Cho \(P=\left(\dfrac{1}{ax-2}+\dfrac{1}{ax+2}+\dfrac{2ax}{a^2x^2+4}+\dfrac{4a^3x^3}{a^4x^4}\right).\dfrac{a^4x^4+16}{a^4x^4}\)
Rút gọn P
Cho \(P=\left(\dfrac{1}{ax-2}+\dfrac{1}{ax+2}+\dfrac{2ax}{a^2x^2+4}+\dfrac{4a^3x^3}{a^4x^4+16}\right).\dfrac{a^4x^4+16}{a^4x^4}\)
a. Rút gọn
b. tìm P biết \(\dfrac{a^2+4}{x^2+9}=\dfrac{a^2}{9}\)
A=\(\left(\dfrac{x-1}{x^2-2x}+\dfrac{x+1}{x^2+2x}-\dfrac{4}{x^3-4x}\right)\div\dfrac{2x+4}{x^2-3x}\)
Rút gọn A
\(A=\left(\dfrac{x-1}{x\left(x-2\right)}+\dfrac{x+1}{x\left(x+2\right)}-\dfrac{4}{x\left(x-2\right)\left(x+2\right)}\right)\cdot\dfrac{x\left(x-3\right)}{2\left(x+2\right)}\)
\(=\dfrac{x^2+x-2+x^2-x+2-4}{x\left(x-2\right)\left(x+2\right)}\cdot\dfrac{x\left(x-3\right)}{2\left(x+2\right)}\)
\(=\dfrac{2x^2-4}{x\left(x-2\right)\left(x+2\right)}\cdot\dfrac{x\left(x-3\right)}{2\left(x+2\right)}\)
\(=\dfrac{2x\left(x^2-2\right)\left(x-3\right)}{2x\left(x-2\right)\cdot\left(x+2\right)^2}=\dfrac{\left(x^2-2\right)\left(x-3\right)}{\left(x-2\right)\left(x+2\right)^2}\)
Rút gọn biểu thức. Chứng minh rằng biểu thức rút gọn không âm vs mọi giá trị của biến thuộc tập xác định (coi a là hằng):
1 - (\(\dfrac{a+x}{ax-x^2}\) + \(\dfrac{2a+3x}{x^2-a^2}\)) : \(\dfrac{a^4-4x^4}{a^4x-a^2x^3}\)
P=(1/ax-2 + 1/ax+2 +2ax/a^2x^2+4 + 4a^3x^3/a^4x^4+16) a^4x^4+16/a^4x^4
Rút gọn P
Tính P biết a^2+4/x^2=a^2/9
Không ai trả lời luôn
Cho biểu thức:
A\(=\left(\dfrac{\left(x+1\right)^2}{\left(x+1\right)^2-3x}-\dfrac{2x^2+4x-1}{x^3+1}-\dfrac{1}{x+1}\right):\dfrac{x^2-4}{3x^2+6x}\)
a/ Rút gọn A
b/ Tìm x ∈ Z để A nguyên
ĐKXĐ: \(x\notin\left\{-1;2;-2\right\}\)
a) Ta có: \(A=\left(\dfrac{\left(x+1\right)^2}{\left(x+1\right)^2-3x}-\dfrac{2x^2+4x-1}{x^3+1}-\dfrac{1}{x+1}\right):\dfrac{x^2-4}{3x^2+6x}\)
\(=\left(\dfrac{\left(x+1\right)^2}{x^2-x+1}-\dfrac{2x^2+4x-1}{\left(x+1\right)\left(x^2-x+1\right)}-\dfrac{1}{x+1}\right):\dfrac{x^2-4}{3x^2+6x}\)
\(=\left(\dfrac{x^3+3x^2+3x+1}{\left(x+1\right)\left(x^2-x+1\right)}-\dfrac{2x^2+4x-1}{\left(x+1\right)\left(x^2-x+1\right)}-\dfrac{x^2-x+1}{\left(x+1\right)\left(x^2-x+1\right)}\right):\dfrac{\left(x-2\right)\left(x+2\right)}{3x\left(x+2\right)}\)
