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Let $𝕂$ be a quadratic field over the rational field and $a_{𝕂}\left(n\right)$ be the number of nonzero integral ideals with norm $n$. We establish Erdős-Kac type theorems weighted by $a_{𝕂}{\left(n\right)}^l$ and $a_{𝕂}{\left(n^2\right)}^l$ of quadratic field in short intervals with $l\in {\mathbb{Z}}^{+}$. We also get asymptotic formulae for the average behavior of $a_{𝕂}{\left(n\right)}^l$ and $a_{𝕂}{\left(n^2\right)}^l$ in short intervals.