8. For a function $f(x)$ defined on $\mathbf{R}$, for any $a, b \in \mathbf{R}$, it holds that $f(a \cdot f(b))=a b$, then $\sqrt{f^{2}(2009)}$ $=$ $\qquad$ .
8. 2009 Detailed Explanation: Taking $a=1$ yields $f(f(b))=b \quad$ (1), taking $a=b$ yields $f(b \cdot f(b))=b^{2}$ (2), taking $a=f(b)$ yields $f\left(f^{2}(b)\right)=b \cdot f(b)$ (3), thus $f\left(f\left(f^{2}(b)\right)\right)=f(b \cdot f(b))$ (4), from (1) we get $f\left(f\left(f^{2}(b)\right)\right)=f^{2}(b)$, from (4) we get $f\left(f\left(f^{2}(b)\right)\right)=f(b \cdot f(b))$, so $f^{2}(b)=b^{2}$, then $|f(b)|=|b|$, hence $\sqrt{f^{2}(2009)}=2009$.