The value of `(3^(2-x) xx 9^(x- 2))/(3^x)` is equal to

Options

A. `1/3`
B. `1/24`
C. 1
D. `1/9`

Answer

To solve the expression \((3^{2-x} \cdot 9^{x-2}) / (3^x)\), we will follow these steps:

### Step 1: Rewrite \(9\) in terms of \(3\)
We know that \(9\) can be expressed as \(3^2\). Therefore, we can rewrite \(9^{x-2}\) as:
\[ 9^{x-2} = (3^2)^{x-2} = 3^{2(x-2)} = 3^{2x - 4} \]

### Step 2: Substitute back into the expression
Now we can substitute this back into the original expression:
\[ \frac{3^{2-x} \cdot 9^{x-2}}{3^x} = \frac{3^{2-x} \cdot 3^{2x-4}}{3^x} \]

### Step 3: Combine the powers of \(3\)
Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can combine the powers in the numerator:
\[ 3^{2-x + 2x - 4} = 3^{(2 - 4) + (2x - x)} = 3^{2x - 2} \]

### Step 4: Simplify the expression
Now we can rewrite the expression:
\[ \frac{3^{2x - 2}}{3^x} \]
Using the property of exponents again, we can simplify this:
\[ 3^{(2x - 2) - x} = 3^{2x - 2 - x} = 3^{x - 2} \]

### Step 5: Rewrite \(3^{x-2}\)
We can express \(3^{x-2}\) in a different form:
\[ 3^{x-2} = \frac{3^x}{3^2} = \frac{3^x}{9} \]

### Final Result
Thus, the value of the original expression is:
\[ \frac{1}{9} \cdot 3^x \]