a: \(\sqrt{1-4a+4a^2}-2a\)
\(=\sqrt{\left(2a-1\right)^2}-2a\)
\(=\left[{}\begin{matrix}2a-1-2a\left(a>=\dfrac{1}{2}\right)\\1-2a-2a\left(a< \dfrac{1}{2}\right)\end{matrix}\right.\)
\(=\left[{}\begin{matrix}-1\left(a>=\dfrac{1}{2}\right)\\1-4a\left(a< \dfrac{1}{2}\right)\end{matrix}\right.\)
b: \(x-2y-\sqrt{x^2-4xy+4y^2}\)
\(=x-2y-\sqrt{\left(x-2y\right)^2}\)
=x-2y-|x-2y|
\(=\left[{}\begin{matrix}x-2y-\left(x-2y\right)\left(x>=2y\right)\\x-2y-\left(2y-x\right)\left(x< 2y\right)\end{matrix}\right.\)
\(=\left[{}\begin{matrix}x-2y-x+2y=0\\x-2y-2y+x=2x-4y\end{matrix}\right.\)
c: \(x^2+\sqrt{x^4-8x^2+16}\)
\(=x^2+\sqrt{\left(x^2-4\right)^2}\)
\(=\left[{}\begin{matrix}x^2+x^2-4\left(x^2>=4\right)\\x^2+4-x^2\left(x^2< 4\right)\end{matrix}\right.=\left[{}\begin{matrix}2x^2-4\left(\left[{}\begin{matrix}x>=2\\x< =-2\end{matrix}\right.\right)\\4\left(-2< x< 2\right)\end{matrix}\right.\)
d: \(2x-1-\dfrac{\sqrt{x^2-10x+25}}{x-5}\)
\(=2x-1-\dfrac{\sqrt{\left(x-5\right)^2}}{x-5}\)
\(=2x-1-\dfrac{\left|x-5\right|}{x-5}\)
\(=\left[{}\begin{matrix}2x-1-\dfrac{x-5}{x-5}\left(x>5\right)\\2x-1-\dfrac{5-x}{x-5}\left(x< 5\right)\end{matrix}\right.=\left[{}\begin{matrix}2x-1-1=2x-2\\2x-1+1=2x\end{matrix}\right.\)
e: \(\dfrac{\sqrt{x^4-4x^2+4}}{x^2-2}=\dfrac{\sqrt{\left(x^2-2\right)^2}}{x^2-2}=\dfrac{\left|x^2-2\right|}{x^2-2}\)
\(=\left[{}\begin{matrix}\dfrac{x^2-2}{x^2-2}\left(x^2>2\right)\\\dfrac{-x^2+2}{x^2-2}\left(x^2< 2\right)\end{matrix}\right.=\left[{}\begin{matrix}1\left(\left[{}\begin{matrix}x>\sqrt{2}\\x< -\sqrt{2}\end{matrix}\right.\right)\\-1\left(-\sqrt{2}< x< \sqrt{2}\right)\end{matrix}\right.\)
f: \(\sqrt{\left(x-4\right)^2}+\dfrac{x-4}{\sqrt{x^2-8x+16}}=\left|x-4\right|+\dfrac{x-4}{\left|x-4\right|}\)
\(=\left[{}\begin{matrix}x-4+\dfrac{x-4}{x-4}\left(x>4\right)\\4-x+\dfrac{x-4}{4-x}\left(x< 4\right)\end{matrix}\right.=\left[{}\begin{matrix}x-4+1=x-3\\4-x-1=3-x\end{matrix}\right.\)
