If `log_(2) x = a and log_(3) y = a`, write `72^(@)` in terms of x and y.

Answer

To solve the problem, we need to express \( 72^a \) in terms of \( x \) and \( y \), given that \( \log_2 x = a \) and \( \log_3 y = a \).

### Step-by-Step Solution:

1. **Start with the expression**: We need to express \( 72^a \).

\[ 72^a \]

**Hint**: Identify the prime factorization of 72 to simplify the expression.

2. **Factor 72**: The prime factorization of 72 is \( 72 = 8 \times 9 = 2^3 \times 3^2 \).

\[ 72 = 2^3 \times 3^2 \]

**Hint**: Break down 72 into its prime factors to make it easier to work with.

3. **Raise the factors to the power of \( a \)**: Now we can write \( 72^a \) as:

\[ 72^a = (2^3 \times 3^2)^a \]

**Hint**: Use the property of exponents that states \( (ab)^n = a^n \times b^n \).

4. **Distribute the exponent**: Apply the exponent to both factors:

\[ 72^a = (2^3)^a \times (3^2)^a = 2^{3a} \times 3^{2a} \]

**Hint**: Remember that when you raise a power to another power, you multiply the exponents.

5. **Express \( 2^{3a} \) and \( 3^{2a} \) in terms of \( x \) and \( y \)**: We know from the problem that:

- \( \log_2 x = a \) implies \( x = 2^a \) or \( 2^a = x \)
- \( \log_3 y = a \) implies \( y = 3^a \) or \( 3^a = y \)

Therefore, we can express \( 2^{3a} \) and \( 3^{2a} \) as follows:

\[ 2^{3a} = (2^a)^3 = x^3 \]
\[ 3^{2a} = (3^a)^2 = y^2 \]

**Hint**: Use the relationships between \( x, y \) and the logarithmic definitions to substitute.

6. **Combine the results**: Now substitute back into the expression for \( 72^a \):

\[ 72^a = x^3 \times y^2 \]

**Final Answer**:

\[ 72^a = x^3 y^2 \]

### Summary of Steps:
1. Start with \( 72^a \).
2. Factor 72 into \( 2^3 \times 3^2 \).
3. Raise each factor to the power of \( a \).
4. Express \( 2^{3a} \) and \( 3^{2a} \) in terms of \( x \) and \( y \).
5. Combine the results to get \( 72^a = x^3 y^2 \).