Sure! Let's simplify the expression step by step:
Given expression:
[tex]\[
\frac{3^{2x} \cdot 27^{x-1}}{9^{2x+2}}
\][/tex]
### Step 1: Simplify each term using the properties of exponents.
Simplify [tex]\(27^{x-1}\)[/tex]:
Recall that [tex]\(27\)[/tex] can be written as [tex]\(3^3\)[/tex]. Hence,
[tex]\[
27^{x-1} = (3^3)^{x-1}
\][/tex]
Using the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex],
[tex]\[
(3^3)^{x-1} = 3^{3(x-1)}
\][/tex]
[tex]\[
3^{3(x-1)} = 3^{3x - 3}
\][/tex]
Simplify [tex]\(9^{2x+2}\)[/tex]:
Recall that [tex]\(9\)[/tex] can be written as [tex]\(3^2\)[/tex]. Hence,
[tex]\[
9^{2x+2} = (3^2)^{2x+2}
\][/tex]
Using the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex],
[tex]\[
(3^2)^{2x+2} = 3^{2(2x+2)}
\][/tex]
[tex]\[
3^{2(2x+2)} = 3^{4x + 4}
\][/tex]
Now we can substitute these simplified forms into the original expression.
### Step 2: Substitute simplified forms back into the expression.
[tex]\[
\frac{3^{2x} \cdot 3^{3x - 3}}{3^{4x + 4}}
\][/tex]
### Step 3: Combine exponents in the numerator.
Using the product of powers property [tex]\(a^m \cdot a^n = a^{m + n}\)[/tex],
[tex]\[
3^{2x} \cdot 3^{3x - 3} = 3^{2x + 3x - 3} = 3^{5x - 3}
\][/tex]
So the expression now is:
[tex]\[
\frac{3^{5x - 3}}{3^{4x + 4}}
\][/tex]
### Step 4: Subtract the exponent in the denominator from the exponent in the numerator.
Using the quotient of powers property [tex]\(\frac{a^m}{a^n} = a^{m - n}\)[/tex],
[tex]\[
\frac{3^{5x - 3}}{3^{4x + 4}} = 3^{(5x - 3) - (4x + 4)}
\][/tex]
[tex]\[
3^{(5x - 3) - (4x + 4)} = 3^{5x - 3 - 4x - 4} = 3^{x - 7}
\][/tex]
### Final simplified form:
[tex]\[
\boxed{3^{x - 7}}
\][/tex]
