The term independent of x in `((1)/(60)-(x^(8))/(81))(2x^(2)-(3)/(x^(2)))^(6)` is

Options

A. `-36`
B. `36`
C. `72`
D. `108`

Answer

To find the term independent of \( x \) in the expression

\[ \left( \frac{1}{60} - \frac{x^8}{81} \right) \left( 2x^2 - \frac{3}{x^2} \right)^6, \]

we will break down the problem into manageable steps.

### Step 1: Expand the expression
We can rewrite the expression as:

\[ \frac{1}{60} \left( 2x^2 - \frac{3}{x^2} \right)^6 - \frac{1}{81} x^8 \left( 2x^2 - \frac{3}{x^2} \right)^6. \]

### Step 2: Find the general term of the binomial expansion
The general term \( T_{r+1} \) in the expansion of \( \left( 2x^2 - \frac{3}{x^2} \right)^6 \) can be expressed using the binomial theorem:

\[ T_{r+1} = \binom{6}{r} (2x^2)^r \left(-\frac{3}{x^2}\right)^{6-r} = \binom{6}{r} 2^r (-3)^{6-r} x^{2r - 2(6-r)} = \binom{6}{r} 2^r (-3)^{6-r} x^{4r - 12}. \]

### Step 3: Identify the term independent of \( x \)
To find the term independent of \( x \), we set the exponent of \( x \) to zero:

\[ 4r - 12 = 0 \implies r = 3. \]

### Step 4: Substitute \( r \) into the general term
Now, substitute \( r = 3 \) into the general term:

\[ T_{4} = \binom{6}{3} 2^3 (-3)^{3} = 20 \cdot 8 \cdot (-27) = -4320. \]

### Step 5: Calculate the contribution of the first term
Now, we need to consider the contribution of the first term \( \frac{1}{60} \):

\[ \frac{1}{60} T_{4} = \frac{1}{60} \cdot (-4320) = -72. \]

### Step 6: Find the contribution from the second term
Next, we find the term independent of \( x \) from the second part:

For \( r = 1 \) (since we have \( x^8 \) multiplying the term):

\[ T_{2} = \binom{6}{1} 2^1 (-3)^{5} = 6 \cdot 2 \cdot (-243) = -2916. \]

Thus, the contribution from the second term is:

\[ -\frac{1}{81} \cdot x^8 \cdot (-2916) = \frac{2916}{81} = 36. \]

### Step 7: Combine both contributions
Now, we combine both contributions to find the term independent of \( x \):

\[ -72 + 36 = -36. \]

### Final Answer
Thus, the term independent of \( x \) in the given expression is

\[ \boxed{-36}. \]