Binomial Theorem
Rational and Irrational Terms
IIT-JEE 1995
Grade 11

Question:

The number of integral terms in the expansion of $(\sqrt{3} + \sqrt[8]{5})^{256}$ is:
(1) $32$
(2) $33$
(3) $34$
(4) $64$

Step-by-Step Solution

Key Concept: Write the general term $T_{r+1} = \binom{256}{r} 3^{(256-r)/2} 5^{r/8}$, and find for how many integer values of $r$ (with $0 \leq r \leq 256$) both exponents $\frac{256 - r}{2}$ and $\frac{r}{8}$ are simultaneously integers.
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Correct Answer: (2)

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