Certainly! Let's simplify the given expression step-by-step:
Given expression:
[tex]\[
\frac{3^{2x+1}}{9^{x+1}}
\][/tex]
Step 1: Rewrite the base of the denominator using powers of 3. Note that \(9\) can be expressed as \(3^2\):
[tex]\[
9 = 3^2
\][/tex]
So, we can substitute \(9\) with \((3^2)\):
[tex]\[
\frac{3^{2x+1}}{(3^2)^{x+1}}
\][/tex]
Step 2: Simplify the exponent in the denominator:
[tex]\[
(3^2)^{x+1} = 3^{2(x+1)} = 3^{2x+2}
\][/tex]
Step 3: Substitute back into the expression:
[tex]\[
\frac{3^{2x+1}}{3^{2x+2}}
\][/tex]
Step 4: Simplify the exponents. When dividing exponents with the same base, subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
\frac{3^{2x+1}}{3^{2x+2}} = 3^{(2x+1)-(2x+2)} = 3^{2x+1-2x-2} = 3^{-1}
\][/tex]
Step 5: Simplify the exponent:
[tex]\[
3^{-1} = \frac{1}{3}
\][/tex]
So, the simplified result of the expression is:
[tex]\[
\frac{1}{3}
\][/tex]
Since none of the given choices match this result, it appears there was a mistake in the list of possible answers. Thus, we now have the final result:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
