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\(\left[\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-y^2}+\frac{1}{\left(2x+y\right)^2}\right].\frac{4x^2+4xy+y^2}{16x}\)
\(=\frac{\left(2x+y\right)^22\left(4x^2-y^2\right)+\left(2x-y\right)^2}{\left(2x-y\right)^2\left(2x+y\right)^2}.\frac{\left(2x+y\right)^2}{16x}\)
\(=\frac{16x^2}{16x\left(2x-y\right)^2}=\frac{x}{\left(2x-y\right)^2}\)
\(\left[\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-4^2}+\frac{1}{\left(2x+y\right)^2}\right].\frac{4x^2+4xy+y^2}{16x}\)
\(=\frac{\left(2x+y\right)^22\left(4x^2-y^2\right)+\left(2x-y\right)^2}{\left(2x-y\right)^2\left(2x+y\right)^2}.\frac{\left(2x+y\right)^2}{16x}\)
\(=\frac{16x^2}{16x\left(2x-y\right)^2}=\frac{x}{\left(2x-y\right)^2}\)
\(A=\left[\frac{1}{a^2}+\frac{1}{b^2}+\frac{2}{a+b}\left(\frac{1}{a}+\frac{1}{b}\right)\right].\frac{ab}{\left(a+b\right)^2}\)
\(=\left(\frac{1}{a}+\frac{1}{b}\right)^2.\frac{ab}{\left(a+b\right)^2}\)
\(=\frac{1}{ab}\)
\(B=\left[\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-y^2}+\frac{1}{\left(2x+y\right)^2}\right].\frac{4x^2+14xy+y^2}{16x}\)
\(=\frac{\left(2x+y\right)^2+2\left(2x+y\right)\left(2x-y\right)+\left(2x-y\right)^2}{\left(2x+y\right)^2.\left(2x-y\right)^2}.\frac{\left(2x+y\right)^2}{16x}\)
\(=\frac{\left(2x+y+2x-y\right)^2}{\left(2x+y\right)^2.\left(2x-y\right)^2}.\frac{\left(2x+y\right)^2}{16x}\)
\(=\frac{x}{\left(2x-y\right)^2}\)
\(A=\left[\frac{1}{a^2}+\frac{1}{b^2}+\frac{2}{a+b}.\left(\frac{1}{a}+\frac{1}{b}\right)\right].\frac{ab}{\left(a+b\right)^2}\)
ĐK: a, b khác 0, a khác -b
\(A=\left[\frac{1}{a^2}+\frac{1}{b^2}+\frac{2}{a+b}.\left(\frac{a+b}{ab}\right)\right].\frac{ab}{\left(a+b\right)^2}\)
\(A=\left[\frac{1}{a^2}+\frac{1}{b^2}+\frac{2}{ab}\right].\frac{ab}{\left(a+b\right)^2}=\left(\frac{1}{a}+\frac{1}{b}\right)^2.\frac{ab}{\left(a+b\right)^2}\)
\(A=\frac{\left(a+b\right)^2}{ab}.\frac{ab}{\left(a+b\right)^2}=1\)
\(B=\left[\frac{1}{\left(2x-y\right)^2}+\frac{2}{\left(4x^2-y^2\right)}+\frac{1}{\left(2x+y\right)^2}\right].\frac{4x^2+4xy+y^2}{16xy}\)
ĐK: xy khác 0, y \(\ne\pm\)2x
\(B=\left[\frac{1}{\left(2x-y\right)^2}+\frac{2}{\left(2x-y\right).\left(2x+y\right)}+\frac{1}{\left(2x+y\right)^2}\right].\frac{\left(2x+y\right)^2}{16xy}\)
\(B=\left[\frac{1}{\left(2x-y\right)}+\frac{1}{\left(2x+y\right)}\right]^2.\frac{\left(2x+y\right)^2}{16xy}\)
\(B=\left(\frac{2x+y+2x-y}{\left(2x-y\right).\left(2x+y\right)}\right)^2.\frac{\left(2x+y\right)^2}{16xy}\)
\(B=\frac{16x^2}{\left(2x-y\right)^2.\left(2x+y\right)^2}.\frac{\left(2x+y\right)^2}{16xy}\)
\(B=\frac{x}{\left(2x-y\right)^2.y}\)
Lời giải:
a) ĐK: $a\neq -b\neq 0$
\(A=\left(\frac{a^2+b^2}{a^2b^2}+\frac{2}{a+b}.\frac{a+b}{ab}\right).\frac{ab}{(a+b)^2}\)
\(=\left(\frac{a^2+b^2}{a^2b^2}+\frac{2ab}{a^2b^2}\right).\frac{ab}{(a+b)^2}=\frac{(a+b)^2}{a^2b^2}.\frac{ab}{(a+b)^2}=\frac{1}{ab}\)
b)
\(B=\left[\frac{(2x+y)^2}{(2x-y)^2(2x+y)^2}+\frac{(2x-y)^2}{(2x-y)^2(2x+y)^2}+\frac{2}{(2x-y)(2x+y)}\right].\frac{(2x+y)^2}{16x}\)
\(=\left[\frac{8x^2+2y^2}{(2x-y)^2(2x+y)^2}+\frac{2(2x-y)(2x+y)}{(2x-y)^2(2x+y)^2}\right].\frac{(2x+y)^2}{16x}\)
