The number of irrational terms in the binomia | vedclass

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(D) The general term of $(3^{1/5} + 7^{1/3})^{100}$ is given by $T_{r+1} = {}^{100}C_{r} (3^{1/5})^{100-r} (7^{1/3})^{r}$.For the term to be rational,

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$94$

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(D) The general term of $(3^{1/5} + 7^{1/3})^{100}$ is given by $T_{r+1} = {}^{100}C_{r} (3^{1/5})^{100-r} (7^{1/3})^{r}$.For the term to be rational,the exponents of $3$ and $7$ must be integers.Thus,$\frac{100-r}{5}$ must be an integer,which implies $r$ must be a multiple of $5$.Also,$\frac{r}{3}$ must be an integer,which implies $r$ must be a multiple of $3$.Therefore,$r$ must be a multiple of $	ext{lcm}(5, 3) = 15$.Given $0 \le r \le 100$,the possible values for $r$ are $0, 15, 30, 45, 60, 75, 90$.There are $7$ such values,so there are $7$ rational terms.The total number of terms in the expansion is $100 + 1 = 101$.Number of irrational terms = $	ext{Total terms} - 	ext{Rational terms} = 101 - 7 = 94$.

The number of irrational terms in the binomial expansion of $(3^{1/5} + 7^{1/3})^{100}$ is

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