If `3^(10)xx27^(2)=9^(2)xx3^(n)` then the value of n is

Options

A. 10
B. 12
C. 15
D. 30

Answer

To solve the equation \(3^{10} \times 27^{2} = 9^{2} \times 3^{n}\), we will express all terms in the base of 3.

### Step-by-Step Solution:

1. **Rewrite the Bases**:
- We know that \(27\) can be expressed as \(3^3\) and \(9\) can be expressed as \(3^2\).
- Therefore, we rewrite the equation:
\[ 3^{10} \times (3^3)^{2} = (3^2)^{2} \times 3^{n} \]

2. **Apply the Power of a Power Rule**:
- Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we simplify:
\[ 3^{10} \times 3^{6} = 3^{4} \times 3^{n} \]

3. **Combine the Exponents on the Left Side**:
- On the left side, we can add the exponents since the bases are the same:
\[ 3^{10 + 6} = 3^{4} \times 3^{n} \]
- This simplifies to:
\[ 3^{16} = 3^{4} \times 3^{n} \]

4. **Combine the Exponents on the Right Side**:
- On the right side, we can also add the exponents:
\[ 3^{16} = 3^{4 + n} \]

5. **Set the Exponents Equal**:
- Since the bases are the same, we can set the exponents equal to each other:
\[ 16 = 4 + n \]

6. **Solve for \(n\)**:
- To find \(n\), we subtract 4 from both sides:
\[ n = 16 - 4 = 12 \]

Thus, the value of \(n\) is \(12\).