Lời giải:

\(P=\frac{a^4-a-b^4+b}{(b^3-1)(a^3-1)}+\frac{2(a-b)}{a^2b^2+3}\)

\(=\frac{(a^4-b^4)-(a-b)}{a^3b^3-(a^3+b^3)+1}+\frac{2(a-b)}{a^2b^2+3}=\frac{(a-b)[(a+b)(a^2+b^2)-1]}{a^3b^3-[(a+b)^3-3ab(a+b)]+1}+\frac{2(a-b)}{a^2b^2+3}\)

\(=\frac{(a-b)[(a^2+b^2)-(a+b)^2]}{a^3b^3-[1-3ab]+1}+\frac{2(a-b)}{a^2b^2+3}=\frac{-2ab(a-b)}{a^3b^3+3ab}+\frac{2(a-b)}{a^2b^2+3}\)

\(=\frac{-2(a-b)}{a^2b^2+3}+\frac{2(a-b)}{a^2b^2+3}=0\)

Lời giải:

\(P=\frac{a^4-a-b^4+b}{(b^3-1)(a^3-1)}+\frac{2(a-b)}{a^2b^2+3}\)

\(=\frac{(a^4-b^4)-(a-b)}{a^3b^3-(a^3+b^3)+1}+\frac{2(a-b)}{a^2b^2+3}=\frac{(a-b)[(a+b)(a^2+b^2)-1]}{a^3b^3-[(a+b)^3-3ab(a+b)]+1}+\frac{2(a-b)}{a^2b^2+3}\)

\(=\frac{(a-b)[(a^2+b^2)-(a+b)^2]}{a^3b^3-[1-3ab]+1}+\frac{2(a-b)}{a^2b^2+3}=\frac{-2ab(a-b)}{a^3b^3+3ab}+\frac{2(a-b)}{a^2b^2+3}\)

\(=\frac{-2(a-b)}{a^2b^2+3}+\frac{2(a-b)}{a^2b^2+3}=0\)

Giải

a, 2A+3B=0 <=> \(\dfrac{10}{2m+1}+\dfrac{12}{2m-1}=0\)

<=>10(2m-1)+ 12(2m+1) =0

<=> 44m +2 =0 

<=> m=-1/22

b, AB= A+B <=> \(\dfrac{20}{\left(2m-1\right)\left(2m+1\right)}=\dfrac{5}{2m+1}+\dfrac{4}{2m-1}\)

<=> 20 = 5(2m -1) + 4(2m+1) 

<=> 20 = 18m - 1

<=> m=7/6

\(a,\)

\(B=\dfrac{x-1}{x+1}-\dfrac{x+1}{x-1}-\dfrac{4}{1-x^2}\) (Điều kiện xác định: \(x\ne\pm1\))

\(=\dfrac{\left(x-1\right)^2}{\left(x+1\right)\left(x-1\right)}-\dfrac{\left(x+1\right)^2}{\left(x+1\right)\left(x-1\right)}+\dfrac{4}{\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{x^2-2x+1-\left(x^2+2x+1\right)+4}{\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{x^2-2x+1-x^2-2x-1+4}{\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{-4x+4}{\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{-4\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}\)

\(=-\dfrac{4}{x+1}\)

\(b,\)

\(x^2-x=0\)

\(\Leftrightarrow x\left(x-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x-1=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\)

Với \(x=0\Rightarrow B=-\dfrac{4}{0+1}=-4\)

Với \(x=1\Rightarrow B=-\dfrac{4}{1+1}=-2\)

\(c,\)

\(B=-3\Rightarrow-\dfrac{4}{x+1}=-3\)

\(\Leftrightarrow-3\left(x+1\right)=-4\)

\(\Leftrightarrow-3x-3+4=0\)

\(\Leftrightarrow-3x+1=0\)

\(\Leftrightarrow-3x=-1\)

\(\Leftrightarrow x=\dfrac{1}{3}\)

\(d,\)

\(B< 0\Rightarrow-\dfrac{4}{x+1}< 0\)

\(\Leftrightarrow x+1>0\)

\(\Leftrightarrow x>-1\)

Kết hợp điều kiện \(x\ne\pm1\) 

\(\Rightarrow-1< x< 1\)

a: ĐKXĐ: \(\left\{{}\begin{matrix}a>0\\a\ne4\end{matrix}\right.\)

\(A=\left(\dfrac{\sqrt{a}}{\sqrt{a}-2}+\dfrac{\sqrt{a}}{\sqrt{a}-2}\right)\cdot\dfrac{a-4}{\sqrt{4a}}\)

\(=\dfrac{2\sqrt{a}}{\sqrt{a}-2}\cdot\dfrac{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}{2a}\)

\(=\sqrt{a}+2\)

b: A-2<0

=>\(\sqrt{a}+2-2< 0\)

=>\(\sqrt{a}< 0\)

=>\(a\in\varnothing\)

c: Bạn ghi đầy đủ đề đi bạn

a: |x-1|=3

=>x-1=3 hoặc x-1=-3

=>x=-2(nhận) hoặc x=4(loại)

Khi x=-2 thì \(A=\dfrac{4+4}{-2-4}=\dfrac{8}{-6}=\dfrac{-4}{3}\)

b: ĐKXĐ: x<>4; x<>-4

\(B=\dfrac{-\left(x+4\right)}{x-4}+\dfrac{x-4}{x+4}-\dfrac{4x^2}{\left(x-4\right)\left(x+4\right)}\)

\(=\dfrac{-x^2-8x-16+x^2-8x+16-4x^2}{\left(x-4\right)\left(x+4\right)}=\dfrac{-4x^2-16x}{\left(x-4\right)\left(x+4\right)}\)

=-4x/x-4

c: A+B

=-4x/x-4+x^2+4/x-4

=(x-2)^2/(x-4)
A+B>0

=>x-4>0

=>x>4

a)

A=\(\left(\dfrac{x+1}{x-1}-\dfrac{x-1}{x+1}\right)\div\dfrac{2x}{5x-5}\)

\(\Leftrightarrow\left(\dfrac{x+1}{x-1}-\dfrac{x-1}{x+1}\right)\div\dfrac{2x}{5\left(x-1\right)}\)

ĐKXĐ: \(\left\{{}\begin{matrix}x-1\ne0\\x+1\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=0+1\\x=0-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\x=-1\end{matrix}\right.\)

MTC: 5(x-1)(x+1)

\([\dfrac{5\left(x+1\right)\left(x+1\right)}{5\left(x-1\right)\left(x+1\right)}-\dfrac{5\left(x-1\right)\left(x-1\right)}{5\left(x-1\right)\left(x+1\right)}]\div\dfrac{2x\left(x+1\right)}{5\left(x-1\right)\left(x+1\right)}\)

\(\Rightarrow[5\left(x+1\right)\left(x+1\right)-5\left(x-1\right)\left(x-1\right)]\div2x\left(x+1\right)\)

\(\Leftrightarrow[5\left(x+1\right)^2-5\left(x-1\right)^2]\div2x^2+2x\)

\(\Leftrightarrow[5\left(x^2+2x+1\right)-5\left(x^2-2x+1\right)]\div2x^2+2x\)

\(\Leftrightarrow(5x^2+10x+5-5x^2+10x-5)\div2x^2+2x\)

\(\Leftrightarrow20x\div\left(2x^2+2x\right)\)

\(\Leftrightarrow10x+10\)

a: \(P=\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)=a-b\)