What is the number of <strong>RATIONAL</strong> terms in the expansion of (3^{\frac{1}{2}}+5^{\frac{1}{4}})^{12}?
- A. 2
- B. 3
- C. 4 ✓
- D. 6
Correct Answer: C. 4
Explanation
The general term is T_{r+1} = \binom{12}{r} (3^{\frac{1}{2}})^{12-r} (5^{\frac{1}{4}})^r = \binom{12}{r} 3^{\frac{12-r}{2}} 5^{\frac{r}{4}}. For the term to be rational, 12-r must be a multiple of 2 and r must be a multiple of 4. This holds when r \in \{0, 4, 8, 12\}, yielding 4 rational terms.
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