If `3x-y=12,` what is the value of `(8^(x))/(2^(y))`?

Options

A. `2^(12)`
B. `4^(4)`
C. `8^(2)`
D. The value cannot be determined from the information given.

Answer

To solve the problem, we need to find the value of \(\frac{8^x}{2^y}\) given the equation \(3x - y = 12\).

### Step-by-Step Solution:

1. **Rewrite \(8^x\) in terms of base 2**:
\[ 8 = 2^3 \implies 8^x = (2^3)^x = 2^{3x} \]
**Hint**: Remember that \(a^m = (a^n)^k\) can be rewritten as \(a^{n \cdot k}\).

2. **Substitute \(8^x\) into the expression**:
\[ \frac{8^x}{2^y} = \frac{2^{3x}}{2^y} \]
**Hint**: When dividing powers with the same base, subtract the exponents.

3. **Apply the property of exponents**:
\[ \frac{2^{3x}}{2^y} = 2^{3x - y} \]
**Hint**: Use the property \(a^m / a^n = a^{m-n}\).

4. **Substitute the value of \(3x - y\)**:
Since we know from the problem statement that \(3x - y = 12\), we can substitute this into our expression:
\[ 2^{3x - y} = 2^{12} \]
**Hint**: Always look for given equations or relationships that can simplify your calculations.

5. **Final Result**:
Therefore, the value of \(\frac{8^x}{2^y}\) is:
\[ 2^{12} \]

### Summary:
The final answer is \(2^{12}\).