To simplify the expression \( \frac{5 \frac{1}{2}}{125 \sqrt{3}} \) using exponent properties, we first rewrite \( 5 \frac{1}{2} \) as \( \frac{11}{2} \).
Next, we can simplify \( 125 \) as \( 5^3 \) and \( \sqrt{3} \) as \( 3^{1/2} \):
So, we have:
\[ \frac{5 \frac{1}{2}}{125 \sqrt{3}} = \frac{5^{11/2}}{5^3 \cdot 3^{1/2}} \]
Using the properties of exponents to simplify the fraction:
\[ = \frac{5^{11/2}}{5^{3}} \cdot \frac{1}{3^{1/2}} = 5^{11/2 - 3} \cdot 3^{-1/2} = 5^{11/2 - 6/2} \cdot 3^{-1/2} = 5^{5/2} \cdot 3^{-1/2} \]
Now we can express this in terms of roots:
\[ = \frac{5^{5/2}}{\sqrt{3}} = \frac{\sqrt{5^5}}{\sqrt{3}} = \frac{5^2 \sqrt{5}}{\sqrt{3}} = \frac{25 \sqrt{5}}{\sqrt{3}} \]
Now, looking at the responses provided:
- \( 5\sqrt{5} \)
- \( \frac{\sqrt{5}}{5} \)
- \( 1 \)
- \( 50 \)
- \( 5^0 \)
- \( 15 \)
None of the options directly match this exact form, but upon checking for simplifications they suggest that none seem directly equivalent to the final result.
However, notice that no clear equivalent options represent this result out of all provided choices. The correct computation led to \( \frac{25 \sqrt{5}}{\sqrt{3}} \), which doesn't match any mentioned answers.
Given the nature of multiple-choice questions, if forced to choose one that could be closest to a reasonable answer based on experimental computational reductions (or provided descriptive discrepancies), I would lean towards the representation of one of the simplified forms but remain aware that no explicit options validate against the computed resolution. None are precise solutions from the calculations or simplifications above.
Conclusively, the answer remains undetermined without a proper equivalent expression in the options given.