To simplify the expression \( 512125\sqrt{3} \) using exponent properties, we can rewrite it in a clearer form.
First, we notice that \( 512125 \) is a number that can be simplified with base denominators:
\[ \sqrt{3} = 3^{1/2} \]
Now we rewrite:
\[ 512125\sqrt{3} = 512125 \cdot 3^{1/2} \]
However, to find an equivalent expression from the given options, we need to express 512125 in a simplified form, potentially using factors of 5 since many of the response options involve \(5\).
Breaking down \(512125\):
- We can try factorizing \(512125\) into its prime factors, but it is evident that it's a larger number.
If we factor \(512125\):
\[ 512125 = 5^2 \cdot 3^1 \cdot 7^3 \]
This specific factorization is complex, but essentially, what you need to focus on is rewriting \(512125\) according to the response options and see if any of them can be factors of \(5\).
However, as it's not clear if \(5^k\) would yield exact simplifications leading to a precise equivalent from the options provided without specific computation.
Let’s check the responses:
- \( 15 \) — not equivalent.
- \( \frac{1}{5} \) — not equivalent.
- \( 1 \) — not equivalent.
- \( 5\sqrt{5} \) — might have \(5k = \sqrt{3}\) as part, not 512125.
- \( 50 \) — not equivalent.
In complexity and general simplification, as none of them represent exact factional reduction from \( 512125\sqrt{3}\), consider command for accuracy in selection if numerical equivalency does not appear.
Thus, I suggest rechecking the factorization or calculating numerically to arrive directly from increasing proportions of base factors represented.
That being said, we can conclude that without further factor evaluations, \(5\sqrt{5}\) provides a rough continuous equivalency solved from algebra factoring; otherwise, all others remain strictly non-equivalent directly.