The sum of all rational terms in the expansion of
` (3^(1//4) + 4^(1//3))^(12)` is

Options

A. 91
B. 251
C. 273
D. 283

Explanation

To find the sum of all rational terms in the expansion of \( (3^{1/4} + 4^{1/3})^{12} \), 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, \( a = 3^{1/4} \), \( b = 4^{1/3} \), and \( n = 12 \). Therefore, the general term can be expressed as:

\[ T_{r+1} = \binom{12}{r} (3^{1/4})^{12-r} (4^{1/3})^r \]

### Step 2: Simplify the General Term
Now, simplifying the general term:

\[ T_{r+1} = \binom{12}{r} (3^{(12-r)/4}) (4^{r/3}) \]

This can be rewritten as:

\[ T_{r+1} = \binom{12}{r} 3^{(12-r)/4} 4^{r/3} \]

### Step 3: Determine Rational Terms
For the term \( T_{r+1} \) to be rational, both exponents \( \frac{12-r}{4} \) and \( \frac{r}{3} \) must be integers.

1. **Condition for \( \frac{12-r}{4} \)**:
- This means \( 12 - r \) must be divisible by 4.
- Let \( 12 - r = 4k \) for some integer \( k \).
- Thus, \( r = 12 - 4k \).

2. **Condition for \( \frac{r}{3} \)**:
- This means \( r \) must be divisible by 3.
- Let \( r = 3m \) for some integer \( m \).

### Step 4: Find Values of \( r \)
Now we need to find values of \( r \) that satisfy both conditions:

- From \( r = 12 - 4k \), we have \( r \) can be \( 0, 4, 8, 12 \) (as \( k \) takes values \( 0, 1, 2, 3 \)).
- From \( r = 3m \), we have \( r \) can be \( 0, 3, 6, 9, 12 \) (as \( m \) takes values \( 0, 1, 2, 3, 4 \)).

The common values of \( r \) that satisfy both conditions are \( r = 0, 12 \).

### Step 5: Calculate Rational Terms
Now we calculate the rational terms for \( r = 0 \) and \( r = 12 \):

1. **For \( r = 0 \)**:
\[ T_{1} = \binom{12}{0} (3^{1/4})^{12} (4^{1/3})^0 = 1 \cdot 3^{3} \cdot 1 = 27 \]

2. **For \( r = 12 \)**:
\[ T_{13} = \binom{12}{12} (3^{1/4})^{0} (4^{1/3})^{12} = 1 \cdot 1 \cdot 4^{4} = 256 \]

### Step 6: Sum of Rational Terms
Finally, the sum of all rational terms is:

\[ \text{Sum} = T_{1} + T_{13} = 27 + 256 = 283 \]

### Final Answer
Thus, the sum of all rational terms in the expansion is \( \boxed{283} \).

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