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The given expression is:

\[
\frac{3^{-10} \cdot 9^8}{(-3)^2} - 9^{-6} \cdot 27^5
\]

Let’s break this down step by step to solve it.

### Step 1: Simplifying each part
#### 1. \( 3^{-10} \cdot 9^8 \)
We know that \( 9 = 3^2 \), so:

\[
9^8 = (3^2)^8 = 3^{16}
\]

Now, the first part of the numerator becomes:

\[
3^{-10} \cdot 3^{16} = 3^{(-10 + 16)} = 3^6
\]

#### 2. \( (-3)^2 \)
This is simply:

\[
(-3)^2 = 9
\]

So, the first term is:

\[
\frac{3^6}{9}
\]

Since \( 9 = 3^2 \), we can simplify further:

\[
\frac{3^6}{3^2} = 3^{6-2} = 3^4
\]

#### 3. \( 9^{-6} \cdot 27^5 \)
We know that \( 9 = 3^2 \) and \( 27 = 3^3 \), so:

\[
9^{-6} = (3^2)^{-6} = 3^{-12}
\]
\[
27^5 = (3^3)^5 = 3^{15}
\]

Thus, the second term becomes:

\[
3^{-12} \cdot 3^{15} = 3^{(-12 + 15)} = 3^3
\]

### Step 2: Combining the terms
Now, we have:

\[
3^4 - 3^3
\]

Factoring this expression gives:

\[
3^3(3 - 1) = 3^3 \cdot 2 = 27 \cdot 2 = 54
\]

### Final Answer:
The value of the expression is \( \boxed{54} \).

Would you like more details on any step or have any questions?

---

Here are 5 related questions:
1. How do we handle negative exponents like \( 3^{-10} \)?
2. Can you explain how to simplify expressions with powers of the same base?
3. What are the rules for multiplying and dividing exponential expressions?
4. Why does \( (-3)^2 = 9 \) and not \( -9 \)?
5. How can we factor expressions like \( 3^4 - 3^3 \)?

**Tip:** Always check for ways to rewrite numbers in terms of prime factors when simplifying exponents!

Simplify the expression: \(\frac{3^{-10} \cdot 9^8}{(-3)^2} - 9^{-6} \cdot 27^5\)

Learn how to simplify a complex expression involving exponents and powers, including steps like breaking down base 9 and base 27 into powers of 3, and applying exponent rules. This detailed solution covers exponentiation and simplification techniques, suitable for students in Grades 10-12.

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