The sum of all rational terms in the expansion of `(2^(1/5) + 3^(1/3))^(15)` is equal to

Options

A. 6131
B. 3133
C. 931
D. 633

Answer

To find the sum of all rational terms in the expansion of \((2^{1/5} + 3^{1/3})^{15}\), 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:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
For our expression, we have \(a = 2^{1/5}\), \(b = 3^{1/3}\), and \(n = 15\). Thus, the general term becomes:
\[ T_{r+1} = \binom{15}{r} (2^{1/5})^{15-r} (3^{1/3})^r \]
This simplifies to:
\[ T_{r+1} = \binom{15}{r} 2^{(15-r)/5} 3^{r/3} \]

### Step 2: Determine Conditions for Rational Terms
For \(T_{r+1}\) to be rational, both exponents \((15-r)/5\) and \(r/3\) must be integers.

1. **Condition for \(2^{(15-r)/5}\)**:
\((15 - r)/5\) is an integer if \(15 - r\) is a multiple of 5. This implies:
\[ 15 - r = 5k \quad \text{for some integer } k \]
Therefore, \(r = 15 - 5k\). The possible values for \(k\) are \(0, 1, 2, 3\) (since \(r\) must be non-negative), giving us \(r = 15, 10, 5, 0\).

2. **Condition for \(3^{r/3}\)**:
\(r/3\) is an integer if \(r\) is a multiple of 3. The possible values of \(r\) that are multiples of 3 within our earlier findings are \(0, 3, 6, 9, 12, 15\).

### Step 3: Find Common Values of \(r\)
Now we need to find the common values of \(r\) from both conditions:
- From \(2^{(15-r)/5}\): \(r = 0, 5, 10, 15\)
- From \(3^{r/3}\): \(r = 0, 3, 6, 9, 12, 15\)

The common values are:
- \(r = 0\)
- \(r = 15\)

### Step 4: Calculate the Rational Terms
Now we calculate the rational terms for \(r = 0\) and \(r = 15\):

1. **For \(r = 0\)**:
\[ T_{1} = \binom{15}{0} (2^{1/5})^{15} (3^{1/3})^0 = 1 \cdot 2^{3} \cdot 1 = 8 \]

2. **For \(r = 15\)**:
\[ T_{16} = \binom{15}{15} (2^{1/5})^{0} (3^{1/3})^{15} = 1 \cdot 1 \cdot 3^{5} = 243 \]

### Step 5: Sum of Rational Terms
Finally, we sum the rational terms:
\[ \text{Sum} = 8 + 243 = 251 \]

### Conclusion
Thus, the sum of all rational terms in the expansion of \((2^{1/5} + 3^{1/3})^{15}\) is:
\[ \boxed{251} \]