\(A=\left(a+b+c\right)^3+\left(a-b-c\right)^3-6a\left(b+c\right)^2\)

\(=\left[a+\left(b+c\right)\right]^3+\left[a-\left(b+c\right)\right]^3-6a\left(b+c\right)^2\)

\(=a^3+3a^2\left(b+c\right)+3a\left(b+c\right)^2+\left(b+c\right)^3+a^3-3a^2\left(b+c\right)+3a\left(b+c\right)^2-\left(b+c\right)^3-6a\left(b+c\right)^2\)

\(=2a^3\)

\(M=\sqrt{\left(a-3\right)^2}-\dfrac{\sqrt{\left(a-3\right)^2}}{a-3}=\left|a-3\right|-\dfrac{\left|a-3\right|}{a-3}\)  

+) Với \(a\ge3\) \(\Rightarrow M=a-3-1=a-4\)

+) Với \(a< 3\) \(\Rightarrow M=3-a+1=4-a\)

\(a,=\left|2-\sqrt{3}\right|=2-\sqrt{3}\\ b,=\left|3-\sqrt{11}\right|=\sqrt{11}-3\\ c,=2\left|a\right|=2a\\ d,=3\left|a-2\right|=3\left(2-a\right)\left(a< 0\Leftrightarrow a-2< 0\right)\)

Bài 1:

b: \(B=1:\left(\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}+1}{x+\sqrt{x}+1}-\frac{\sqrt{x}+1}{x-1}\right)\)

\(=1:\left(\frac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\sqrt{x}+1}{x+\sqrt{x}+1}-\frac{1}{\sqrt{x}-1}\right)\)

\(=1:\frac{x+2+\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(=1:\frac{x+2+x-1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=1:\frac{x-\sqrt{x}}{\left.\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)\right.}\)

\(=\frac{\left(\sqrt{x}\right)-1)\left(x+\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{x+\sqrt{x}+1}{\sqrt{x}}\)

Bài 2:

a: \(\left(\frac{6a+1}{a^2-6a}+\frac{6a-1}{a^2+6a}\right)\cdot\frac{a^2-36}{a^2+1}\)

\(=\left(\frac{6a+1}{a\left(a-6\right)}+\frac{6a-1}{a\left(a+6\right)}\right)\cdot\frac{\left(a-6\right)\left(a+6\right)}{a^2+1}\)

\(=\frac{\left(6a+1\right)\left(a+6\right)+\left(6a-1\right)\left(a-6\right)}{a\left(a-6\right)\left(a+6\right)}\cdot\frac{\left(a-6\right)\left(a+6\right)}{a^2+1}\)

\(=\frac{6a^2+37a+6+6a^2-37a+6}{a}\cdot\frac{1}{a^2+1}=\frac{12a^2+12}{a\left(a^2+1\right)}\)

\(=\frac{12\left(a^2+1\right)}{a\left(a^2+1\right)}=\frac{12}{a}\)

b: \(\left(\frac{1}{x}+\frac{1}{y}\right)\cdot\frac{1}{x+2\sqrt{xy}+y}+\frac{2}{\left.\left(\sqrt{x}+\sqrt{y}\right)^3\right.}\cdot\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\)

\(=\frac{x+y}{xy}\cdot\frac{1}{\left(\sqrt{x}+\sqrt{y}\right)^2}+\frac{2}{\left(\sqrt{x}+\sqrt{y}\right)^3}\cdot\frac{\sqrt{x}+\sqrt{y}}{\sqrt{xy}}\)

\(=\frac{x+y}{xy\left(\sqrt{x}+\sqrt{y}\right)^2}+\frac{2}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)^2}=\frac{x+y+2\sqrt{xy}}{xy\left(\sqrt{x}+\sqrt{y}\right)^2}=\frac{1}{xy}\)

\(\frac{\sqrt{x}-\sqrt{y}}{xy\cdot\sqrt{xy}}:\left\lbrack\left(\frac{1}{x}+\frac{1}{y}\right)\cdot\frac{1}{x+2\sqrt{xy}+y}+\frac{2}{\left.\left(\sqrt{x}+\sqrt{y}\right)^3\right.}\cdot\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\right\rbrack\)

\(=\frac{\sqrt{x}-\sqrt{y}}{xy\cdot\sqrt{xy}}:\frac{1}{xy}\)

