If `3^(n)=n^(6), n^(18)=`

Options

A. `3^(n)n^(3)`
B. `3^(n)n^(12)`
C. `9^(n)`
D. `3^(12n)`

Explanation

To solve the equation \(3^n = n^6\) and find the value of \(n^{18}\), we can follow these steps:

### Step 1: Understand the given equation
We start with the equation:
\[ 3^n = n^6 \]
This tells us that \(n^6\) can be expressed as \(3^n\).

### Step 2: Express \(n^{18}\) in terms of \(n^6\)
We need to find \(n^{18}\). We can express \(n^{18}\) as:
\[ n^{18} = n^{6 + 12} = n^6 \cdot n^{12} \]

### Step 3: Substitute \(n^6\) with \(3^n\)
From the first equation, we know that \(n^6 = 3^n\). Therefore, we can substitute \(n^6\) in our expression for \(n^{18}\):
\[ n^{18} = n^6 \cdot n^{12} = 3^n \cdot n^{12} \]

### Step 4: Final expression for \(n^{18}\)
Thus, we have:
\[ n^{18} = 3^n \cdot n^{12} \]

### Conclusion
The value of \(n^{18}\) is:
\[ n^{18} = 3^n \cdot n^{12} \]