What is the value of `3^(1) + 3^(-1) + 3^(2) + 3^(-2)` ?

Options

A. `112//9`
B. `16//3`
C. `4//3`
D. `109//9`

Answer

To solve the expression \(3^{1} + 3^{-1} + 3^{2} + 3^{-2}\), we will break it down step by step.

### Step 1: Calculate each term individually
1. **Calculate \(3^{1}\)**:
\[ 3^{1} = 3 \]

2. **Calculate \(3^{-1}\)**:
\[ 3^{-1} = \frac{1}{3} \]

3. **Calculate \(3^{2}\)**:
\[ 3^{2} = 9 \]

4. **Calculate \(3^{-2}\)**:
\[ 3^{-2} = \frac{1}{3^{2}} = \frac{1}{9} \]

### Step 2: Combine the results
Now we can combine these results:
\[ 3 + \frac{1}{3} + 9 + \frac{1}{9} \]

### Step 3: Find a common denominator
To add these fractions, we will find a common denominator. The least common multiple (LCM) of \(1\), \(3\), and \(9\) is \(9\).

### Step 4: Convert each term to have the common denominator
1. Convert \(3\) to a fraction:
\[ 3 = \frac{27}{9} \]

2. Convert \(\frac{1}{3}\) to have a denominator of \(9\):
\[ \frac{1}{3} = \frac{3}{9} \]

3. \(9\) as a fraction:
\[ 9 = \frac{81}{9} \]

4. \(\frac{1}{9}\) is already in the correct form.

### Step 5: Add the fractions
Now we can add all the fractions:
\[ \frac{27}{9} + \frac{3}{9} + \frac{81}{9} + \frac{1}{9} = \frac{27 + 3 + 81 + 1}{9} = \frac{112}{9} \]

### Final Result
Thus, the value of \(3^{1} + 3^{-1} + 3^{2} + 3^{-2}\) is:
\[ \frac{112}{9} \]

---