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]
