To determine which of the given expressions is equivalent to \(3^{x+2}\), let's evaluate each option step-by-step.
Starting with the expression \(3^{x+2}\), we recognize that we can expand this as follows:
[tex]\[ 3^{x+2} = 3^x \cdot 3^2 \][/tex]
Now, let's evaluate each option:
1. Option 1: \(\frac{3^x}{6}\)
[tex]\[
\frac{3^x}{6}
\][/tex]
Here, we are dividing \(3^x\) by 6. To understand this better, note that \(3^{x+2}\) can be expanded as \(9 \cdot 3^x\):
[tex]\[
3^{x+2} = 3^2 \cdot 3^x = 9 \cdot 3^x
\][/tex]
Comparing \(\frac{3^x}{6}\) and \(9 \cdot 3^x\), it's evident they are different. Therefore, this option is not equivalent to \(3^{x+2}\).
2. Option 2: \(\frac{3^x}{9}\)
[tex]\[
\frac{3^x}{9} = \frac{3^x}{3^2} = 3^{x-2}
\][/tex]
Comparing \(3^{x-2}\) with \(3^{x+2}\), clearly, they are not the same. Therefore, this option is not equivalent to \(3^{x+2}\).
3. Option 3: \(6 \cdot 3^x\)
[tex]\[
6 \cdot 3^x
\][/tex]
Here, we are multiplying \(3^x\) by 6. But \(3^{x+2} = 9 \cdot 3^x\):
[tex]\[
6 \cdot 3^x \neq 9 \cdot 3^x
\][/tex]
So, this option is not equivalent to \(3^{x+2}\).
4. Option 4: \(9 \cdot 3^x\)
[tex]\[
9 \cdot 3^x = 3^2 \cdot 3^x = 3^{x+2}
\][/tex]
This matches perfectly with our expanded form of \(3^{x+2}\). Therefore, this option is equivalent to \(3^{x+2}\).
Based on the evaluations, we conclude that the expression equivalent to \(3^{x+2}\) is:
[tex]\[
\boxed{9 \cdot 3^x}
\][/tex]
