The number of rational terms in the expansion of `(2^(1/2) + 3^(1/3))^(15)`

Answer

To find the number of rational terms in the expansion of \( (2^{1/2} + 3^{1/3})^{15} \), we can use the Binomial Theorem. Let's break down the solution step by step.

### Step 1: Identify the Binomial Expansion
The expression can be expanded using the Binomial Theorem, which states:
\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = 2^{1/2} \), \( b = 3^{1/3} \), and \( n = 15 \).

### Step 2: Write the General Term
The general term \( T_r \) in the expansion is given by:
\[ T_r = \binom{15}{r} (2^{1/2})^{15-r} (3^{1/3})^r \]
This simplifies to:
\[ T_r = \binom{15}{r} 2^{(15-r)/2} 3^{r/3} \]

### Step 3: Determine Conditions for Rationality
For \( T_r \) to be a rational number, both exponents \( \frac{15-r}{2} \) and \( \frac{r}{3} \) must be integers. This leads us to the following conditions:

1. \( \frac{15 - r}{2} \) is an integer, which implies \( 15 - r \) must be even. Therefore, \( r \) must be odd.
2. \( \frac{r}{3} \) is an integer, which implies \( r \) must be a multiple of 3.

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

- Since \( r \) must be odd and a multiple of 3, we can list the odd multiples of 3 that are less than or equal to 15:
- The odd multiples of 3 are: \( 3, 9, 15 \).

### Step 5: Check Each Value
Let's check each of these values to ensure they satisfy both conditions:
- For \( r = 3 \):
- \( 15 - 3 = 12 \) (even)
- \( 3 \div 3 = 1 \) (integer)
- For \( r = 9 \):
- \( 15 - 9 = 6 \) (even)
- \( 9 \div 3 = 3 \) (integer)
- For \( r = 15 \):
- \( 15 - 15 = 0 \) (even)
- \( 15 \div 3 = 5 \) (integer)

### Step 6: Conclusion
The values of \( r \) that yield rational terms are \( r = 3, 9, 15 \). Therefore, there are **3 rational terms** in the expansion of \( (2^{1/2} + 3^{1/3})^{15} \).

### Final Answer
The number of rational terms in the expansion is **3**.

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