Find the term independent of x in the expansion of `(2x^2-3/x^3)^(25)`

Answer

To find the term independent of \( x \) in the expansion of \( (2x^2 - \frac{3}{x^3})^{25} \), we can follow these steps:

### Step 1: Identify the General Term
The general term \( T_{r+1} \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = 2x^2 \), \( b = -\frac{3}{x^3} \), and \( n = 25 \). Therefore, the general term becomes:
\[ T_{r+1} = \binom{25}{r} (2x^2)^{25-r} \left(-\frac{3}{x^3}\right)^r \]

### Step 2: Simplify the General Term
We can simplify \( T_{r+1} \) as follows:
\[ T_{r+1} = \binom{25}{r} (2^{25-r} x^{2(25-r)}) \left(-3^r x^{-3r}\right) \]
This simplifies to:
\[ T_{r+1} = \binom{25}{r} (-1)^r 2^{25-r} 3^r x^{50 - 5r} \]

### Step 3: Find the Term Independent of \( x \)
To find the term independent of \( x \), we need the exponent of \( x \) to be zero:
\[ 50 - 5r = 0 \]
Solving for \( r \):
\[ 5r = 50 \implies r = 10 \]

### Step 4: Find the Specific Term
Now that we have \( r = 10 \), we can find the term \( T_{11} \) (since \( r \) starts from 0):
\[ T_{11} = \binom{25}{10} (-1)^{10} 2^{25-10} 3^{10} x^{50 - 5 \cdot 10} \]
This simplifies to:
\[ T_{11} = \binom{25}{10} 2^{15} 3^{10} \]

### Conclusion
The term independent of \( x \) is given by:
\[ T_{11} = \binom{25}{10} 2^{15} 3^{10} \]