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 \( T_r \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = 2x^2 \), \( b = -\frac{3}{x^3} \), and \( n = 25 \). Thus, the general term becomes:
\[ T_r = \binom{25}{r} (2x^2)^{25-r} \left(-\frac{3}{x^3}\right)^r \]

### Step 2: Simplify the General Term
Now, simplify \( T_r \):
\[ T_r = \binom{25}{r} (2^{25-r} (x^2)^{25-r}) \left(-3^r (x^{-3})^r\right) \]
This can be rewritten as:
\[ T_r = \binom{25}{r} 2^{25-r} (-3)^r x^{2(25-r) - 3r} \]
Calculating the exponent of \( x \):
\[ T_r = \binom{25}{r} 2^{25-r} (-3)^r x^{50 - 2r - 3r} = \binom{25}{r} 2^{25-r} (-3)^r x^{50 - 5r} \]

### 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 \( r \):
\[ 50 = 5r \implies r = \frac{50}{5} = 10 \]

### Step 4: Substitute \( r \) Back into the General Term
Now substitute \( r = 10 \) back into the general term:
\[ T_{10} = \binom{25}{10} 2^{25-10} (-3)^{10} x^{50 - 5 \cdot 10} \]
This simplifies to:
\[ T_{10} = \binom{25}{10} 2^{15} (-3)^{10} \]

### Step 5: Calculate the Coefficient
Now we calculate the coefficient:
\[ T_{10} = \binom{25}{10} 2^{15} \cdot 3^{10} \]

### Final Answer
Thus, the term independent of \( x \) in the expansion is:
\[ \binom{25}{10} 2^{15} \cdot 3^{10} \]

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