Find `x`, if `8^(2-x) xx ((1)/(2))^(4-3x) = (0.0625)^(x)`.

Options

A. 0
B. 4
C. 2
D. 1

Answer

To solve the equation \( 8^{(2-x)} \times \left(\frac{1}{2}\right)^{(4-3x)} = (0.0625)^{x} \), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 2
First, we can express \( 8 \) and \( \frac{1}{2} \) in terms of base \( 2 \):
- \( 8 = 2^3 \)
- \( \frac{1}{2} = 2^{-1} \)

Thus, we can rewrite the left side of the equation:
\[ 8^{(2-x)} = (2^3)^{(2-x)} = 2^{3(2-x)} = 2^{6 - 3x} \]
\[ \left(\frac{1}{2}\right)^{(4-3x)} = (2^{-1})^{(4-3x)} = 2^{-(4-3x)} = 2^{-(4 - 3x)} = 2^{3x - 4} \]

Now, substituting these into the equation gives:
\[ 2^{(6 - 3x)} \times 2^{(3x - 4)} = (0.0625)^{x} \]

### Step 2: Combine the left side
Using the property of exponents \( a^m \times a^n = a^{m+n} \), we can combine the left side:
\[ 2^{(6 - 3x) + (3x - 4)} = 2^{(6 - 4)} = 2^{2} \]

### Step 3: Simplify the right side
Next, we need to simplify \( (0.0625)^{x} \). We can express \( 0.0625 \) as a power of \( 2 \):
\[ 0.0625 = \frac{625}{10000} = \frac{25^2}{100^2} = \frac{(5^2)^2}{(10^2)^2} = \frac{5^4}{(2 \cdot 5)^4} = \frac{5^4}{2^4 \cdot 5^4} = \frac{1}{2^4} = 2^{-4} \]
Thus, we have:
\[ (0.0625)^{x} = (2^{-4})^{x} = 2^{-4x} \]

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

### Step 5: Solve for \( x \)
Now, we solve for \( x \):
\[ x = -\frac{2}{4} = -\frac{1}{2} \]

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