Let's break down and solve the given expression step by step. The expression is:

$\frac{\frac{1}{5^2} \times (-5)^6 \div 5^2}{\left(7^6 \times \frac{1}{7}\right)^0}$

Step-by-Step Solution:

1. Numerator: $\frac{1}{5^2} \times (-5)^6 \div 5^2$

   • $\frac{1}{5^2} = \frac{1}{25}$
   • $(-5)^6 = 15625$ (since any even power of a negative number is positive)
   • $5^2 = 25$

   Now calculate this: $\frac{1}{25} \times 15625 \div 25 = \frac{15625}{25} \div 25 = 625 \div 25 = 25$

2. Denominator: $\left(7^6 \times \frac{1}{7}\right)^0$

   • $7^6 \times \frac{1}{7} = 7^{6-1} = 7^5$
   • Anything raised to the power of 0 equals 1, so $\left(7^5\right)^0 = 1$.

3. Final Expression: $\frac{25}{1} = 25$

Final Answer:

$25$

Would you like further details or explanations on any step?

Here are five related questions to expand on this topic:

1. How do the properties of exponents apply when dividing two numbers with the same base?
2. Why does raising any number to the power of 0 result in 1?
3. How can you simplify expressions involving negative numbers raised to an even or odd power?
4. What are some common mistakes when working with fractions and exponents together?
5. How can you approach simplifying complex expressions like this more efficiently?

Tip: When simplifying expressions, always look for opportunities to cancel terms or apply exponent rules early on to reduce complexity!