To simplify the expression \( 5 \frac{1}{2} 125\sqrt{3} \) using exponent properties, let's break it down.
First, we'll convert \( 5 \frac{1}{2} \) into an improper fraction:
\[ 5 \frac{1}{2} = \frac{11}{2} \]
Now, we can rewrite the whole expression:
\[ \frac{11}{2} \times 125 \times \sqrt{3} \]
Next, we can express 125 as a power of 5:
\[ 125 = 5^3 \]
So, we have:
\[ \frac{11}{2} \times 5^3 \times \sqrt{3} \]
To express \(\sqrt{3}\) with exponents, we write it as:
\[ \sqrt{3} = 3^{1/2} \]
Now putting it all together, we have:
\[ \frac{11}{2} \times 5^3 \times 3^{1/2} \]
Now we can analyze the provided answer choices:
- \( 5\sqrt{5} \)
- \( \frac{ \sqrt{5} }{ 5 } \)
- \( \frac{1}{5} \)
- \( 5^0 \)
None of these options seem directly equivalent to the expression we derived, which still keeps \( \frac{11}{2} \), and has a factor of \( 3^{1/2} \).
Let's see if any of these choices correspond to a simplification or result that would match an equivalent form based on some assumptions:
- \( 5^0 = 1 \) which isn't relevant.
- \( \frac{1}{5} \) isn't related to our product.
- \( 5\sqrt{5} \) and \( \frac{ \sqrt{5} }{ 5 } \) don’t relate to larger multiplicative forms of \( 125\sqrt{3} \) and \( \frac{11}{2} \).
Thus, without additional clarity or options being specifically reduced correctly, the expression as analyzed doesn't clearly equate to any of the original choices without additional manipulation or if interpretations differ.
Given the simplifications made from the original construct of \( 5 \frac{1}{2} \times 125 \sqrt{3} \), no choice completely reflects the algebraic simplification directly via exponent properties without more revisions of forms.
Therefore, given the listed options:
- None can be deemed equivalent directly based on simplification to \( 5 \frac{1}{2} 125\sqrt{3} \) concerning available potential responses.
Correctly formulated choices will depend on clarity of instructions or intended operational assessment reflecting intended equivalent expressions matching appropriately.