The problem is to simplify the expression: $\frac{(10^2)^4 \times (5^3)^4}{(5^4)^2 \times (10^2)^5}$

1. Problem Description

The problem is to simplify the expression:

$\frac{(10^2)^4 \times (5^3)^4}{(5^4)^2 \times (10^2)^5}$

2. Solution Steps

First, we use the rule $(a^m)^n = a^{m \times n}$ to simplify the exponents.

$(10^2)^4 = 10^{2 \times 4} = 10^8$

$(5^3)^4 = 5^{3 \times 4} = 5^{12}$

$(5^4)^2 = 5^{4 \times 2} = 5^8$

$(10^2)^5 = 10^{2 \times 5} = 10^{10}$

So the expression becomes:

$\frac{10^8 \times 5^{12}}{5^8 \times 10^{10}}$

We can rewrite this as:

$\frac{10^8}{10^{10}} \times \frac{5^{12}}{5^8}$

Now we use the rule $\frac{a^m}{a^n} = a^{m-n}$ to simplify.

$\frac{10^8}{10^{10}} = 10^{8-10} = 10^{-2}$

$\frac{5^{12}}{5^8} = 5^{12-8} = 5^4$

So the expression simplifies to:

$10^{-2} \times 5^4$

Since $10 = 2 \times 5$, we can write $10^{-2} = (2 \times 5)^{-2} = 2^{-2} \times 5^{-2}$.

Then the expression becomes:

$2^{-2} \times 5^{-2} \times 5^4 = 2^{-2} \times 5^{4-2} = 2^{-2} \times 5^2 = \frac{1}{2^2} \times 5^2 = \frac{1}{4} \times 25 = \frac{25}{4}$

Alternatively, we can rewrite $10^{-2} \times 5^4 = \frac{5^4}{10^2} = \frac{5^4}{(2\times5)^2} = \frac{5^4}{2^2\times5^2} = \frac{5^{4-2}}{2^2} = \frac{5^2}{2^2} = \frac{25}{4}$.

3. Final Answer

$\frac{25}{4}$