If `((1)/(16))^(4-3x) xx 8^(x-2)= (0.25)^(x)`, then find the value of `(17x)/(22)+1`

Options

A. 2
B. `(39)/(22)`
C. 1
D. `(22)/(17)`

Answer

To solve the equation \(\left(\frac{1}{16}\right)^{4-3x} \times 8^{x-2} = (0.25)^{x}\), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 2
We know:
- \(16 = 2^4\) so \(\frac{1}{16} = 2^{-4}\)
- \(8 = 2^3\)
- \(0.25 = \frac{1}{4} = 2^{-2}\)

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

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

This simplifies to:
\[ 2^{-16 + 12x} \times 2^{3x - 6} = 2^{-2x} \]

### Step 3: Combine the exponents on the left side
Using the property \(a^m \times a^n = a^{m+n}\), we combine the exponents:
\[ 2^{(-16 + 12x + 3x - 6)} = 2^{-2x} \]

This simplifies to:
\[ 2^{15x - 22} = 2^{-2x} \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 15x - 22 = -2x \]

### Step 5: Solve for \(x\)
Now, we will solve for \(x\):
\[ 15x + 2x = 22 \]
\[ 17x = 22 \]
\[ x = \frac{22}{17} \]

### Step 6: Find the value of \(\frac{17x}{22} + 1\)
Now we substitute \(x\) back into the expression:
\[ \frac{17x}{22} + 1 = \frac{17 \cdot \frac{22}{17}}{22} + 1 \]
\[ = \frac{22}{22} + 1 = 1 + 1 = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]