The total number of irrational terms in the b | vedclass

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(D) The general term in the expansion of $(7^{1/5} - 3^{1/10})^{60}$ is given by $T_{r+1} = ^{60}C_{r} (7^{1/5})^{60-r} (-3^{1/10})^{r}$.Simplifying t

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$54$

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(D) The general term in the expansion of $(7^{1/5} - 3^{1/10})^{60}$ is given by $T_{r+1} = ^{60}C_{r} (7^{1/5})^{60-r} (-3^{1/10})^{r}$.Simplifying the exponents,we get $T_{r+1} = ^{60}C_{r} (7)^{(60-r)/5} (-1)^{r} (3)^{r/10} = ^{60}C_{r} (-1)^{r} (7)^{12 - r/5} (3)^{r/10}$.For the term to be rational,the exponents of $7$ and $3$ must be integers.This requires $r/5$ and $r/10$ to be integers,which implies $r$ must be a multiple of $10$.Since $0 \le r \le 60$,the possible values for $r$ are $0, 10, 20, 30, 40, 50, 60$.There are $7$ such values,so there are $7$ rational terms.The total number of terms in the expansion is $60 + 1 = 61$.Therefore,the number of irrational terms is $61 - 7 = 54$.

The total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is:

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