If `3^(9)+3^(12)+3^(15)+3^(n)` is a perfect cube, `n in N`,then the value of n is

Options

A. 18
B. 17
C. 14
D. 16

Answer

To solve the problem, we need to determine the value of \( n \) such that the expression \( 3^9 + 3^{12} + 3^{15} + 3^n \) is a perfect cube, where \( n \) is a natural number.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ 3^9 + 3^{12} + 3^{15} + 3^n \]
We can factor out the smallest power of 3, which is \( 3^9 \):
\[ 3^9(1 + 3^3 + 3^6 + 3^{n-9}) \]

2. **Simplify the Expression Inside the Parentheses**:
Next, we simplify the terms inside the parentheses:
\[ 1 + 3^3 + 3^6 = 1 + 27 + 729 = 757 \]
So, we can rewrite our expression as:
\[ 3^9(757 + 3^{n-9}) \]

3. **Set the Expression to be a Perfect Cube**:
For the entire expression \( 3^9(757 + 3^{n-9}) \) to be a perfect cube, both \( 3^9 \) and \( 757 + 3^{n-9} \) must be perfect cubes. Since \( 3^9 = (3^3)^3 \), it is already a perfect cube. Therefore, we need to ensure that \( 757 + 3^{n-9} \) is also a perfect cube.

4. **Let \( 757 + 3^{n-9} = k^3 \)**:
We can set up the equation:
\[ 3^{n-9} = k^3 - 757 \]
where \( k \) is some integer.

5. **Find Possible Values of \( k \)**:
We need \( k^3 \) to be greater than 757. The smallest integer \( k \) that satisfies this is \( k = 10 \) because \( 10^3 = 1000 \).

6. **Calculate \( n \) for \( k = 10 \)**:
Plugging \( k = 10 \) into the equation gives:
\[ 3^{n-9} = 1000 - 757 = 243 \]
Since \( 243 = 3^5 \), we have:
\[ n - 9 = 5 \]
Therefore:
\[ n = 14 \]

7. **Conclusion**:
The value of \( n \) that makes the expression a perfect cube is:
\[ \boxed{14} \]