The number of irrational terms in the expansi | vedclass

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(B) The general term in the expansion of $(3^{\frac{1}{8}}+5^{\frac{1}{4}})^{84}$ is given by $T_{r+1} = {}^{84}C_{r} (3^{\frac{1}{8}})^{84-r} (5^{\fr

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$74$

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(B) The general term in the expansion of $(3^{\frac{1}{8}}+5^{\frac{1}{4}})^{84}$ is given by $T_{r+1} = {}^{84}C_{r} (3^{\frac{1}{8}})^{84-r} (5^{\frac{1}{4}})^{r} = {}^{84}C_{r} \cdot 3^{\frac{84-r}{8}} \cdot 5^{\frac{r}{4}}$.For the term to be rational, the exponents of $3$ and $5$ must be integers.Thus, $\frac{84-r}{8}$ must be an integer and $\frac{r}{4}$ must be an integer.From $\frac{r}{4} = k$, we have $r = 4k$, where $0 \le r \le 84$.Substituting $r = 4k$ into $\frac{84-4k}{8} = \frac{21-k}{2}$, we see that $21-k$ must be divisible by $2$, which means $k$ must be odd.Since $0 \le 4k \le 84$, we have $0 \le k \le 21$.The odd values for $k$ are $1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21$.There are $11$ such values of $k$, so there are $11$ rational terms.The total number of terms in the expansion is $84+1 = 85$.Therefore, the number of irrational terms is $85 - 11 = 74$.

The number of irrational terms in the expansion of $(3^{\frac{1}{8}}+5^{\frac{1}{4}})^{84}$ is

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