\(x\ge0\)

\(\left(\sqrt{x}-4\right)\left(|x+2|-1\right)\left(x^2-3\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x}-4=0\Rightarrow x=16\left(tm\right)\\|x+2|-1=0\Leftrightarrow\left[{}\begin{matrix}x+2=1\Rightarrow x=-1\\x+2=-1\Rightarrow x=-3\end{matrix}\right.\\x^2-3=0\Rightarrow x=\pm\sqrt{3}\end{matrix}\right.\)

\(B=\dfrac{1}{3}+\dfrac{1}{3^2}-\dfrac{1}{3^3}+...+\dfrac{1}{3^{50}}-\dfrac{1}{3^{51}}\)

\(=\dfrac{1}{\left(-3\right)}+\dfrac{1}{\left(-3\right)^2}+\dfrac{1}{\left(-3\right)^3}+...+\dfrac{1}{\left(-3\right)^{50}}+\dfrac{1}{\left(-3\right)^{51}}-\dfrac{1}{3}\)

\(=\dfrac{1}{\left(3\right)^2}+\dfrac{1}{\left(3\right)^3}+...+\dfrac{1}{\left(-3\right)^{51}}+\dfrac{1}{\left(-3\right)^{52}}\)

\(\Rightarrow\dfrac{4}{3}B=\dfrac{1}{-3}-\dfrac{1}{\left(-3\right)^{52}}=\dfrac{-3^{51}-1}{3^{52}}\Rightarrow B=\dfrac{-3^{51}-1}{4.3^{51}}\)

\(P=\dfrac{1}{x^2+y^2}+\dfrac{2}{xy}+4xy=\left(\dfrac{1}{x^2+y^2}+\dfrac{1}{2xy}\right)+\left(4xy+\dfrac{1}{4xy}\right)+\dfrac{5}{4xy}\)

\(=\left(\dfrac{2xy}{x^2+y^2}+\dfrac{x^2+y^2}{2xy}+2\right)+\left(4xy+\dfrac{1}{4xy}\right)+\dfrac{5}{4xy}\)

+) \(\dfrac{2xy}{x^2+y^2}+\dfrac{x^2+y^2}{2xy}\ge2\)

+) \(4xy+\dfrac{1}{4xy}\ge2\)

+) \(\dfrac{1}{xy}\ge\dfrac{4}{\left(x+y\right)^2}\Leftrightarrow\dfrac{1}{xy}\ge\dfrac{4}{1}\Rightarrow\dfrac{5}{4xy}\ge5\)

\(\Rightarrow minP=11,t\) khi \(x=y=\dfrac{1}{2}\)

\(A=\dfrac{3}{1^2.2^2}+\dfrac{5}{2^2.3^2}+\dfrac{7}{3^2.4^2}+...+\dfrac{19}{9^2.10^2}\)

\(=\dfrac{2^2-1^2}{1^2.2^2}+\dfrac{3^2-2^2}{2^2.3^2}+\dfrac{4^2-3^2}{3^2.4^2}+...+\dfrac{10^2-9^2}{9^2.10^2}\)

\(=\dfrac{1}{1^2}-\dfrac{1}{2^2}+\dfrac{1}{2^2}-\dfrac{1}{3^2}+\dfrac{1}{3^2}-\dfrac{1}{4^2}+...+\dfrac{1}{9^2}-\dfrac{1}{10^2}\)

\(=1-\dfrac{1}{10^2}< 1\)

\(\left(3x-5\right)^{2006}+\left(y^2-1\right)^{2008}+\left(x-z\right)^{2100}=0\)

\(\Rightarrow\left\{{}\begin{matrix}\left(3x-5\right)^{2006}=0\\\left(y^2-1\right)^{2008}=0\\\left(x-z\right)^{2100}=0\end{matrix}\right.\Leftrightarrow x=z=\dfrac{5}{3}\)

\(\Rightarrow\left[{}\begin{matrix}y=1\\y=-1\end{matrix}\right.\)