Question

The number of irrational terms in the expansion of ${(5^{\frac{1}{3}}+2^{\frac{1}{4}})}^{100}$ is

• A. 94
• B. 92
• C. 93
• D. 91

Answer 1

The correct option is B

92

${(5^{\frac{1}{3}}+2^{\frac{1}{4}})}^{100}$ = ${(2^{\frac{1}{4}}+5^{\frac{1}{3}})}^{100}$

$T_{r+1}$ = ${}^{100}{C}_r$ ${\left(2^{\frac{1}{4}}\right)}^{100-r}$ . ${\left(5^{\frac{1}{3}}\right)}^r$ = ${}^{100}{C}_r{2}^{25-\frac{r}{4}}{.5}^{\frac{r}{3}}$

For rational terms, 'r' should be divisible by 12.

$\therefore$ No.of rational terms = 9

$\therefore$ No.of irrational terms = 101 - 9 = 92