The number of rational terms in the binomial | vedclass

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(D) The general term of the expansion ${\left( 7^{\frac{1}{7}} + 11^{\frac{1}{11}} \right)^{711}}$ is given by $T_{r+1} = {}^{711}C_r \cdot (7^{\frac{

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$10$

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(D) The general term of the expansion ${\left( 7^{\frac{1}{7}} + 11^{\frac{1}{11}} \right)^{711}}$ is given by $T_{r+1} = {}^{711}C_r \cdot (7^{\frac{1}{7}})^{711-r} \cdot (11^{\frac{1}{11}})^r = {}^{711}C_r \cdot 7^{\frac{711-r}{7}} \cdot 11^{\frac{r}{11}}$.For the term to be rational,the exponents of $7$ and $11$ must be integers.$1$) $\frac{r}{11} = k_1 \Rightarrow r = 11k_1$,where $k_1 \in \{0, 1, 2, \dots, 64\}$.$2$) $\frac{711-r}{7} = k_2$ $\Rightarrow 711-r = 7k_2$ $\Rightarrow r \equiv 711 \pmod{7}$ $\Rightarrow r \equiv 4 \pmod{7}$.Substituting $r = 11k_1$ into the second condition: $11k_1 \equiv 4 \pmod{7}$ $\Rightarrow 4k_1 \equiv 4 \pmod{7}$ $\Rightarrow k_1 \equiv 1 \pmod{7}$.So,$k_1$ can be $1, 8, 15, 22, 29, 36, 43, 50, 57, 64$.There are $10$ such values of $k_1$,which correspond to $10$ rational terms.

The number of rational terms in the binomial expansion of ${\left( 7^{\frac{1}{7}} + 11^{\frac{1}{11}} \right)^{711}}$ is:

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