The number of integral terms in the expansion | vedclass

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Question

(b) ${T_{r + 1}} = {}^{256}{C_r}{(\sqrt 3 )^{256 - r}}{(\sqrt[8]{5})^r}$ $ = {}^{256}{C_r}{(3)^{\frac{{256 - r}}{2}}}{(5)^{r/8}}$

Terms would be in

Vedclass · Accepted Answer

$33$

Vedclass · Answer

(b) ${T_{r + 1}} = {}^{256}{C_r}{(\sqrt 3 )^{256 - r}}{(\sqrt[8]{5})^r}$ $ = {}^{256}{C_r}{(3)^{\frac{{256 - r}}{2}}}{(5)^{r/8}}$

Terms would be integral if $\frac{{256 - r}}{2}$ and $\frac{r}{8}$ both are positive integer.

As $0 \leq r \leq 256$, $	herefore r = 0,\,8,\,16,\,24,.....,256$

For above values of $r$, $\left( {\frac{{256 - r}}{2}} ight)$ is also an integer.

 Total number of values of $r = 33.$

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

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