Find the value of a, ` ( a ne ` integer ) if :
` 2^(a-5) xx 6^(2a-4) = (1)/( 12^(4) xx 2)`

Answer

To solve the equation \( 2^{(a-5)} \times 6^{(2a-4)} = \frac{1}{12^4 \times 2} \), we will follow these steps:

### Step 1: Rewrite the right-hand side
We start by rewriting the right-hand side of the equation. We know that \( 12 = 2^2 \times 3 \), so we can express \( 12^4 \) as:
\[ 12^4 = (2^2 \times 3)^4 = 2^8 \times 3^4 \]
Thus, we can rewrite the right-hand side:
\[ 12^4 \times 2 = 2^8 \times 3^4 \times 2 = 2^9 \times 3^4 \]
Now, the equation becomes:
\[ 2^{(a-5)} \times 6^{(2a-4)} = \frac{1}{2^9 \times 3^4} \]

### Step 2: Rewrite \( 6^{(2a-4)} \)
Next, we rewrite \( 6^{(2a-4)} \) in terms of its prime factors:
\[ 6 = 2 \times 3 \implies 6^{(2a-4)} = (2 \times 3)^{(2a-4)} = 2^{(2a-4)} \times 3^{(2a-4)} \]
Now substituting this back into the equation gives us:
\[ 2^{(a-5)} \times (2^{(2a-4)} \times 3^{(2a-4)}) = \frac{1}{2^9 \times 3^4} \]
This simplifies to:
\[ 2^{(a-5 + 2a-4)} \times 3^{(2a-4)} = \frac{1}{2^9 \times 3^4} \]
Combining the powers of 2:
\[ 2^{(3a-9)} \times 3^{(2a-4)} = \frac{1}{2^9 \times 3^4} \]

### Step 3: Express the right-hand side with negative exponents
We can express the right-hand side using negative exponents:
\[ \frac{1}{2^9 \times 3^4} = 2^{-9} \times 3^{-4} \]
Now we have:
\[ 2^{(3a-9)} \times 3^{(2a-4)} = 2^{-9} \times 3^{-4} \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
1. For the base 2:
\[ 3a - 9 = -9 \]
2. For the base 3:
\[ 2a - 4 = -4 \]

### Step 5: Solve the equations
**From the first equation:**
\[ 3a - 9 = -9 \implies 3a = 0 \implies a = 0 \]

**From the second equation:**
\[ 2a - 4 = -4 \implies 2a = 0 \implies a = 0 \]

### Conclusion
Both equations give us the same solution:
\[ a = 0 \]