If `x_(1), x_(2)(x_(1) gt x_(2))` are the two solutions of the equation
`3^(log_(2)x)-12(x^(log_(16)9))=log_(3)((1)/(3))^(3^(3))`, then the value of `x_(1)-2x_(2)` is :

Answer

To solve the equation \( 3^{\log_2 x} - 12 \left( x^{\log_{16} 9} \right) = \log_3 \left( \frac{1}{3} \right)^{3^3} \), we will follow these steps:

### Step 1: Simplify the Right Side
The right side of the equation can be simplified as follows:
\[ \log_3 \left( \frac{1}{3} \right)^{3^3} = \log_3 \left( 3^{-27} \right) = -27 \]
Thus, the equation becomes:
\[ 3^{\log_2 x} - 12 \left( x^{\log_{16} 9} \right) = -27 \]

### Step 2: Simplify the Left Side
Next, we will simplify the term \( x^{\log_{16} 9} \):
\[ \log_{16} 9 = \frac{\log_2 9}{\log_2 16} = \frac{\log_2 9}{4} \]
So, we can rewrite \( x^{\log_{16} 9} \) as:
\[ x^{\log_{16} 9} = x^{\frac{\log_2 9}{4}} = \left( x^{\log_2 9} \right)^{\frac{1}{4}} = \sqrt[4]{x^{\log_2 9}} \]

### Step 3: Substitute \( t = \log_2 x \)
Let \( t = \log_2 x \). Then, we can rewrite \( 3^{\log_2 x} \) as \( 3^t \) and \( x^{\log_{16} 9} \) as \( 2^{t \cdot \frac{\log_2 9}{4}} \):
\[ 3^t - 12 \cdot 2^{\frac{t \cdot \log_2 9}{4}} = -27 \]

### Step 4: Rewrite the Equation
Now, we can rewrite the equation in terms of \( t \):
\[ 3^t - 12 \cdot 2^{\frac{t \cdot \log_2 9}{4}} + 27 = 0 \]

### Step 5: Substitute \( a = 3^{\frac{t}{2}} \)
Let \( a = 3^{\frac{t}{2}} \). Then \( 3^t = a^2 \) and \( 2^{\frac{t \cdot \log_2 9}{4}} = 3^{\frac{t}{2}} \) can be expressed as:
\[ a^2 - 12a + 27 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 27}}{2 \cdot 1} = \frac{12 \pm \sqrt{144 - 108}}{2} = \frac{12 \pm \sqrt{36}}{2} = \frac{12 \pm 6}{2} \]
Thus, we have:
\[ a = \frac{18}{2} = 9 \quad \text{or} \quad a = \frac{6}{2} = 3 \]

### Step 7: Find \( t \)
Now, we can find \( t \):
1. If \( a = 9 \):
\[ 3^{\frac{t}{2}} = 9 \implies \frac{t}{2} = 2 \implies t = 4 \]
2. If \( a = 3 \):
\[ 3^{\frac{t}{2}} = 3 \implies \frac{t}{2} = 1 \implies t = 2 \]

### Step 8: Find \( x \)
Now, substituting \( t \) back to find \( x \):
1. If \( t = 4 \):
\[ x = 2^4 = 16 \]
2. If \( t = 2 \):
\[ x = 2^2 = 4 \]

### Step 9: Calculate \( x_1 - 2x_2 \)
Let \( x_1 = 16 \) and \( x_2 = 4 \):
\[ x_1 - 2x_2 = 16 - 2 \cdot 4 = 16 - 8 = 8 \]

### Final Answer
The value of \( x_1 - 2x_2 \) is \( \boxed{8} \).