Find the value of n, given : `(2xx4^(3)xx2^(n-4)xx3xx2^(n+2))/(3^(3)xx2^(16))=(2)/(9)`

Answer

To solve the equation

\[ \frac{2 \times 4^3 \times 2^{n-4} \times 3 \times 2^{n+2}}{3^3 \times 2^{16}} = \frac{2}{9} \]

we will follow these steps:

### Step 1: Simplify the expression
First, we can rewrite \(4\) as \(2^2\). Therefore, \(4^3\) can be expressed as:

\[ 4^3 = (2^2)^3 = 2^{6} \]

Now, substituting this back into the equation gives:

\[ \frac{2 \times 2^6 \times 2^{n-4} \times 3 \times 2^{n+2}}{3^3 \times 2^{16}} = \frac{2}{9} \]

### Step 2: Combine the powers of 2
Next, we can combine the powers of \(2\) in the numerator:

\[ 2 \times 2^6 \times 2^{n-4} \times 2^{n+2} = 2^{1 + 6 + (n-4) + (n+2)} = 2^{2n + 5} \]

So the equation now looks like:

\[ \frac{2^{2n + 5} \times 3}{3^3 \times 2^{16}} = \frac{2}{9} \]

### Step 3: Simplify the fraction
Now, we can simplify the fraction:

\[ \frac{2^{2n + 5} \times 3}{3^3 \times 2^{16}} = \frac{2^{2n + 5}}{3^2 \times 2^{16}} = \frac{2^{2n + 5 - 16}}{9} = \frac{2^{2n - 11}}{9} \]

### Step 4: Set the equation equal to \(\frac{2}{9}\)
Now we have:

\[ \frac{2^{2n - 11}}{9} = \frac{2}{9} \]

### Step 5: Eliminate the denominators
Multiplying both sides by \(9\):

\[ 2^{2n - 11} = 2 \]

### Step 6: Express \(2\) as a power of \(2\)
We can express \(2\) as \(2^1\):

\[ 2^{2n - 11} = 2^1 \]

### Step 7: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:

\[ 2n - 11 = 1 \]

### Step 8: Solve for \(n\)
Now, solving for \(n\):

\[ 2n = 1 + 11 \]
\[ 2n = 12 \]
\[ n = 6 \]

### Final Answer
Thus, the value of \(n\) is:

\[ \boxed{6} \]