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(B) The general term in the expansion of $( \sqrt{2} + \sqrt[4]{3} )^{100}$ is given by $T_{r+1} = ^{100}C_r (\sqrt{2})^{100-r} (\sqrt[4]{3})^r$.$T_{r

Vedclass · Accepted Answer

$26$

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(B) The general term in the expansion of $( \sqrt{2} + \sqrt[4]{3} )^{100}$ is given by $T_{r+1} = ^{100}C_r (\sqrt{2})^{100-r} (\sqrt[4]{3})^r$.$T_{r+1} = ^{100}C_r (2)^{50 - r/2} (3)^{r/4}$.For the term to be rational,the exponents of $2$ and $3$ must be integers.Thus,$r/2$ must be an integer and $r/4$ must be an integer.This implies that $r$ must be a multiple of $4$.Since $0 \le r \le 100$,the possible values for $r$ are $0, 4, 8, \dots, 100$.This is an arithmetic progression where $a = 0$,$d = 4$,and the last term $l = 100$.Using the formula $l = a + (n-1)d$,we get $100 = 0 + (n-1)4$.$25 = n - 1$,so $n = 26$.Therefore,there are $26$ rational terms.

Number of rational terms in the expansion of $( \sqrt{2} + \sqrt[4]{3} )^{100}$ is:

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