To simplify the expression \( 125^3 \div 75^3 \) using the Quotient Rule of Integer Exponents, we can rewrite it as:
\[ \frac{125^3}{75^3} = \left(\frac{125}{75}\right)^3 \]
Next, we simplify \( \frac{125}{75} \):
\[ \frac{125}{75} = \frac{125 \div 25}{75 \div 25} = \frac{5}{3} \]
Now we can substitute this back into our expression:
\[ \left(\frac{125}{75}\right)^3 = \left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27} \]
None of the provided options directly correspond to \( \frac{125}{27} \). If there are any mistakes in the options provided, please check if \( 125^3 \div 75^3 \) was meant to lead to any of the given numbers.
If you meant \( 125 = 5^3 \) and \( 75 = 3 \cdot 5^2 \), we can recognize that:
\[ 125 = 5^3 \quad \text{and} \quad 75 = 3 \cdot 5^2 \]
This leads us back to our original simplification outcome:
\[ \frac{(5^3)^3}{(3 \cdot 5^2)^3} = \frac{5^9}{3^3 \cdot 5^6} = \frac{5^{9-6}}{3^3} = \frac{5^3}{3^3} = \frac{125}{27} \]
We can conclude that \( \frac{125^3}{75^3} = \left(\frac{5}{3}\right)^3 \), but it simplifies to \( \frac{125}{27} \).
If you're picking one of the multiple-choice options you provided, it's best to confirm the values were correctly listed as I can't recognize the correct match from provided options.