Question

The sum of the rational terms in the binomial expansion of \(\left(2^{\frac{1}{2}} + 3^{\frac{1}{5}}\right)^{10}\) is :

• A. 25
• B. 32
• C. 9
• D. 41

Hint:

General term has \(2^{(10-r)/2}\,3^{r/5}\); it is rational only if both exponents are integers.

Answer 1

The Correct Option is D

\(T_{r+1} = \binom{10}{r}\,(2^{1/2})^{10-r}\,(3^{1/5})^{r} = \binom{10}{r}\,2^{(10-r)/2}\,3^{r/5}\).
For a rational term, \(r\) must be a multiple of 5 and \(10-r\) must be even. With \(0 \le r \le 10\): \(r = 0, 5, 10\).
\(r=5\) gives \(10-5 = 5\) (odd), rejected. So \(r = 0\) and \(r = 10\).
\(r=0\): \(\binom{10}{0}\,2^5 = 32\).
\(r=10\): \(\binom{10}{10}\,3^2 = 9\).
Sum \(= 32 + 9 = 41\).
Answer: (4) 41