Answer :
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]
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]
