To simplify the expression \( 1253 \div 753 \) using the Quotient Rule of Integer Exponents, we first rewrite the bases:
- We can express \( 125 \) and \( 75 \) as powers of \( 5 \) and \( 3 \):
- \( 125 = 5^3 \)
- \( 75 = 5^2 \times 3^1 \)
So, rewriting the expression \( \frac{125^3}{75^3} \):
\[ \frac{(5^3)^3}{(5^2 \cdot 3)^3} \]
Using the power of a product property:
\[ = \frac{5^{3 \cdot 3}}{(5^2)^3 \cdot 3^3} = \frac{5^9}{5^6 \cdot 3^3} \]
Now applying the Quotient Rule of Integer Exponents:
\[ = \frac{5^9}{5^6} \cdot \frac{1}{3^3} \]
This simplifies to:
\[ = 5^{9-6} \cdot \frac{1}{3^3} = 5^3 \cdot \frac{1}{3^3} \]
Thus, we can express it as:
\[ = \frac{5^3}{3^3} = \left( \frac{5}{3} \right)^3 \]
Given this simplification, let's look back at your provided multiple-choice answers and match:
None of the provided responses seem to match exactly. To clarify:
- The final simplified expression is \( \left( \frac{5}{3} \right)^{3} \) or \( 125/27 \) when computed numerically.
Hopefully, this helps clarify the steps taken to simplify the expression! If you'd like the final answer to match any specific formats from your responses, further context is needed.