Question

The number of rational terms in the binomial expansion of \(\left(4^{\frac{1}{4}} + 5^{\frac{1}{6}}\right)^{120}\) is _______________.

Hint:

Write the general term, note \(4^{1/4} = 2^{1/2}\), and require both exponents to be integers.

Answer 1

Correct Answer: 21

General term: \(T_{r+1} = {}^{120}C_r\,(4^{1/4})^{120-r}(5^{1/6})^r = {}^{120}C_r\,2^{\frac{120-r}{2}}\,5^{\frac{r}{6}}\).
It is rational iff \(\frac{120-r}{2}\) and \(\frac{r}{6}\) are both integers.
\(\frac{r}{6}\) integer means r is a multiple of 6; then r is even so \(\frac{120-r}{2}\) is automatically an integer.
r = 0, 6, 12, ..., 120 (r = 6k, k = 0, 1, ..., 20), giving 21 values.
Answer: 21