Find n if `(5^(3))^(4)=(5^(2))^(n)`

Answer

To solve the equation \( (5^3)^4 = (5^2)^n \), we will use the power of a power property of exponents, which states that \( (a^m)^n = a^{m \cdot n} \).

### Step-by-Step Solution:

1. **Apply the Power of a Power Property**:
\[ (5^3)^4 = 5^{3 \cdot 4} \]
This simplifies to:
\[ 5^{12} \]

2. **Apply the Power of a Power Property on the Right Side**:
\[ (5^2)^n = 5^{2 \cdot n} \]

3. **Set the Exponents Equal to Each Other**:
Since the bases are the same, we can set the exponents equal:
\[ 12 = 2n \]

4. **Solve for \( n \)**:
To find \( n \), divide both sides by 2:
\[ n = \frac{12}{2} = 6 \]

### Final Answer:
The value of \( n \) is \( 6 \).

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