If `(sqrt(3))^(5) xx 81 = 3^(n) xx 3sqrt(3)` then the value of n is

Options

A. 4
B. 5
C. 6
D. None of these

Answer

To solve the equation \( (\sqrt{3})^5 \times 81 = 3^n \times 3\sqrt{3} \), we will express everything in terms of base 3.

### Step-by-step Solution:

1. **Rewrite \(\sqrt{3}\) and \(81\) in terms of base \(3\)**:
\[ \sqrt{3} = 3^{1/2} \]
Therefore,
\[ (\sqrt{3})^5 = (3^{1/2})^5 = 3^{(1/2) \times 5} = 3^{5/2} \]
Also, \(81\) can be expressed as:
\[ 81 = 3^4 \]

2. **Substitute back into the equation**:
The left-hand side becomes:
\[ 3^{5/2} \times 3^4 \]

3. **Combine the exponents on the left-hand side**:
Using the property of exponents \(a^m \times a^n = a^{m+n}\):
\[ 3^{5/2 + 4} = 3^{5/2 + 8/2} = 3^{(5 + 8)/2} = 3^{13/2} \]

4. **Rewrite the right-hand side**:
The right-hand side is:
\[ 3^n \times 3\sqrt{3} = 3^n \times 3^{1} \times 3^{1/2} = 3^{n + 1 + 1/2} = 3^{n + 2/2 + 1/2} = 3^{n + 3/2} \]

5. **Set the exponents equal to each other**:
Since the bases are the same, we can equate the exponents:
\[ \frac{13}{2} = n + \frac{3}{2} \]

6. **Solve for \(n\)**:
Subtract \(\frac{3}{2}\) from both sides:
\[ \frac{13}{2} - \frac{3}{2} = n \]
\[ \frac{10}{2} = n \]
\[ n = 5 \]

### Final Answer:
The value of \(n\) is \(5\).