To simplify \( 512125\sqrt{3} \) using exponent properties, we first need to express \( 512125 \) in terms of its prime factors, specifically \( 5 \), because it appears in the options.
We can divide \( 512125 \) by \( 5 \) repeatedly to find its prime factorization:
- \( 512125 \div 5 = 102425 \)
- \( 102425 \div 5 = 20485 \)
- \( 20485 \div 5 = 4097 \)
- \( 4097 \) is not divisible by \( 5 \). So, we stop here.
Now we know \( 512125 = 5^3 \times 4097 \).
So, we have:
\[ 512125\sqrt{3} = 5^3 \times 4097 \sqrt{3} \]
However, since none of the options seem to directly fit this expression, we likely don't want to fully factor \( 4097 \) if it doesn't need to be simplified to match the options provided.
Since \( 512125 \) doesn't simplify neatly to one of the answer choices, let's evaluate the options given:
- 15 – Not equivalent.
- \( \frac{1}{5} \) – Not equivalent.
- \( 5\sqrt{5} \) – Not equivalent.
- \( 50 \) – Not equivalent.
- \( 5^0 \) – \( 1 \), which does not represent \( 512125\sqrt{3} \).
Thus, none of the provided options are equivalent expressions when simplified.
If we revisit if \( \sqrt{3} \) should play a role, it's apparent that options should have \( \sqrt{3} \) not accounted for but rather focus on \( 512125 \).
Based on the simplifications, without a clear correspondence to the given options focused on the roots/multipliers, we should consider that the needed expressions are invalid by the prime breakdown.
If you had concluded to choose from this, please consider providing a clarification or adjustment for pursuing alternative ways to align from the simplifications beyond the linear fractional choice.
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