The sum of rational term(s) in `(sqrt3 + 2^(1/3) + 5^(1/4))^8` is equal to

Options

A. 3150
B. 336
C. 3486
D. 3592

Explanation

To find the sum of the rational terms in the expression \((\sqrt{3} + 2^{1/3} + 5^{1/4})^8\), we will use the multinomial expansion and identify the terms that are rational.

### Step-by-Step Solution:

1. **Identify the Expression**:
The expression is \((\sqrt{3} + 2^{1/3} + 5^{1/4})^8\).

2. **Rewrite the Terms**:
We can rewrite the terms in the expression as:
\[ \sqrt{3} = 3^{1/2}, \quad 2^{1/3} = 2^{1/3}, \quad 5^{1/4} = 5^{1/4} \]
Thus, the expression becomes:
\[ (3^{1/2} + 2^{1/3} + 5^{1/4})^8 \]

3. **Use the Multinomial Theorem**:
According to the multinomial theorem, we can expand this as:
\[ \sum_{a+b+c=8} \frac{8!}{a!b!c!} (3^{1/2})^a (2^{1/3})^b (5^{1/4})^c \]
This simplifies to:
\[ \sum_{a+b+c=8} \frac{8!}{a!b!c!} 3^{a/2} 2^{b/3} 5^{c/4} \]

4. **Determine Conditions for Rational Terms**:
For the terms to be rational:
- \(a/2\) must be an integer, which means \(a\) must be even.
- \(b/3\) must be an integer, which means \(b\) must be a multiple of 3.
- \(c/4\) must be an integer, which means \(c\) must be a multiple of 4.

5. **Find Possible Values of \(a\), \(b\), and \(c\)**:
We will find combinations of \(a\), \(b\), and \(c\) such that \(a + b + c = 8\):
- Let \(a = 0, 2, 4, 6, 8\) (even values).
- Let \(b = 0, 3, 6\) (multiples of 3).
- Let \(c = 0, 4, 8\) (multiples of 4).

6. **Calculate Each Case**:
- **Case 1**: \(a = 8, b = 0, c = 0\)
\[ \text{Term} = \frac{8!}{8!0!0!} 3^{4} = 81 \]

- **Case 2**: \(a = 2, b = 6, c = 0\)
\[ \text{Term} = \frac{8!}{2!6!0!} 3^{1} 2^{4} = \frac{40320}{2 \cdot 720} \cdot 3 \cdot 16 = 28 \cdot 48 = 1344 \]

- **Case 3**: \(a = 4, b = 0, c = 4\)
\[ \text{Term} = \frac{8!}{4!0!4!} 3^{2} 5^{1} = \frac{40320}{24 \cdot 24} \cdot 9 \cdot 5 = 70 \cdot 45 = 3150 \]

- **Case 4**: \(a = 0, b = 0, c = 8\)
\[ \text{Term} = \frac{8!}{0!0!8!} 5^{2} = 25 \]

7. **Sum of All Rational Terms**:
Now, we sum all the rational terms calculated:
\[ 81 + 1344 + 3150 + 25 = 3592 \]

### Final Answer:
The sum of the rational terms in \((\sqrt{3} + 2^{1/3} + 5^{1/4})^8\) is **3592**.