if 2x-3y=10, what is the value of `(4^(x))/(8^(y))`?

Options

A. `((1)/(2))^(10)`
B. `2^(10)`
C. `4^(6)`
D. `8^(5)`

Answer

To solve the equation \( \frac{4^x}{8^y} \) given that \( 2x - 3y = 10 \), we can follow these steps:

### Step 1: Rewrite the bases in terms of powers of 2
We know that:
- \( 4 = 2^2 \)
- \( 8 = 2^3 \)

Thus, we can rewrite \( 4^x \) and \( 8^y \) as:
\[ 4^x = (2^2)^x = 2^{2x} \]
\[ 8^y = (2^3)^y = 2^{3y} \]

### Step 2: Substitute into the expression
Now, substituting these into our original expression, we have:
\[ \frac{4^x}{8^y} = \frac{2^{2x}}{2^{3y}} \]

### Step 3: Apply the property of exponents
Using the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify the expression:
\[ \frac{2^{2x}}{2^{3y}} = 2^{2x - 3y} \]

### Step 4: Substitute the known value
From the problem, we know that \( 2x - 3y = 10 \). Therefore, we can substitute this value into our expression:
\[ 2^{2x - 3y} = 2^{10} \]

### Step 5: Final result
Thus, the value of \( \frac{4^x}{8^y} \) is:
\[ 2^{10} \]

### Conclusion
The final answer is \( 2^{10} \).

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