Question

The number of rational terms in the binomial expansion of $ \left( 5^{1/2} + 7^{1/8} \right)^{1016} $ is:

• A. 129
• B. 128
• C. 127
• D. 130

Hint:

For binomial expansions, check the conditions for rational terms by ensuring that all exponents are integers.

Answer 1

The Correct Option is B

The general term in the binomial expansion of \( (a + b)^n \) is given by: \[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \] Here, \( a = 5^{1/2} \), \( b = 7^{1/8} \), and \( n = 1016 \).
The general term becomes: \[ T_{r+1} = \binom{1016}{r} \left(5^{1/2}\right)^{1016-r} \left(7^{1/8}\right)^r = \binom{1016}{r} 5^{(1016-r)/2} 7^{r/8} \] For the term to be rational, both exponents \( (1016 - r)/2 \) and \( r/8 \) must be integers.
Therefore, \( r \) must be a multiple of 8 and even.
The number of multiples of 8 in the range \( 0 \leq r \leq 1016 \) is: \[ \frac{1016}{8} + 1 = 128 \] Thus, the number of rational terms is 128.