Find the term independent of `x` in the expansion of the following binomials:
`(2x^(2) - (1)/(x) )^12` What is its value?

Answer

To find the term independent of \( x \) in the expansion of \( (2x^2 - \frac{1}{x})^{12} \), we will follow these steps:

### Step 1: Identify the General Term
The general term \( T_{r+1} \) in the expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, we have:
- \( n = 12 \)
- \( a = 2x^2 \)
- \( b = -\frac{1}{x} \)

### Step 2: Write the General Term
Substituting the values into the formula, we get:
\[ T_{r+1} = \binom{12}{r} (2x^2)^{12-r} \left(-\frac{1}{x}\right)^r \]
This simplifies to:
\[ T_{r+1} = \binom{12}{r} (2^{12-r} (x^2)^{12-r}) \left(-1^r \cdot x^{-r}\right) \]
\[ = \binom{12}{r} (-1)^r 2^{12-r} x^{2(12-r) - r} \]
\[ = \binom{12}{r} (-1)^r 2^{12-r} x^{24 - 3r} \]

### Step 3: Find the Term Independent of \( x \)
For the term to be independent of \( x \), the exponent of \( x \) must be zero:
\[ 24 - 3r = 0 \]
Solving for \( r \):
\[ 3r = 24 \implies r = 8 \]

### Step 4: Identify the Term
The term independent of \( x \) corresponds to \( r = 8 \), which is the \( (8 + 1) \)th term, or the 9th term:
\[ T_{9} = \binom{12}{8} (-1)^8 2^{12-8} \]

### Step 5: Calculate the Value of the Term
Now we calculate:
\[ T_{9} = \binom{12}{8} \cdot 2^4 \]
Using the property \( \binom{n}{r} = \binom{n}{n-r} \):
\[ \binom{12}{8} = \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = \frac{11880}{24} = 495 \]
Now substituting back:
\[ T_{9} = 495 \cdot 16 = 7920 \]

### Conclusion
The term independent of \( x \) in the expansion of \( (2x^2 - \frac{1}{x})^{12} \) is \( 7920 \).

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