Given a positive rational integer \(n\) and its prime factorization \(n = \prod_{i=1}^k p_i^{\alpha _i}\), the irrational factor of \(n\) is defined as \(I(n) = \prod_{i=1}^k p_i^{1/{\alpha _i}}\). If \(G(n): = \prod_{i=1}^n I(i)^{1/n}\), then the following statements are proven to be true: 1. There is an absolute constant \(c_1>0\) such that \(G(n) = c_1n + O(\sqrt n)\), for any \(n \geq 1\). 2. There is an absolute constant \(c_2>0\) such that \[ \sum_{n\leq x}I(n) = c_2x^2 + O(x^{3/2}(\log x)^{9/4}). \] 3. There is an absolute constant \(c_3>0\) such that \[ \sum_{n\leq x}\left( 1-\frac{n}{x}\right) I(n) = \frac{c_2}{3}x^2 + O\left( x^{3/2}e^{-c_3(\log x)^{3/5}(\log\log x)^{-1/5}}\right). \]