The sum of the all rational terms in the expansion of `(3^(1//7) + 5^(1//2))^(14)` is

Options

A. `3^(2)`
B. `3^(2) + 5^(7)`
C. `3^(7) + 5^(2)`
D. `5^(7)`

Answer

To find the sum of all rational terms in the expansion of \( (3^{1/7} + 5^{1/2})^{14} \), we will follow these steps:

### Step 1: Identify the General Term
The general term 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, \( a = 3^{1/7} \), \( b = 5^{1/2} \), and \( n = 14 \). Thus, the general term becomes:
\[ T_{r+1} = \binom{14}{r} (3^{1/7})^{14-r} (5^{1/2})^r \]
This simplifies to:
\[ T_{r+1} = \binom{14}{r} 3^{(14-r)/7} 5^{r/2} \]

### Step 2: Determine Conditions for Rationality
For the term \( T_{r+1} \) to be rational, both \( \frac{14 - r}{7} \) and \( \frac{r}{2} \) must be integers. This leads us to two conditions:
1. \( 14 - r \) must be divisible by 7.
2. \( r \) must be divisible by 2.

### Step 3: Solve the Conditions
1. **Condition 1**: \( 14 - r = 7k \) for some integer \( k \).
- This implies \( r = 14 - 7k \).
2. **Condition 2**: \( r = 2m \) for some integer \( m \).

Now, substituting \( r = 14 - 7k \) into \( r = 2m \):
\[ 14 - 7k = 2m \]
Rearranging gives:
\[ 7k + 2m = 14 \]

### Step 4: Find Integer Solutions
We can find integer solutions for \( k \) and \( m \):
- If \( k = 0 \), then \( 2m = 14 \) → \( m = 7 \) → \( r = 14 \).
- If \( k = 1 \), then \( 2m = 7 \) → \( m = 3.5 \) (not an integer).
- If \( k = 2 \), then \( 2m = 0 \) → \( m = 0 \) → \( r = 14 - 14 = 0 \).

Thus, the valid values of \( r \) are \( r = 0 \) and \( r = 14 \).

### Step 5: Calculate the Rational Terms
1. **For \( r = 0 \)**:
\[ T_1 = \binom{14}{0} (3^{1/7})^{14} (5^{1/2})^0 = 1 \cdot 3^{2} \cdot 1 = 9 \]

2. **For \( r = 14 \)**:
\[ T_{15} = \binom{14}{14} (3^{1/7})^{0} (5^{1/2})^{14} = 1 \cdot 1 \cdot 5^{7} = 78125 \]

### Step 6: Sum the Rational Terms
Now, we sum the rational terms:
\[ \text{Sum} = T_1 + T_{15} = 9 + 78125 = 78134 \]

### Final Answer
The sum of all rational terms in the expansion is \( \boxed{78134} \).