Question

Find the total number of integers term in the expansion of :

${\left(9^{\frac{1}{4}}+8^{\frac{1}{6}}\right)}^{500}$

Answer 1

$\begin{array}{c}{\left(9^{\frac{1}{4}}+8^{\frac{1}{6}}\right)}^{500}={\left(3^{\frac{1}{2}}+2^{\frac{1}{2}}\right)}^{500}\\ {=}^{500}{C}_0{\left(3^{\frac{1}{2}}\right)}^{500}{+}^{500}{C}_1{\left(3^{\frac{1}{2}}\right)}^{499}.\left(2^{\frac{1}{2}}\right)+......{+}^{500}{C}_{500}{\left(2^{\frac{1}{2}}\right)}^{500}\\ Totalterm=501\\ Integersterms=501-250\\ =251\end{array}$