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

Options

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

Answer

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

### Step 1: Rewrite the bases in terms of powers of 2
We know that:
- \(8 = 2^3\)
- \(\frac{1}{2} = 2^{-1}\)
- \(0.0625 = \frac{625}{10000} = \frac{25^2}{10^4} = \frac{(5^2)^2}{(10^2)^2} = \frac{5^4}{10^4} = 2^{-4}\) (since \(10^4 = 2^4 \cdot 5^4\))

Thus, we can rewrite the equation as:
\[ (2^3)^{(x-2)} \cdot (2^{-1})^{(4-3x)} = (2^{-4})^{x} \]

### Step 2: Simplify the left side using the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ 2^{3(x-2)} \cdot 2^{-(4-3x)} = 2^{-4x} \]

### Step 3: Combine the powers on the left side
Since the bases are the same, we can add the exponents:
\[ 2^{3(x-2) - (4-3x)} = 2^{-4x} \]

### Step 4: Expand and simplify the exponent
Now, let's expand the exponent on the left:
\[ 3(x-2) = 3x - 6 \]
So, we have:
\[ 3x - 6 - 4 + 3x = 6x - 10 \]
Thus, the equation becomes:
\[ 2^{6x - 10} = 2^{-4x} \]

### Step 5: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ 6x - 10 = -4x \]

### Step 6: Solve for \(x\)
Now, we will isolate \(x\):
\[ 6x + 4x = 10 \]
\[ 10x = 10 \]
\[ x = 1 \]

### Final Answer
Thus, the solution is:
\[ \boxed{1} \]