The number of rational terms in the binomial | vedclass

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(C) The general term in the expansion of $(\sqrt[4]{5}+\sqrt[5]{4})^{100}$ is given by:$T_{r+1} = {}^{100}C_r (5^{1/4})^{100-r} (4^{1/5})^r$$T_{r+1} =

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(C) The general term in the expansion of $(\sqrt[4]{5}+\sqrt[5]{4})^{100}$ is given by:$T_{r+1} = {}^{100}C_r (5^{1/4})^{100-r} (4^{1/5})^r$$T_{r+1} = {}^{100}C_r 5^{\frac{100-r}{4}} 4^{\frac{r}{5}}$For the term to be rational,the exponents of $5$ and $4$ must be integers.Thus,$\frac{100-r}{4}$ must be an integer,which implies $r$ must be a multiple of $4$ (i.e.,$r \in \{0, 4, 8, \dots, 100\}$).Also,$\frac{r}{5}$ must be an integer,which implies $r$ must be a multiple of $5$ (i.e.,$r \in \{0, 5, 10, \dots, 100\}$).Therefore,$r$ must be a multiple of $	ext{lcm}(4, 5) = 20$.The possible values for $r$ are $0, 20, 40, 60, 80, 100$.Counting these values,we get $6$ terms.Hence,the number of rational terms is $6$.

The number of rational terms in the binomial expansion of $(\sqrt[4]{5}+\sqrt[5]{4})^{100}$ is

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