The number of integral terms in the expansion of -3-5256 is

General term, T_r+1 = ^256C_r(√(3))^256 r(√(5))^r =^256C_r(3)^(256 r)/2 (5)^r/8 Now this term is an integer if 256 r2, r8 will be an integer, for which r=0 ...

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Standard XII

Mathematics

General Term of Binomial Expansion

The number of...

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The number of integral terms in the expansion of ${(\sqrt{3} + \sqrt[8]{5})}^{256}$ is

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Solution

General term,
$T_{r + 1} =^{256} C_{r} (\sqrt{3})^{256 - r} (\sqrt[8]{5})^{r}$
$=^{256} C_{r} (3)^{(256 - r) / 2} (5)^{r / 8}$
Now this term is an integer if $\frac{256 - r}{2}, \frac{r}{8}$ will be an integer, for which $r = 0, 8, 16, \dots, 256$
For A.P., $0, 8, 16, 24, . . ., 256$ number of terms will be
$= \frac{256}{8} + 1 = 33$
Hence, there are $33$ integral terms.

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General Term of Binomial Expansion

Standard XII Mathematics

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The number of integral terms in the expansion of (√3+8√5)256 is

​​​General term, Tr+1=256Cr(√3)256−r(8√5)r =256Cr(3)(256−r)/2(5)r/8 Now this term is an integer if 256−r2,r8 will be an integer, for which r=0,8,16,…,256 For A.P., 0,8,16,24,...,256 number of terms will be =2568+1=33 Hence, there are 33 integral terms.

The number of integral terms in the expansion of ${(\sqrt{3} + \sqrt[8]{5})}^{256}$ is