\sqrt{2\times2\times2\times2\times3}=\sqrt{(2\times2\times2\times2)\times3}=\sqrt{16\times3}=4\sqrt{3}\) (correct simplification after prime factorization).

For Option D: \(\sqrt{48}=\sqrt{16\times3}\), 16 is \(2^4\) (prime factors), so it's also using prime - related factorization. But let's check the options again.

Wait, the key is "uses prime factorization to write \(\sqrt{48}\) in simplest form".

Let's re - check each option:

• Option A: Incorrect simplification.
• Option B: Not prime factorization.
• Option C: Prime factorization: \(48 = 2\times2\times2\times2\times3\), then \(\sqrt{48}=\sqrt{(2\times2)\times(2\times2)\times3}=2\times2\times\sqrt{3}=4\sqrt{3}\). This uses prime factorization.
• Option D: \(\sqrt{16\times3}\), 16 is \(2^4\) (from prime factorization), but is \(16\times3\) considered prime factorization? No, prime factorization is expressing as a product of primes. So Option D is using a factorization into a perfect square and a prime, not prime factorization directly.

So the correct answer is C.

Wait, no. Let's see the definition of prime factorization: expressing a number as a product of prime numbers.

In Option C: \(\sqrt{48}=\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}\) (prime factorization) and then simplifies to \(4\sqrt{3}\) correctly.

In Option D: \(\sqrt{16\cdot3}\), 16 is not a prime number, so it's not prime factorization. So the correct answer is C.

Final Answer: C. \(\boldsymbol{\sqrt{48}=\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}=4\sqrt{3}}\)