a) \(...\Leftrightarrow H=\sqrt{ab}\left(\sqrt{a^2}+\sqrt{b^2}-\sqrt{ab}\right):\sqrt{ab}\left(ab\ge0\right)\)
\(\Leftrightarrow H=\left|a\right|+\left|b\right|-\sqrt{ab}=\left[{}\begin{matrix}a+b-\sqrt{ab}\left(a\ge0;b\ge0\right)\\-a-b-\sqrt{ab}\left(a< 0;b< 0\right)\end{matrix}\right.\)
b) \(...\Leftrightarrow E=\sqrt{\dfrac{\left(x-3\right)^2}{\left(x-3\right)^2}}\left(x>3\right)\)
\(\Leftrightarrow E=\sqrt{\left(\dfrac{x-3}{x-3}\right)^2}=\left|\dfrac{x-3}{x-3}\right|=\dfrac{x-3}{x-3}=1\)
c) \(...\Leftrightarrow F=\left(x-y\right)\dfrac{\sqrt{xy}}{\left|x-y\right|}=\left(x-y\right)\dfrac{\sqrt{xy}}{-\left(x-y\right)}=-\sqrt{xy}\left(x< y< 0\right)\)
d) \(...\Leftrightarrow T=\dfrac{\left(x-1\right)^2}{\left|2-x\right|}+\dfrac{x^2-2}{x-2}\)
\(\Leftrightarrow T=\dfrac{\left(x-1\right)^2}{-\left(x-2\right)}+\dfrac{x^2-2}{x-2}\left(x< 2\right)\)
\(\Leftrightarrow T=\dfrac{-\left(x^2-2x+1\right)+x^2-2}{x-2}=\dfrac{-x^2+2x-1+x^2-2}{x-2}\)
\(\Leftrightarrow T=\dfrac{2x-3}{x-2}\)