\(=\frac{\sqrt{x}-\sqrt{y}}{xy\cdot\sqrt{xy}}\cdot xy=\frac{\sqrt{x}-\sqrt{y}}{\sqrt{xy}}\)

Bài 1:

b: \(B=1:\left(\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}+1}{x+\sqrt{x}+1}-\frac{\sqrt{x}+1}{x-1}\right)\)

\(=1:\left(\frac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\sqrt{x}+1}{x+\sqrt{x}+1}-\frac{1}{\sqrt{x}-1}\right)\)

\(=1:\frac{x+2+\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(=1:\frac{x+2+x-1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=1:\frac{x-\sqrt{x}}{\left.\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)\right.}\)

\(=\frac{\left(\sqrt{x}\right)-1)\left(x+\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{x+\sqrt{x}+1}{\sqrt{x}}\)

Bài 2:

a: \(\left(\frac{6a+1}{a^2-6a}+\frac{6a-1}{a^2+6a}\right)\cdot\frac{a^2-36}{a^2+1}\)

\(=\left(\frac{6a+1}{a\left(a-6\right)}+\frac{6a-1}{a\left(a+6\right)}\right)\cdot\frac{\left(a-6\right)\left(a+6\right)}{a^2+1}\)

\(=\frac{\left(6a+1\right)\left(a+6\right)+\left(6a-1\right)\left(a-6\right)}{a\left(a-6\right)\left(a+6\right)}\cdot\frac{\left(a-6\right)\left(a+6\right)}{a^2+1}\)

\(=\frac{6a^2+37a+6+6a^2-37a+6}{a}\cdot\frac{1}{a^2+1}=\frac{12a^2+12}{a\left(a^2+1\right)}\)

\(=\frac{12\left(a^2+1\right)}{a\left(a^2+1\right)}=\frac{12}{a}\)

b: \(\left(\frac{1}{x}+\frac{1}{y}\right)\cdot\frac{1}{x+2\sqrt{xy}+y}+\frac{2}{\left.\left(\sqrt{x}+\sqrt{y}\right)^3\right.}\cdot\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\)

\(=\frac{x+y}{xy}\cdot\frac{1}{\left(\sqrt{x}+\sqrt{y}\right)^2}+\frac{2}{\left(\sqrt{x}+\sqrt{y}\right)^3}\cdot\frac{\sqrt{x}+\sqrt{y}}{\sqrt{xy}}\)

\(=\frac{x+y}{xy\left(\sqrt{x}+\sqrt{y}\right)^2}+\frac{2}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)^2}=\frac{x+y+2\sqrt{xy}}{xy\left(\sqrt{x}+\sqrt{y}\right)^2}=\frac{1}{xy}\)

\(\frac{\sqrt{x}-\sqrt{y}}{xy\cdot\sqrt{xy}}:\left\lbrack\left(\frac{1}{x}+\frac{1}{y}\right)\cdot\frac{1}{x+2\sqrt{xy}+y}+\frac{2}{\left.\left(\sqrt{x}+\sqrt{y}\right)^3\right.}\cdot\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\right\rbrack\)

\(=\frac{\sqrt{x}-\sqrt{y}}{xy\cdot\sqrt{xy}}:\frac{1}{xy}\)

\(=\frac{\sqrt{x}-\sqrt{y}}{xy\cdot\sqrt{xy}}\cdot xy=\frac{\sqrt{x}-\sqrt{y}}{\sqrt{xy}}\)

Bài 3:

a: \(A=\left(\frac{x}{x-y}-\frac{y}{x+y}\right):\left(\frac{x+y}{x-y}-\frac{2xy}{x^2-y^2}\right)\)

\(=\frac{x\left(x+y\right)-y\left(x-y\right)}{\left(x-y\right)\left(x+y\right)}:\frac{\left(x+y\right)^2-2xy}{\left(x-y\right)\left(x+y\right)}\)

\(=\frac{x^2+xy-xy+y^2}{\left(x-y\right)\left(x+y\right)}\cdot\frac{\left(x-y\right)\left(x+y\right)}{x^2+2xy+y^2-2xy}=\frac{x^2+y^2}{x^2+y^2}=1\)

b: \(\left(\frac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right)\left(\frac{1-\sqrt{a}}{1-a}\right)^2\)

\(=\left(\frac{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}+a\right)}{1-\sqrt{a}}+\sqrt{a}\right)\cdot\left(\frac{1-\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}\right)^2\)

