To simplify \( \frac{5^{\frac{11}{2}}}{125\sqrt{3}} \), we can start by rewriting the components in the expression using exponents.
- Rewrite \( 125 \) as \( 5^3 \) (since \( 125 = 5^3 \)).
- Rewrite \( \sqrt{3} \) as \( 3^{\frac{1}{2}} \) (since \( \sqrt{3} = 3^{1/2} \)).
- So we can rewrite \( \frac{5^{\frac{11}{2}}}{125\sqrt{3}} \) as:
\[ \frac{5^{\frac{11}{2}}}{5^3 \cdot 3^{\frac{1}{2}}} \]
- Using the properties of exponents \(\left(\frac{a^m}{a^n} = a^{m-n}\right)\), we can simplify the \( 5 \) terms:
\[ 5^{\frac{11}{2} - 3} = 5^{\frac{11}{2} - \frac{6}{2}} = 5^{\frac{5}{2}} \]
- So, the expression becomes:
\[ \frac{5^{\frac{5}{2}}}{3^{\frac{1}{2}}} \]
- This can be rewritten as:
\[ \frac{5^{\frac{5}{2}}}{\sqrt{3}} \]
Since \(5^{\frac{5}{2}}\) can be expressed as \(5^2 \cdot 5^{\frac{1}{2}} = 25\sqrt{5}\), the final simplified result is:
\[ \frac{25\sqrt{5}}{\sqrt{3}} = 25\cdot\frac{\sqrt{5}}{\sqrt{3}} = 25\sqrt{\frac{5}{3}} \]
Now checking the choices provided in your question:
- \(50\)
- \(5^0\) (which is 1)
- \(1\)
- \(\frac{1}{5}\)
- \(5\sqrt{5}\)
None of these expressions are equivalent to \(25\sqrt{\frac{5}{3}}\) directly.
The closest equivalent, however, is likely evaluating the term generally; nonetheless, based on the characteristics of your question, the answer would be needed to recognize that none of the options correctly represent the original simplified expression.
So if you were looking for a clear equivalence, among the provided choices, the accurate selection hasn’t been given. Nonetheless, all options do not satisfy replacing \( \frac{5^{\frac{5}{2}}}{\sqrt{3}} \).
Would you like to proceed with another type of question, or clarify any parts?