\(\frac{5^{5}.}{5^{3}.}\frac{6^{3}.}{6.}\frac{8^{10}}{8^{9}}\)

To simplify the given expression:

\[
\frac{5^{5}}{5^{3}} \cdot \frac{6^{3}}{6} \cdot \frac{8^{10}}{8^{9}}
\]

we can use the property of exponents: 

\[
\frac{a^m}{a^n} = a^{m-n}.
\]

### Step-by-step simplification:

1. Simplify \(\frac{5^{5}}{5^{3}}\):

   \[
   \frac{5^{5}}{5^{3}} = 5^{5-3} = 5^{2}.
   \]

2. Simplify \(\frac{6^{3}}{6}\):

   \[
   \frac{6^{3}}{6} = \frac{6^{3}}{6^{1}} = 6^{3-1} = 6^{2}.
   \]

3. Simplify \(\frac{8^{10}}{8^{9}}\):

   \[
   \frac{8^{10}}{8^{9}} = 8^{10-9} = 8^{1}.
   \]

### Combining the simplified results:

\[
5^{2} \cdot 6^{2} \cdot 8^{1}.
\]

Now, compute the values:

- \( 5^{2} = 25 \)
- \( 6^{2} = 36 \)
- \( 8^{1} = 8 \)

Multiplying these:

\[
25 \cdot 36 \cdot 8.
\]

Calculating step-by-step:

1. \( 25 \cdot 36 = 900 \).
2. \( 900 \cdot 8 = 7200 \).

### Final Simplified Answer:

\[
\frac{5^{5}}{5^{3}} \cdot \frac{6^{3}}{6} \cdot \frac{8^{10}}{8^{9}} = 7200.
\]

---

Would you like to see more details or have any questions? Here are some related questions:

1. How do exponent rules apply when dividing powers with the same base?
2. What happens when multiplying exponents with different bases?
3. How can we further simplify products of powers when bases differ?
4. How do changes in exponents affect the overall value of an expression?
5. How do exponent rules relate to logarithmic functions?

**Tip:** Always simplify exponents first before multiplying for an efficient approach!

Learn how to simplify exponents using the division and multiplication rules. This problem involves dividing and multiplying powers with the same and different bases, covering key concepts in exponents for Grades 9-12.

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