If `x^4` occurs in the rth term in the expansion of `(x^4+1/(x^3))^(15),` then find the value of `r`.

Explanation

To find the value of \( r \) such that \( x^4 \) occurs in the \( r \)th term of the expansion of \( (x^4 + \frac{1}{x^3})^{15} \), we will follow these steps:

### Step 1: Identify the general term in the binomial expansion
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, \( a = x^4 \), \( b = \frac{1}{x^3} \), and \( n = 15 \).

### Step 2: Write the \( r \)th term
Thus, the \( r \)th term \( T_r \) can be expressed as:
\[ T_r = \binom{15}{r-1} (x^4)^{15-(r-1)} \left(\frac{1}{x^3}\right)^{r-1} \]
This simplifies to:
\[ T_r = \binom{15}{r-1} (x^4)^{16-r} \left(\frac{1}{x^3}\right)^{r-1} \]

### Step 3: Simplify the expression
Now, simplifying the powers of \( x \):
\[ T_r = \binom{15}{r-1} x^{4(16-r)} \cdot x^{-3(r-1)} = \binom{15}{r-1} x^{64 - 4r + 3r - 3} = \binom{15}{r-1} x^{67 - r} \]

### Step 4: Set the exponent equal to 4
To find when \( x^4 \) occurs, we set the exponent equal to 4:
\[ 67 - r = 4 \]

### Step 5: Solve for \( r \)
Now, solving for \( r \):
\[ 67 - 4 = r \implies r = 63 \]

### Conclusion
Thus, the value of \( r \) is:
\[ \boxed{63} \]