Find the coefficient of `x^11` in `(2x^2 + (3)/(x^3))^13`

Answer

To find the coefficient of \( x^{11} \) in the expression \( (2x^2 + \frac{3}{x^3})^{13} \), we can use the Binomial Theorem. Let's break down the solution step by step:

### Step 1: Identify the terms in the binomial expansion
The expression can be rewritten as:
\[ (2x^2 + 3x^{-3})^{13} \]
Here, we identify \( a = 2x^2 \) and \( b = 3x^{-3} \) with \( n = 13 \).

### Step 2: Use the Binomial Theorem
According to the Binomial Theorem, the expansion of \( (a + b)^n \) is given by:
\[ \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]
In our case, we have:
\[ \sum_{r=0}^{13} \binom{13}{r} (2x^2)^{13-r} (3x^{-3})^r \]

### Step 3: Simplify the terms
Now, we simplify the terms:
\[ = \sum_{r=0}^{13} \binom{13}{r} (2^{13-r} x^{2(13-r)}) (3^r x^{-3r}) \]
This can be rewritten as:
\[ = \sum_{r=0}^{13} \binom{13}{r} 2^{13-r} 3^r x^{2(13-r) - 3r} \]
\[ = \sum_{r=0}^{13} \binom{13}{r} 2^{13-r} 3^r x^{26 - 5r} \]

### Step 4: Set the exponent of \( x \) to 11
We need to find the value of \( r \) such that the exponent of \( x \) equals 11:
\[ 26 - 5r = 11 \]
Solving for \( r \):
\[ 5r = 26 - 11 \]
\[ 5r = 15 \implies r = 3 \]

### Step 5: Calculate the coefficient for \( r = 3 \)
Now, we substitute \( r = 3 \) back into the binomial expansion to find the coefficient:
\[ \text{Coefficient} = \binom{13}{3} 2^{13-3} 3^3 \]
Calculating each part:
\[ \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \]
\[ 2^{10} = 1024 \]
\[ 3^3 = 27 \]
Now, multiply these values together:
\[ \text{Coefficient} = 286 \times 1024 \times 27 \]

### Step 6: Final calculation
Calculating \( 286 \times 1024 \):
\[ 286 \times 1024 = 292864 \]
Now, multiply by 27:
\[ 292864 \times 27 = 7907528 \]

### Final Answer
Thus, the coefficient of \( x^{11} \) in the expansion of \( (2x^2 + \frac{3}{x^3})^{13} \) is:
\[ \boxed{7907528} \]