Find the value of `[2xx 3^(n+4) - 9xx 3^(n)]/3^(n+2)`.

Options

A. 18
B. 3
C. 1
D. 17

Answer

To solve the expression \(\frac{2 \cdot 3^{n+4} - 9 \cdot 3^n}{3^{n+2}}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{2 \cdot 3^{n+4} - 9 \cdot 3^n}{3^{n+2}} \]

### Step 2: Simplify the numerator
We can factor out \(3^n\) from the numerator:
\[ = \frac{3^n(2 \cdot 3^4 - 9)}{3^{n+2}} \]

### Step 3: Calculate \(3^4\)
Now, calculate \(3^4\):
\[ 3^4 = 81 \]
So, we can substitute this value into the expression:
\[ = \frac{3^n(2 \cdot 81 - 9)}{3^{n+2}} \]

### Step 4: Simplify inside the parentheses
Now simplify inside the parentheses:
\[ 2 \cdot 81 = 162 \]
Thus, we have:
\[ = \frac{3^n(162 - 9)}{3^{n+2}} \]
Calculating \(162 - 9\):
\[ 162 - 9 = 153 \]
So, we can rewrite the expression as:
\[ = \frac{3^n \cdot 153}{3^{n+2}} \]

### Step 5: Simplify the fraction
Now, we can simplify the fraction by using the property of exponents:
\[ = 153 \cdot \frac{3^n}{3^{n+2}} = 153 \cdot 3^{n - (n+2)} = 153 \cdot 3^{-2} \]

### Step 6: Calculate \(3^{-2}\)
Calculating \(3^{-2}\):
\[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \]
So, we have:
\[ = 153 \cdot \frac{1}{9} \]

### Step 7: Final calculation
Now, we can calculate:
\[ = \frac{153}{9} = 17 \]

Thus, the final value of the expression is:
\[ \boxed{17} \]