First slide

Binomial theorem for positive integral Index

Question

$T h e n u m b e r o f i n t e g r a l t e r m s i n t h e e x p a n s i o n o f {(\sqrt{3} + \sqrt[8]{5})}^{256} i s$

Moderate

A

32
B

33
C

34
D

35

Solution

$\begin{matrix} General term of the binomial expansion {(\sqrt{3} + \sqrt[8]{5})}^{256} is \\ T_{r + 1} = C (256, r) {(\sqrt{3})}^{256 - r} {(\sqrt[8]{5})}^{r} \\ = C (256, r) 3^{128 - \frac{r}{2}} 5^{\frac{r}{8}} \end{matrix}$
$T e r m s a r e i n t e g e r s, t h e n t h e v a l u e o f r s u c h t h a t i t i s m u l t i p l e o f 8 a n d l e s s t h a n o r e q u a l t o 256 H e n c e, t h e p o s s i b l e v a l u e s o f r a r e 0, 8, 16, . . . 256 . T h e n u m b e r o f v a l u e o f r i s 33 . h e n c e t h e n u m b e r o f r a t i o n a l t e r m s i s 33 .$

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