Find the term independent of x in `(3x^2 + (5)/(x^3) )^12`

Answer

To find the term independent of \( x \) in the expression \( (3x^2 + \frac{5}{x^3})^{12} \), we will use the Binomial Theorem.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \( T_{r+1} \) in the expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Here, \( a = 3x^2 \), \( b = \frac{5}{x^3} \), and \( n = 12 \).

Thus, the general term becomes:
\[ T_{r+1} = \binom{12}{r} (3x^2)^{12-r} \left(\frac{5}{x^3}\right)^r \]

2. **Simplify the General Term**:
Expanding the general term:
\[ T_{r+1} = \binom{12}{r} (3^{12-r} x^{2(12-r)}) \left(\frac{5^r}{x^{3r}}\right) \]
This simplifies to:
\[ T_{r+1} = \binom{12}{r} 3^{12-r} 5^r x^{2(12-r) - 3r} \]
Simplifying the exponent of \( x \):
\[ T_{r+1} = \binom{12}{r} 3^{12-r} 5^r x^{24 - 2r - 3r} = \binom{12}{r} 3^{12-r} 5^r x^{24 - 5r} \]

3. **Find the Term Independent of \( x \)**:
For the term to be independent of \( x \), the exponent of \( x \) must be zero:
\[ 24 - 5r = 0 \]
Solving for \( r \):
\[ 5r = 24 \implies r = \frac{24}{5} \]

4. **Check if \( r \) is an Integer**:
Since \( r = \frac{24}{5} \) is not an integer, there is no integer value of \( r \) that satisfies this equation.

5. **Conclusion**:
Therefore, there is no term independent of \( x \) in the expansion of \( (3x^2 + \frac{5}{x^3})^{12} \). The answer is:
\[ \text{The term independent of } x \text{ is } 0. \]