Get comprehensive solutions to your questions with the help of IDNLearn.com's experts. Discover prompt and accurate answers from our experts, ensuring you get the information you need quickly.

Which of the following expressions is equivalent to [tex]$3^{x+2}$[/tex]?

A. [tex]\frac{3^x}{6}[/tex]
B. [tex]\frac{3^x}{9}[/tex]
C. [tex]6(3)^x[/tex]
D. [tex]9(3)^x[/tex]


Sagot :

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]