\(=\frac{8x^2+2y^2+2(4x^2-y^2)}{(2x-y)^2(2x+y)^2}.\frac{(2x+y)^2}{16x}\)
\(=\frac{16x^2}{(2x-y)^2(2x+y)^2}.\frac{(2x+y)^2}{16x}=\frac{x}{(2x-y)^2}\)
Ta có: \(VT=\frac{4x^2-4xy+y^2}{y^3-6y^2x+12ỹ^2-8x^3}\)
\(=\frac{\left(2x-y\right)^2}{\left(y-2x\right)^3}=-\frac{\left(2x-y\right)^2}{\left(2x-y\right)^3}=\frac{-1}{2x-y}=VP\)(đpcm)
Bài 1:
a) Ta có: \(\left(\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-y^2}+\frac{1}{\left(2x+y\right)^2}\right):\frac{16x}{4x^2+4xy+y^2}\)
\(=\left(\frac{\left(2x+y\right)^2}{\left(2x-y\right)^2\cdot\left(2x+y\right)^2}+\frac{2\cdot\left(2x+y\right)\left(2x-y\right)}{\left(2x+y\right)^2\cdot\left(2x-y\right)^2}+\frac{\left(2x-y\right)^2}{\left(2x+y\right)^2\cdot\left(2x-y\right)^2}\right)\cdot\frac{\left(2x+y\right)^2}{16x}\)
\(=\frac{\left(2x+y+2x-y\right)^2}{\left(2x-y\right)^2\cdot\left(2x+y\right)^2}\cdot\frac{\left(2x+y\right)^2}{16x}\)
\(=\frac{\left(4x\right)^2}{\left(2x-y\right)^2}\cdot\frac{1}{16x}\)
\(=\frac{16x^2}{16x\cdot\left(2x-y\right)^2}\)
\(=\frac{x}{\left(2x-y\right)^2}\)
b) Ta có: \(\frac{3}{3x+3}+\frac{10}{5-5x}+\frac{5x-1}{x^2-1}\)
\(=\frac{1}{x+1}-\frac{2}{x-1}+\frac{5x-1}{x^2-1}\)
\(=\frac{x-1}{\left(x+1\right)\left(x-1\right)}-\frac{2\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{5x-1}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x-1-2\left(x+1\right)+5x-1}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x-1-2x-2+5x-1}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{4x-4}{\left(x-1\right)\left(x+1\right)}=\frac{4\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{4}{x+1}\)
c) Ta có: \(A=\left(x^4-x^2+2x-1\right):\left(x^2+x-1\right)-\left(x^2-x\right)\)
\(=\frac{\left(x^2\right)^2-\left(x^2-2x+1\right)}{x^2+x-1}-x^2+x\)
\(=\frac{\left(x^2\right)^2-\left(x-1\right)^2}{x^2+x-1}-x^2+x\)
\(=\frac{\left(x^2-x+1\right)\left(x^2+x-1\right)}{x^2+x-1}-x^2+x\)
\(=x^2-x+1-x^2+x\)
=1
\(a,\frac{x+1}{x-2}+\frac{x-1}{x+2}=\frac{2\left(x^2+2\right)}{x^2-4}\)\(\Leftrightarrow\frac{x^2+3x+2+x^2-3x+2}{x^2-4}=\frac{2\left(x^2+2\right)}{x^2-4}\)
\(\Leftrightarrow2\left(x^2+2\right)=2\left(x^2+2\right)\)(luôn đúng)
Vậy pt có vô số nghiệm
\(b,\Leftrightarrow\left(2x+3\right)\left(\frac{3x+8}{2-7x}+1\right)=\left(x-5\right)\left(\frac{3x+8}{2-7x}+1\right)\)
\(\Leftrightarrow\left(\frac{3x+8}{2-7x}+1\right)\left(2x+3-x+5\right)=0\)\(\Leftrightarrow\left(\frac{-4x+10}{2-7x}\right)\left(x+8\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}-4x+10=0\\x+8=0\end{cases}\Rightarrow}\orbr{\begin{cases}x=\frac{5}{2}\\x=-8\end{cases}}\)
Mấy câu rút gọn bạn quy đồng nha
Lời giải:
a)
\(A=\frac{x^2y(y-x)-xy^2(x-y)}{3y^2-2x^2}=\frac{x^2y(y-x)+xy^2(y-x)}{3y^2-2x^2}=\frac{(xy^2+x^2y)(y-x)}{3y^2-2x^2}\)
\(=\frac{xy(x+y)(y-x)}{3y^2-2x^2}=\frac{xy(y^2-x^2)}{3y^2-2x^2}\)
Với $x=-3; y=\frac{1}{2}$ thì:
$xy=\frac{-3}{2}; x^2=9; y^2=\frac{1}{4}$
Do đó $A=\frac{-35}{46}$
b)
\(B=\frac{(8x^3-y^3)(4x^2-y^2)}{(2x+y)(4x^2-4xy+y^2)}=\frac{(2x-y)(4x^2+2xy+y^2)(2x-y)(2x+y)}{(2x+y)(2x-y)^2}\)
\(=4x^2+2xy+y^2=4.2^2+2.2.\frac{-1}{2}+(\frac{-1}{2})^2=\frac{57}{4}\)