If `sqrt(3^(n)) = 81`. Then, n is equal to:

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the equation \( \sqrt{3^n} = 81 \), we will follow these steps:

### Step 1: Square both sides
To eliminate the square root, we will square both sides of the equation.

\[ (\sqrt{3^n})^2 = 81^2 \]

### Step 2: Simplify the left side
When we square the left side, the square and the square root cancel each other out.

\[ 3^n = 81^2 \]

### Step 3: Calculate \( 81^2 \)
Next, we need to calculate \( 81^2 \). We know that \( 81 = 3^4 \), so:

\[ 81^2 = (3^4)^2 \]

Using the exponent rule \( (a^m)^n = a^{m \cdot n} \):

\[ (3^4)^2 = 3^{4 \cdot 2} = 3^8 \]

### Step 4: Set the exponents equal
Now we have:

\[ 3^n = 3^8 \]

Since the bases are the same, we can set the exponents equal to each other:

\[ n = 8 \]

### Final Answer
Thus, the value of \( n \) is:

\[ \boxed{8} \]

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