If `x - 27 " and " y = "log"_(3) 4, "then" x^(y)` equals

Options

A. 64
B. 16
C. `(3)/(7)`
D. `(1)/(16)`

Explanation

To solve the problem where \( x = 27 \) and \( y = \log_3 4 \), we need to find \( x^y \).

### Step-by-Step Solution:

1. **Identify the values of x and y**:
\[ x = 27, \quad y = \log_3 4 \]

2. **Express \( x^y \)**:
\[ x^y = 27^{\log_3 4} \]

3. **Rewrite 27 in terms of base 3**:
\[ 27 = 3^3 \]
Therefore,
\[ x^y = (3^3)^{\log_3 4} \]

4. **Use the power of a power property**:
\[ (a^m)^n = a^{m \cdot n} \]
Applying this property, we get:
\[ x^y = 3^{3 \cdot \log_3 4} \]

5. **Use the property of logarithms**:
We know that \( a^{\log_a b} = b \). Thus,
\[ 3^{3 \cdot \log_3 4} = 4^3 \]

6. **Calculate \( 4^3 \)**:
\[ 4^3 = 64 \]

### Final Answer:
\[ x^y = 64 \]