Let us simplify the given mathematical expression step by step:

The expression is:

$\frac{\left(-1 \times \left((-5)^2\right)^{-3}\right)}{\left(\left(5^2\right)^{-3} : (-1)^{10}\right)}$

Step 1: Simplify the powers and terms

1. $(-5)^2 = 25$, so: $\left((-5)^2\right)^{-3} = 25^{-3} = \frac{1}{25^3} = \frac{1}{15625}$

2. $(5^2)^{-3} = 25^{-3} = \frac{1}{25^3} = \frac{1}{15625}$

3. $(-1)^{10} = 1$, because any even power of $-1$ equals $1$.

Step 2: Rewrite the fraction

The numerator becomes: $-1 \times \frac{1}{15625} = -\frac{1}{15625}.$

The denominator becomes: $\frac{1}{15625} : 1 = \frac{1}{15625}.$

Thus, the entire fraction is: $\frac{-\frac{1}{15625}}{\frac{1}{15625}}.$

Step 3: Simplify the fraction

Dividing two fractions: $\frac{-\frac{1}{15625}}{\frac{1}{15625}} = -1.$

Final Answer:

$\boxed{-1}$

Would you like a more detailed explanation or additional examples?

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Here are 5 related questions:

1. What is the result of $(-2)^4 : (-2)^2$?
2. How does the parity of an exponent affect the sign of $(-x)^n$?
3. Can you explain the general rule for dividing powers with the same base?
4. What happens when you divide two fractions?
5. What is the value of $(3^{-2})^{-3}$?

Tip: When dealing with negative bases and powers, always simplify the powers first, as signs can influence the outcome!