The number of rational terms in the expansion | vedclass

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(D) The general term $T_{r+1}$ in the expansion of $(3^{1/4} + 7^{1/6})^{144}$ is given by: $T_{r+1} = {}^{144}C_r (3^{1/4})^{144-r} (7^{1/6})^r$ $T_{

Vedclass · Accepted Answer

$13$

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(D) The general term $T_{r+1}$ in the expansion of $(3^{1/4} + 7^{1/6})^{144}$ is given by: $T_{r+1} = {}^{144}C_r (3^{1/4})^{144-r} (7^{1/6})^r$ $T_{r+1} = {}^{144}C_r (3)^{\frac{144-r}{4}} (7)^{\frac{r}{6}}$ For the term to be rational,the exponents of $3$ and $7$ must be integers. Thus,$\frac{144-r}{4} = 36 - \frac{r}{4}$ must be an integer,which implies $r$ must be a multiple of $4$. Also,$\frac{r}{6}$ must be an integer,which implies $r$ must be a multiple of $6$. Therefore,$r$ must be a multiple of $	ext{lcm}(4, 6) = 12$. Since $0 \le r \le 144$,the possible values for $r$ are $0, 12, 24, \dots, 144$. This is an arithmetic progression where $a = 0$,$d = 12$,and the last term $l = 144$. Using the formula $l = a + (n-1)d$: $144 = 0 + (n-1)12$ $12 = n - 1$ $n = 13$. Thus,there are $13$ rational terms.

The number of rational terms in the expansion of $(3^{1/4} + 7^{1/6})^{144}$ is

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