\(=\left(a+\sqrt{a}+1+\sqrt{a}\right)\cdot\frac{1}{\left(\sqrt{a}+1\right)^2}=\frac{\left(\sqrt{a}+1\right)^2}{\left(\sqrt{a}+1\right)^2}=1\)

Bài 3:

\(a,A=\dfrac{x^2+xy-xy+y^2}{\left(x-y\right)\left(x+y\right)}:\dfrac{x^2+2xy+y^2-2xy}{\left(x-y\right)\left(x+y\right)}\\ A=\dfrac{x^2+y^2}{\left(x-y\right)\left(x+y\right)}\cdot\dfrac{\left(x-y\right)\left(x+y\right)}{x^2+y^2}=1\\ b,=\left[\dfrac{\left(1-\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}{1-\sqrt{a}}+\sqrt{a}\right]\left[\dfrac{1-\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}\right]^2\\ =\left(a+2\sqrt{a}+1\right)\left(\dfrac{1}{\sqrt{a}+1}\right)^2\\ =\left(\sqrt{a}+1\right)^2\cdot\dfrac{1}{\left(\sqrt{a}+1\right)^2}=1\)

a: \(B=\left(\frac{a-b}{a^2+ab}-\frac{a}{b^2+ab}\right):\left(\frac{b^3}{a^3-ab^2}+\frac{1}{a+b}\right)\)

\(=\left(\frac{a-b}{a\left(a+b\right)}-\frac{a}{b\left(a+b\right)}\right):\left(\frac{b^3}{a\left(a^2-b^2\right)}+\frac{1}{a+b}\right)\)

\(=\frac{b\left(a-b\right)-a^2}{ab\left(a+b\right)}:\frac{b^3+a\left(a-b\right)}{a\left(a-b\right)\left(a+b\right)}\)

\(=\frac{ba-b^2-a^2}{ab\left(a+b\right)}\cdot\frac{a\left(a-b\right)\left(a+b\right)}{b^3+a^2-ab}\)

\(=\frac{-a^2+ab-b^2}{b}\cdot\frac{a-b}{b^3+a^2-ab}=\frac{-\left(a-b\right)\left(a^2-ab+b^2\right)}{b\left(b^3+a^2-ab\right)}\)

b: \(C=\frac{a}{b-2}-\left\lbrack\frac{\left(a^2+2a+1\right)}{b^2-4}\right\rbrack\cdot\frac{b+2}{a+1}\)

\(=\frac{a}{b-2}-\frac{\left(a+1\right)^2}{\left(b-2\right)\left(b+2\right)}\cdot\frac{b+2}{a+1}=\frac{a}{b-2}-\frac{a+1}{b-2}\)

\(=-\frac{1}{b-2}\)

a: \(B=\left(\frac{a-b}{a^2+ab}-\frac{a}{b^2+ab}\right):\left(\frac{b^3}{a^3-ab^2}+\frac{1}{a+b}\right)\)

\(=\left(\frac{a-b}{a\left(a+b\right)}-\frac{a}{b\left(a+b\right)}\right):\left(\frac{b^3}{a\left(a^2-b^2\right)}+\frac{1}{a+b}\right)\)

\(=\frac{b\left(a-b\right)-a^2}{ab\left(a+b\right)}:\frac{b^3+a\left(a-b\right)}{a\left(a-b\right)\left(a+b\right)}\)

\(=\frac{ba-b^2-a^2}{ab\left(a+b\right)}\cdot\frac{a\left(a-b\right)\left(a+b\right)}{b^3+a^2-ab}\)

\(=\frac{-a^2+ab-b^2}{b}\cdot\frac{a-b}{b^3+a^2-ab}=\frac{-\left(a-b\right)\left(a^2-ab+b^2\right)}{b\left(b^3+a^2-ab\right)}\)

b: \(C=\frac{a}{b-2}-\left\lbrack\frac{\left(a^2+2a+1\right)}{b^2-4}\right\rbrack\cdot\frac{b+2}{a+1}\)

\(=\frac{a}{b-2}-\frac{\left(a+1\right)^2}{\left(b-2\right)\left(b+2\right)}\cdot\frac{b+2}{a+1}=\frac{a}{b-2}-\frac{a+1}{b-2}\)

\(=-\frac{1}{b-2}\)