Let's break down the given expression step by step. We will process each part separately and then combine results together for the final answer.
The given expression is as follows:
[tex]\[
\left(3^2 \cdot 3^4 \cdot 2^6\right) \cdot \left(3^5 \cdot 24 \cdot 2 \cdot 2\right) \cdot \left(2^5 \cdot 5^7 \cdot 5\right)
\][/tex]
### Step 1: Evaluate Each Parenthesis Separately
First Part:
[tex]\[
3^2 \cdot 3^4 \cdot 2^6
\][/tex]
First, let's handle the exponents of 3:
[tex]\[
3^2 \cdot 3^4 = 3^{2+4} = 3^6
\][/tex]
Now, multiply by [tex]\(2^6\)[/tex]:
[tex]\[
3^6 \cdot 2^6
\][/tex]
Evaluating, we get:
[tex]\[
3^6 = 729
\][/tex]
[tex]\[
2^6 = 64
\][/tex]
Thus,
[tex]\[
3^6 \cdot 2^6 = 729 \cdot 64 = 46656
\][/tex]
Second Part:
[tex]\[
3^5 \cdot 24 \cdot 2 \cdot 2
\][/tex]
Group and simplify:
[tex]\[
3^5 \cdot 24 \cdot 2^2
\][/tex]
Evaluate the powers and product:
[tex]\[
3^5 = 243
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
Thus,
[tex]\[
243 \cdot 24 \cdot 4 = 243 \cdot 96 = 23328
\][/tex]
Third Part:
[tex]\[
2^5 \cdot 5^7 \cdot 5
\][/tex]
Simplify the exponents of 5:
[tex]\[
5^7 \cdot 5 = 5^{7+1} = 5^8
\][/tex]
Combine with [tex]\(2^5\)[/tex]:
[tex]\[
2^5 \cdot 5^8
\][/tex]
Evaluate the powers:
[tex]\[
2^5 = 32
\][/tex]
[tex]\[
5^8 = 390625
\][/tex]
Thus,
[tex]\[
2^5 \cdot 5^8 = 32 \cdot 390625 = 12500000
\][/tex]
### Step 2: Combine Intermediate Results
We have the three intermediate results:
[tex]\[
46656, 23328, \text{ and } 12500000
\][/tex]
Multiply these results together for the final product:
[tex]\[
46656 \cdot 23328 \cdot 12500000 = 13604889600000000
\][/tex]
### Final Answer
[tex]\[
\left(3^2 \cdot 3^4 \cdot 2^6\right) \cdot \left(3^5 \cdot 24 \cdot 2 \cdot 2\right) \cdot \left(2^5 \cdot 5^7 \cdot 5\right) = 13604889600000000
\][/tex]
So, the final product of the entire expression is:
[tex]\[
\boxed{13604889600000000}
\][/tex]
