Mathematics - Binomial Theorem Question with Solution | TestHub

Consider the expression:

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

The general term, $T_{r+1}$, is given by:

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

For the terms to be rational, the exponent $r$ must be divisible by both 4 and 3. Therefore, $r$ must be divisible by 12.

The possible values for $r$ are $0,12,24,36,48,60,72,84,96$.

The number of rational terms is 9.

The total number of terms in the expansion is $100+1=101$.

Thus, the number of irrational terms is $101-9=92$.