Let's solve the given mathematical expression step-by-step.
1. First, evaluate the innermost exponentiation for [tex]\(\left(\left((5)^2\right)^3\right)^2\)[/tex]:
- Calculate [tex]\(5^2\)[/tex]:
[tex]\[5^2 = 25\][/tex]
- Next, raise 25 to the power of 3:
[tex]\[25^3 = 15625\][/tex]
- Then, raise 15625 to the power of 2:
[tex]\[15625^2 = 244140625\][/tex]
2. Next, evaluate [tex]\(\left((-5)^5\right)^2\)[/tex]:
- Calculate [tex]\((-5)^5\)[/tex]:
[tex]\[(-5)^5 = -3125\][/tex]
- Then, raise [tex]\(-3125\)[/tex] to the power of 2:
[tex]\[(-3125)^2 = 9765625\][/tex]
3. Finally, evaluate [tex]\((-5)^2\)[/tex]:
- Calculate [tex]\((-5)^2\)[/tex]:
[tex]\[(-5)^2 = 25\][/tex]
4. Now, perform the division of these results as per the given expression:
[tex]\[
\frac{\left(\left((5)^2\right)^3\right)^2}{\left((-5)^5\right)^2} \div (-5)^2
= \frac{244140625}{9765625} \div 25
\][/tex]
5. Perform the first division:
[tex]\[
\frac{244140625}{9765625} = 25
\][/tex]
6. Perform the second division:
[tex]\[
25 \div 25 = 1
\][/tex]
Therefore, the final result of the given expression is:
[tex]\[
\boxed{1}
\][/tex]
