1 + 5^2 + 5^4 + 5^6 + 5^8/1 + 5 + 5^2 + 5^3 +...+ 5^8 + 5^9

To simplify the expression given, we will separate it into its components and solve each part.

1. **Numerator:**
The first part to evaluate is:

\[
1 + 5^2 + 5^4 + 5^6 + 5^8
\]

This can be identified as a geometric series with the first term \(a = 1\) and the common ratio \(r = 5^2\). The number of terms \(n\) is 5 (for the terms \(5^0\) through \(5^8\)).

The sum of a finite geometric series can be calculated using the formula:

\[
S_n = a \frac{r^n - 1}{r - 1}
\]

In this case:
- \(a = 1\)
- \(r = 25\) (since \(r = 5^2\))
- \(n = 5\)

Now, we can calculate:

\[
S = 1 \cdot \frac{25^5 - 1}{25 - 1} = \frac{25^5 - 1}{24}
\]

Calculating \(25^5\):

\[
25^5 = (5^2)^5 = 5^{10} = 9765625
\]

So,

\[
S = \frac{9765625 - 1}{24} = \frac{9765624}{24} = 406901
\]

2. **Denominator:**
Now for the second part:

The series is:

\[
1 + 5 + 5^2 + 5^3 + \ldots + 5^8
\]

This is again a geometric series with:
- \(a = 1\)
- \(r = 5\)
- \(n = 9\)

Thus, we use the formula:

\[
S = 1 \cdot \frac{5^9 - 1}{5 - 1} = \frac{5^9 - 1}{4}
\]

Calculating \(5^9\):

\[
5^9 = 1953125
\]

Thus,

\[
S = \frac{1953125 - 1}{4} = \frac{1953124}{4} = 488281
\]

3. **Combining both computations:**

The next part of the expression mentions some repeated terms \(5^2 + 5^4 + 5^6 + 5^8 \), however, it's better confirmed that they just repeat in sections without affecting the calculation.

The full expression we have is:

\[
\frac{1 + 5^2 + 5^4 + 5^6 + 5^8}{1 + 5 + 5^2 + 5^3 + 5^4 + 5^5 + 5^6 + 5^7 + 5^8} = \frac{406901}{488281}
\]

This fraction appears in a more standard form, thus the final answer can be left as is or computed in the simplest form using a calculator.

So, the final answer is:

\[
\frac{406901}{488281}
\]

This value is approximately equal to 0.833.