If `8^(x) = (4)/(2^(x)) ` then find x
A)1
B)2
C)`(1)/(2)`
D)`(2)/(3)`

Options

A. 1
B. 2
C. `(1)/(2)`
D. `(2)/(3)`

Answer

To solve the equation \( 8^{x} = \frac{4}{2^{x}} \), we will follow these steps:

### Step 1: Rewrite the bases
We know that \( 8 \) can be expressed as \( 2^3 \) and \( 4 \) can be expressed as \( 2^2 \). Therefore, we can rewrite the equation as:
\[ (2^3)^{x} = \frac{2^2}{2^{x}} \]

### Step 2: Simplify the left side
Using the power of a power property, we can simplify the left side:
\[ 2^{3x} = \frac{2^2}{2^{x}} \]

### Step 3: Simplify the right side
The right side can be simplified using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \):
\[ 2^{3x} = 2^{2 - x} \]

### Step 4: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ 3x = 2 - x \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \) by adding \( x \) to both sides:
\[ 3x + x = 2 \]
\[ 4x = 2 \]
Now, divide both sides by \( 4 \):
\[ x = \frac{2}{4} = \frac{1}{2} \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{\frac{1}{2}} \]