Let's simplify the expression $\frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}$ step by step.

1. Rewrite the numbers with common bases:

   • $125 = 5^3$
   • So the expression becomes: $\frac{3^{-5} \times 10^{-5} \times 5^3}{5^{-7} \times 6^{-5}}$

2. Combine the powers of 5 in the denominator:

   • $5^{-7}$ can be moved to the numerator as $5^{7}$
   • The expression becomes: $\frac{3^{-5} \times 10^{-5} \times 5^3 \times 5^7}{6^{-5}}$

3. Combine the powers of 5:

   • $5^3 \times 5^7 = 5^{3+7} = 5^{10}$
   • Now the expression is: $\frac{3^{-5} \times 10^{-5} \times 5^{10}}{6^{-5}}$

4. Express $10^{-5}$ in terms of its prime factors:

   • $10^{-5} = (2 \times 5)^{-5} = 2^{-5} \times 5^{-5}$
   • The expression becomes: $\frac{3^{-5} \times 2^{-5} \times 5^{-5} \times 5^{10}}{6^{-5}}$

5. Combine the powers of 5:

   • $5^{-5} \times 5^{10} = 5^{-5+10} = 5^5$
   • The expression is now: $\frac{3^{-5} \times 2^{-5} \times 5^5}{6^{-5}}$

6. Express $6^{-5}$ in terms of its prime factors:

   • $6^{-5} = (2 \times 3)^{-5} = 2^{-5} \times 3^{-5}$
   • The expression becomes: $\frac{3^{-5} \times 2^{-5} \times 5^5}{2^{-5} \times 3^{-5}}$

7. Cancel out common terms:

   • $3^{-5}$ in the numerator and denominator cancel out.
   • $2^{-5}$ in the numerator and denominator cancel out.
   • This leaves us with: $5^5$

Therefore, the simplified form of the expression is:

$5^5 = 3125$

Would you like more details or have any questions about the steps? Here are 5 more questions you might want to ask:

1. How do you simplify expressions with exponents?
2. What are the properties of exponents?
3. Can you explain the steps to convert to common bases?
4. How do you handle division of powers?
5. Can you solve another exponentiation problem?

Tip: When simplifying expressions with exponents, always look for common bases and apply the properties of exponents to combine them efficiently.