The sum of the rational terms in the expansion of
`(2^(1//5) + sqrt(3))^(20)` , is

Options

A. 71
B. 85
C. 97
D. none of these

Answer

To find the sum of the rational terms in the expansion of \( (2^{1/5} + \sqrt{3})^{20} \), we will 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:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = 2^{1/5} \), \( b = \sqrt{3} \), and \( n = 20 \). Therefore, the general term becomes:
\[ T_{r+1} = \binom{20}{r} (2^{1/5})^{20-r} (\sqrt{3})^r \]
This simplifies to:
\[ T_{r+1} = \binom{20}{r} 2^{(20-r)/5} 3^{r/2} \]

### Step 2: Determine Conditions for Rational Terms
For \( T_{r+1} \) to be a rational term, both \( \frac{20-r}{5} \) and \( \frac{r}{2} \) must be integers.

1. **Condition for \( 2^{(20-r)/5} \)**:
- \( 20 - r \) must be divisible by 5. This gives us:
\[ r = 20, 15, 10, 5, 0 \]

2. **Condition for \( 3^{r/2} \)**:
- \( r \) must be even. This gives us:
\[ r = 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \]

### Step 3: Find Common Values of \( r \)
Now, we need to find the common values of \( r \) from both conditions:
- From the first condition: \( r = 0, 5, 10, 15, 20 \)
- From the second condition: \( r = 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \)

The common values are:
\[ r = 0, 10, 20 \]

### Step 4: Calculate the Rational Terms
Now we will calculate the rational terms for \( r = 0, 10, 20 \):

1. **For \( r = 0 \)**:
\[ T_1 = \binom{20}{0} 2^{20/5} 3^{0/2} = 1 \cdot 2^4 \cdot 1 = 16 \]

2. **For \( r = 10 \)**:
\[ T_{11} = \binom{20}{10} 2^{(20-10)/5} 3^{10/2} = \binom{20}{10} 2^2 3^5 \]
We know \( \binom{20}{10} = 184756 \), so:
\[ T_{11} = 184756 \cdot 4 \cdot 243 = 184756 \cdot 972 = 179216000 \]

3. **For \( r = 20 \)**:
\[ T_{21} = \binom{20}{20} 2^{(20-20)/5} 3^{20/2} = 1 \cdot 1 \cdot 3^{10} = 59049 \]

### Step 5: Sum of the Rational Terms
Now we sum the rational terms:
\[ \text{Sum} = T_1 + T_{11} + T_{21} = 16 + 179216000 + 59049 \]
Calculating this gives:
\[ \text{Sum} = 179216000 + 59049 + 16 = 179275065 \]

### Final Answer
Thus, the sum of the rational terms in the expansion of \( (2^{1/5} + \sqrt{3})^{20} \) is:
\[ \boxed{179275065} \]