The term independent of x in the expanion of `(root(6)(x)-(2)/(root(3)(x)))^(18)` is

Options

A. `.^(18)C_(8)2^(12)`
B. `.^(18)C_(6)2^(6)`
C. `.^(18)C_(6)2^(8)`
D. `.^(18)C_(8)2^(8)`

Answer

To find the term independent of \( x \) in the expansion of \( \left( \sqrt[6]{x} - \frac{2}{\sqrt[3]{x}} \right)^{18} \), we will follow these steps:

### Step 1: Rewrite the expression
We can rewrite the expression in a more manageable form:
\[ \left( x^{1/6} - 2x^{-1/3} \right)^{18} \]

### Step 2: Identify the general term
The general term \( T_r \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = x^{1/6} \), \( b = -2x^{-1/3} \), and \( n = 18 \). Thus, the general term can be expressed as:
\[ T_r = \binom{18}{r} (x^{1/6})^{18-r} (-2x^{-1/3})^r \]

### Step 3: Simplify the general term
Now, simplifying \( T_r \):
\[ T_r = \binom{18}{r} (x^{(18-r)/6}) (-2)^r (x^{-r/3}) \]
Combining the powers of \( x \):
\[ T_r = \binom{18}{r} (-2)^r x^{\frac{18-r}{6} - \frac{r}{3}} \]
To combine the exponents:
\[ \frac{18-r}{6} - \frac{r}{3} = \frac{18-r - 2r}{6} = \frac{18 - 3r}{6} \]
Thus, we have:
\[ T_r = \binom{18}{r} (-2)^r x^{\frac{18 - 3r}{6}} \]

### Step 4: Set the exponent of \( x \) to zero
For the term to be independent of \( x \), we set the exponent equal to zero:
\[ \frac{18 - 3r}{6} = 0 \]
Multiplying through by 6 gives:
\[ 18 - 3r = 0 \]
Solving for \( r \):
\[ 3r = 18 \implies r = 6 \]

### Step 5: Substitute \( r \) back into the general term
Now, we substitute \( r = 6 \) back into the general term:
\[ T_6 = \binom{18}{6} (-2)^6 x^{0} \]
Calculating \( (-2)^6 \):
\[ (-2)^6 = 64 \]
Thus, we have:
\[ T_6 = \binom{18}{6} \cdot 64 \]

### Step 6: Calculate \( \binom{18}{6} \)
Now we compute \( \binom{18}{6} \):
\[ \binom{18}{6} = \frac{18!}{6!(18-6)!} = \frac{18!}{6! \cdot 12!} \]
Calculating this gives:
\[ \binom{18}{6} = 18564 \]

### Step 7: Final calculation
Now, we multiply:
\[ T_6 = 18564 \cdot 64 = 1184256 \]

### Conclusion
The term independent of \( x \) in the expansion is:
\[ \boxed{1184256} \]