Sure, let's solve this step-by-step.
The given expression is:
[tex]\[
\frac{3^7 \cdot 3^4}{(3^2)^4 \cdot 3^3}
\][/tex]
We start by simplifying both the numerator and the denominator separately.
### Step 1: Simplify the Numerator
The numerator is [tex]\(3^7 \cdot 3^4\)[/tex]. When multiplying two exponents with the same base, we add their exponents:
[tex]\[
3^7 \cdot 3^4 = 3^{7+4} = 3^{11}
\][/tex]
### Step 2: Simplify the Denominator
The denominator is [tex]\((3^2)^4 \cdot 3^3\)[/tex]. First, simplify [tex]\((3^2)^4\)[/tex]. When raising a power to another power, we multiply the exponents:
[tex]\[
(3^2)^4 = 3^{2 \cdot 4} = 3^8
\][/tex]
Now, we have:
[tex]\[
3^8 \cdot 3^3
\][/tex]
Again, multiply these exponents with the same base by adding their exponents:
[tex]\[
3^8 \cdot 3^3 = 3^{8 + 3} = 3^{11}
\][/tex]
### Step 3: Divide the Numerator by the Denominator
Now, we divide the combined numerator by the combined denominator:
[tex]\[
\frac{3^{11}}{3^{11}}
\][/tex]
When dividing exponents with the same base, we subtract the exponents:
[tex]\[
3^{11 - 11} = 3^0
\][/tex]
We know that any number raised to the power of 0 is 1:
[tex]\[
3^0 = 1
\][/tex]
Therefore, the simplified value of the given expression is:
[tex]\[
\boxed{1}
\][/tex]
