Select the correct answer.

Which expression is equivalent to [tex]\(\sqrt{48 x^5}\)[/tex], if [tex]\(x \ \textgreater \ 0\)[/tex]?

A. [tex]\(12 x^8 \sqrt{3 x}\)[/tex]
B. [tex]\(4 x^3 \sqrt{3}\)[/tex]
C. [tex]\(4 x^2 \sqrt{3 x}\)[/tex]
D. [tex]\(12 x^2 \sqrt{x}\)[/tex]



Answer :

To determine which expression is equivalent to [tex]\(\sqrt{48 x^5}\)[/tex] when [tex]\(x > 0\)[/tex], let's break it down step by step.

1. Simplify the expression under the square root:

The given expression is [tex]\(\sqrt{48 x^5}\)[/tex].

2. Factor the constants and the [tex]\(x\)[/tex] terms:

[tex]\(48\)[/tex] can be factored into [tex]\(16 \times 3\)[/tex].

[tex]\(x^5\)[/tex] can be broken down as [tex]\(x^4 \times x\)[/tex].

So, we have:
[tex]\[ \sqrt{48 x^5} = \sqrt{16 \times 3 \times x^4 \times x} \][/tex]

3. Separate the square root of the product into the product of square roots:

By applying the property [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we get:
[tex]\[ \sqrt{16 \times 3 \times x^4 \times x} = \sqrt{16} \times \sqrt{3} \times \sqrt{x^4} \times \sqrt{x} \][/tex]

4. Simplify each square root:

[tex]\(\sqrt{16} = 4\)[/tex], [tex]\(\sqrt{3}\)[/tex] remains as it is because it’s already simplified, [tex]\(\sqrt{x^4} = x^2\)[/tex], and [tex]\(\sqrt{x}\)[/tex] remains as it is because it’s already in its simplest radical form.

Therefore, combining these, we get:
[tex]\[ 4 \times \sqrt{3} \times x^2 \times \sqrt{x} \][/tex]

5. Combine the simplified parts into one expression:

Putting it all together, we have:
[tex]\[ 4 x^2 \sqrt{3} \sqrt{x} = 4 x^2 \sqrt{3 x} \][/tex]

The correct expression that is equivalent to [tex]\(\sqrt{48 x^5}\)[/tex], when [tex]\(x > 0\)[/tex], is:
[tex]\[ 4 x^2 \sqrt{3 x} \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{C. 4 x^2 \sqrt{3 x}} \][/tex]