Sure, let's simplify the given expression step-by-step:
We are given the expression:
[tex]$\frac{\left(6 a^a\right)^8 \times 6 a^5 \times \left(a^a\right)^{-4}}{\left(a^9\right)^3}$[/tex]
First, let's simplify each part separately.
1. Simplify [tex]\(\left(6 a^a\right)^8\)[/tex]:
[tex]\[
\left(6 a^a\right)^8 = 6^8 \cdot \left(a^a\right)^8
\][/tex]
2. Simplify [tex]\(6 a^5\)[/tex]:
[tex]\[
6 a^5
\][/tex]
3. Simplify [tex]\(\left(a^a\right)^{-4}\)[/tex]:
[tex]\[
\left(a^a\right)^{-4} = \frac{1}{\left(a^a\right)^4}
\][/tex]
4. Simplify [tex]\(\left(a^9\right)^3\)[/tex]:
[tex]\[
\left(a^9\right)^3 = a^{27}
\][/tex]
Combine the simplified parts in the numerator and denominator:
Numerator:
[tex]\[
6^8 \cdot \left(a^a\right)^8 \cdot 6 \cdot a^5 \cdot \frac{1}{\left(a^a\right)^4} = 6^8 \cdot 6 \cdot a^5 \cdot \frac{\left(a^a\right)^8}{\left(a^a\right)^4}
\][/tex]
Simplify within the numerator:
[tex]\[
6^8 \cdot 6 \cdot a^5 \cdot \left(a^a\right)^{8-4} = 6^9 \cdot a^5 \cdot a^a^4
\][/tex]
Denominator:
[tex]\[
a^{27}
\][/tex]
Now we can combine the numerator and the denominator:
[tex]\[
\frac{6^9 \cdot a^5 \cdot a^a^4}{a^{27}}
\][/tex]
Next, simplify the fraction by combining the [tex]\(a\)[/tex] terms:
[tex]\[
6^9 \cdot \frac{a^5 \cdot a^{4a}}{a^{27}} = 6^9 \cdot a^{5 + 4a - 27} = 6^9 \cdot a^{4a - 22}
\][/tex]
Now simply evaluate [tex]\(6^9\)[/tex]:
[tex]\[
6^9 = 10077696
\][/tex]
Therefore, the simplified expression is:
[tex]\[
10077696 \cdot a^{4a - 22}
\][/tex]
So, the final result is:
[tex]\[
\boxed{10077696 \cdot a^{4a - 22}}
\][/tex]
