Alright, let's simplify the given expression step by step.
The original expression is:
[tex]\[
\frac{a^2 \left(3^{a-8} \cdot 3^{a+9}\right)}{3^{2a} \cdot a}
\][/tex]
### Step 1: Simplify the Expression inside the Parentheses
First, we need to simplify the term inside the parentheses, [tex]\(3^{a-8} \cdot 3^{a+9}\)[/tex].
Recall the property of exponents: [tex]\(b^m \cdot b^n = b^{m+n}\)[/tex]. So,
[tex]\[
3^{a-8} \cdot 3^{a+9} = 3^{(a-8) + (a+9)}
\][/tex]
Now, combine the exponents:
[tex]\[
(a-8) + (a+9) = a + a - 8 + 9 = 2a + 1
\][/tex]
Thus,
[tex]\[
3^{a-8} \cdot 3^{a+9} = 3^{2a+1}
\][/tex]
### Step 2: Substitute the Simplified Expression back
Now we substitute [tex]\( 3^{2a+1} \)[/tex] back into the original fraction:
[tex]\[
\frac{a^2 \cdot 3^{2a+1}}{3^{2a} \cdot a}
\][/tex]
### Step 3: Simplify the Overall Fraction
Next, we simplify the overall fraction [tex]\(\frac{a^2 \cdot 3^{2a+1}}{3^{2a} \cdot a}\)[/tex].
We can split the fraction into two parts for easier simplification:
[tex]\[
\frac{a^2}{a} \cdot \frac{3^{2a+1}}{3^{2a}}
\][/tex]
#### Simplify [tex]\(\frac{a^2}{a}\)[/tex]
The term [tex]\(\frac{a^2}{a}\)[/tex] simplifies to:
[tex]\[
\frac{a^2}{a} = a
\][/tex]
#### Simplify [tex]\(\frac{3^{2a+1}}{3^{2a}}\)[/tex]
Using the properties of exponents again, [tex]\( \frac{b^m}{b^n} = b^{m-n} \)[/tex]:
[tex]\[
\frac{3^{2a+1}}{3^{2a}} = 3^{(2a+1) - 2a} = 3^1 = 3
\][/tex]
### Step 4: Combine the Results
Now we combine the simplified terms:
[tex]\[
a \cdot 3 = 3a
\][/tex]
### Final Answer
Therefore, the simplified form of the original expression is:
[tex]\[
\boxed{3a}
\][/tex]
