The number of integral terms in the expansion of ${(5^{1 / 2} + 7^{1 / 8})}^{1024}$ is................

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Solution

${(5^{\frac{1}{2}} + 7^{\frac{1}{8}})}^{1024} G e n e r a l T e r m f o r t h e a b o v e s e r i e s i s T_{r + 1} =^{1024} C_{r} {(5^{\frac{1}{2}})}^{r} ({7^{\frac{1}{8}})}^{1024 - r} F o r t h e t e r m t o b e i n t e r g r a l r m u s t b e t h e i n t e r g a l m u l i t i p l e o f 8 S o r = 0, 8......., 1024 1024 = 0 + (n - 1) \times 8 n = 129 S o t h e n u m b e r o f i n t e r g a l t e r m s a r e 129.$

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