`[3^(3)+3^(2)+3^(-2)+3^(-3)]` is equal to

Options

A. 0
B. `36+(1)/(36)`
C. `(976)/(27)`
D. `3^(5)+3^(-5)`

Answer

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

### Step 1: Calculate each term
First, we calculate the values of each term in the expression:

1. \( 3^{3} = 27 \)
2. \( 3^{2} = 9 \)
3. \( 3^{-2} = \frac{1}{3^{2}} = \frac{1}{9} \)
4. \( 3^{-3} = \frac{1}{3^{3}} = \frac{1}{27} \)

### Step 2: Write the expression with calculated values
Now, we can rewrite the expression with the calculated values:

\[ 27 + 9 + \frac{1}{9} + \frac{1}{27} \]

### Step 3: Combine the whole numbers
Next, we combine the whole numbers:

\[ 27 + 9 = 36 \]

So, we have:

\[ 36 + \frac{1}{9} + \frac{1}{27} \]

### Step 4: Find a common denominator for the fractions
To add the fractions \( \frac{1}{9} \) and \( \frac{1}{27} \), we need a common denominator. The least common multiple (LCM) of 9 and 27 is 27.

Now, we convert \( \frac{1}{9} \) to have a denominator of 27:

\[ \frac{1}{9} = \frac{3}{27} \]

### Step 5: Add the fractions
Now we can add the fractions:

\[ \frac{3}{27} + \frac{1}{27} = \frac{3 + 1}{27} = \frac{4}{27} \]

### Step 6: Combine with the whole number
Now, we combine this with the whole number:

\[ 36 + \frac{4}{27} \]

### Step 7: Write the final answer
To express this as a single fraction, we can convert 36 into a fraction with a denominator of 27:

\[ 36 = \frac{36 \times 27}{27} = \frac{972}{27} \]

Now, we add the two fractions:

\[ \frac{972}{27} + \frac{4}{27} = \frac{972 + 4}{27} = \frac{976}{27} \]

Thus, the final answer is:

\[ \frac{976}{27} \]