Write the set B={3,9,27,81} in set builder form.

Answer

To write the set \( B = \{3, 9, 27, 81\} \) in set builder form, we will follow these steps:

### Step 1: Identify the elements of the set
The elements of the set \( B \) are \( 3, 9, 27, \) and \( 81 \).

### Step 2: Look for a pattern
We can observe that each element in the set can be expressed as a power of \( 3 \):
- \( 3 = 3^1 \)
- \( 9 = 3^2 \)
- \( 27 = 3^3 \)
- \( 81 = 3^4 \)

### Step 3: Generalize the pattern
From the observations, we can generalize the elements of the set as \( 3^n \), where \( n \) is a natural number.

### Step 4: Define the range of \( n \)
The values of \( n \) that correspond to the elements in the set \( B \) are \( 1, 2, 3, \) and \( 4 \). Thus, \( n \) can take values from \( 1 \) to \( 4 \).

### Step 5: Write the set builder notation
Now we can write the set \( B \) in set builder form:
\[ B = \{ x \mid x = 3^n, n \in \mathbb{N}, 1 \leq n \leq 4 \} \]

### Final Answer
The set \( B \) in set builder form is:
\[ B = \{ x \mid x = 3^n, n \in \mathbb{N}, 1 \leq n \leq 4 \} \]

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