11.28 Simplify the expression

$$
\frac{m}{m^{2}+1} \sqrt{1+\left(\frac{m^{2}-1}{2 m}\right)^{2}}
$$
11.28 For simplification, we will use formulas (1.9), (1.10):

$$
\begin{aligned}
& A=\frac{m}{m^{2}+1} \sqrt{1+\frac{m^{4}-2 m^{2}+1}{4 m^{2}}}=\frac{m}{m^{2}+1} \sqrt{\frac{m^{4}+2 m^{2}+1}{4 m^{2}}}= \\
& =\frac{m}{m^{2}+1} \sqrt{\frac{\left(m^{2}+1\right)^{2}}{4 m^{2}}}=\frac{m\left|m^{2}+1\right|}{\left(m^{2}+1\right) 2|m|}=\frac{m}{2|m|}
\end{aligned}
$$

since $m^{2}+1>0$ for any $m$, i.e., $\left|m^{2}+1\right|=m^{2}+1$. Now, using the definition of the modulus and considering that $m \neq 0$, we get: $A=0.5$, if $m>0 ; A=-0.5$, if $m<0$.