To study the irreducible root systems, the homology groups of the real toric varieties associated with the Weyl groups have been studied. The $ZZ_{2}$-homology of the real toric variety associated with the Weyl group is naturally determined by the combinatorial structure of the Weyl group. However, the rational homology is more complicated. Since 2012, the rational Betti numbers for the classical types and several exceptional types $F_{4}$, $G_{2}$ and $E_{6}$ have been computed by many mathematicians and it has remained only for two types $E_{7}$ and $E_{8}$. In this talk, we will discuss the rational Betti numbers of the real toric varieties associated with the Weyl group of types $E_{7}$ and $E_{8}$. This is a joint work with Suyoung Choi and Younghan Yoon.