The value of `(3^(502) - 3^(500) + 16)/(3^(500) + 2)` is

Options

A. 8
B. 16
C. 2
D. 4

Answer

To solve the expression \((3^{502} - 3^{500} + 16)/(3^{500} + 2)\), we can follow these steps:

### Step 1: Factor out \(3^{500}\)
We start with the expression:
\[ \frac{3^{502} - 3^{500} + 16}{3^{500} + 2} \]
Notice that \(3^{502} = 3^{500} \cdot 3^2\). So we can rewrite the numerator:
\[ 3^{502} - 3^{500} + 16 = 3^{500} \cdot 3^2 - 3^{500} + 16 = 3^{500}(9 - 1) + 16 \]
This simplifies to:
\[ 3^{500}(8) + 16 \]

### Step 2: Rewrite the numerator
Now we can rewrite the numerator as:
\[ 8 \cdot 3^{500} + 16 \]
Thus, the expression becomes:
\[ \frac{8 \cdot 3^{500} + 16}{3^{500} + 2} \]

### Step 3: Factor the numerator
We can factor out the common term in the numerator:
\[ 8 \cdot 3^{500} + 16 = 8(3^{500} + 2) \]
Now the expression is:
\[ \frac{8(3^{500} + 2)}{3^{500} + 2} \]

### Step 4: Cancel the common terms
Since \(3^{500} + 2\) is present in both the numerator and the denominator, we can cancel it out (assuming \(3^{500} + 2 \neq 0\)):
\[ 8 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{8} \]

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