The number of rational terms in the expansion | vedclass

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(A) The general term of the expansion $(3^{1/8} + 5^{1/3})^{400}$ is given by $T_{r+1} = ^{400}C_r (3^{1/8})^{400-r} (5^{1/3})^r$.This simplifies to $

Vedclass · Accepted Answer

$17$

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(A) The general term of the expansion $(3^{1/8} + 5^{1/3})^{400}$ is given by $T_{r+1} = ^{400}C_r (3^{1/8})^{400-r} (5^{1/3})^r$.This simplifies to $T_{r+1} = ^{400}C_r \cdot 3^{(50 - r/8)} \cdot 5^{r/3}$.For the term to be rational,the exponents of $3$ and $5$ must be integers.Thus,$r$ must be divisible by $8$ and $r$ must be divisible by $3$.This implies $r$ must be a multiple of $	ext{lcm}(8, 3) = 24$.Since $0 \le r \le 400$,the possible values for $r$ are $0, 24, 48, \dots, 24k$ where $24k \le 400$.Solving $24k \le 400$ gives $k \le 16.66$,so $k$ can range from $0$ to $16$.The number of such values is $16 - 0 + 1 = 17$.

The number of rational terms in the expansion of $(3^{1/8} + 5^{1/3})^{400}$ is

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