Question

The number of integral terms in the expansion of $(\sqrt{3} + \sqrt[8]{5})^{256}$ is

• A. 32
• B. 33
• C. 34
• D. 35

Answer 1

Answer: (B)

33

Answer 2

$T_{r + 1} =^{256} ⥂ C_{r}. ⥂ 3^{\frac{256 - r}{2}}.5^{\frac{r}{8}}$

First term = $256 ⥂ C_{0}3^{128}5^{0} = \text{integer}$ and after eight terms, i.e., 9^th term = $256 ⥂ ⥂ C_{8}3^{124}.5^{1} = \text{integer}$

Continuing like this, we get an A.P., $1^{\text{st}},9^{th}.......257^{th}$; $T_{n} = a + (n - 1)d \Rightarrow 257 = 1 + (n - 1)8 \Rightarrow n = 33$