\(=\dfrac{x^3+3x^2+3x+1-2x^2-4x+1-x^2+x-1}{\left(x+1\right)\left(x^2-x+1\right)}:\dfrac{x-2}{3x}\)
\(=\dfrac{x^3+1}{\left(x+1\right)\left(x^2-x+1\right)}\cdot\dfrac{3x}{x-2}\)
\(=\dfrac{3x}{x-2}\)
b) Để A nguyên thì \(3x⋮x-2\)
\(\Leftrightarrow3x-6+6⋮x-2\)
mà \(3x-6⋮x-2\)
nên \(6⋮x-2\)
\(\Leftrightarrow x-2\inƯ\left(6\right)\)
\(\Leftrightarrow x-2\in\left\{1;-1;2;-2;3;-3;6;-6\right\}\)
hay \(x\in\left\{3;1;4;0;5;-1;8;-4\right\}\)
Kết hợp ĐKXĐ, ta được:
\(x\in\left\{3;1;4;0;5;8;-4\right\}\)
Vậy: Để A nguyên thì \(x\in\left\{3;1;4;0;5;8;-4\right\}\)
Cho hai biểu thức: A=\(\dfrac{4x-16}{x^2-16}\)và B=\(\dfrac{1}{x}+\dfrac{1}{x+4}+\dfrac{2x-4}{x\left(x+4\right)}\)
a) Rút gọn biểu thức A;
b) Chứng minh: B = A.
\(a,A=\dfrac{4\left(x-4\right)}{\left(x-4\right)\left(x+4\right)}=\dfrac{4}{x+4}\\ b,B=\dfrac{x+4+x+2x-4}{x\left(x+4\right)}=\dfrac{4x}{x\left(x+4\right)}=\dfrac{4}{x+4}=A\)
\(A=\left(\dfrac{4x}{x+2}-\dfrac{x^3-8}{x^3+8}\times\dfrac{4x^2-8x+16}{x^2-4}\right)\div\dfrac{16}{x+2}\times\dfrac{x^2+3x+2}{x^2+x+1}\)
\(B=\dfrac{x^2+x-2}{x^3-1}\)
a) Tìm ĐKXĐ của A, B. Rút gọn A, B
b)Tìm GTLN của A+B
\(A=\left(\dfrac{1+x}{1-x}-\dfrac{1-x}{1+x}+\dfrac{4x^2}{1-x^2}\right):\dfrac{4x^2-4}{x^2-2x+1}\)
a, Rút gọn A
ĐKXĐ: \(x\ne\pm1\)
\(A=\left(\dfrac{\left(1+x\right)^2}{\left(1-x\right)\left(1+x\right)}-\dfrac{\left(1-x\right)^2}{\left(1-x\right)\left(1+x\right)}+\dfrac{4x^2}{\left(1-x\right)\left(1+x\right)}\right):\dfrac{4\left(x-1\right)\left(x+1\right)}{\left(x-1\right)^2}\)
\(=\left(\dfrac{x^2+2x+1-\left(x^2-2x+1\right)+4x^2}{\left(1-x\right)\left(1+x\right)}\right):\dfrac{4\left(x+1\right)}{x-1}\)
\(=\left(\dfrac{4x^2+4x}{\left(1-x\right)\left(1+x\right)}\right):\dfrac{4\left(x+1\right)}{x-1}\)
\(=\dfrac{4x\left(x+1\right)}{\left(1-x\right)\left(1+x\right)}.\dfrac{\left(x-1\right)}{4\left(x+1\right)}=-\dfrac{x}{x+1}\)
Rút gọn:
\(A=\dfrac{2}{x-1}\sqrt{\dfrac{x^2-2x+1}{4x^2}}\)
\(B=\left(x^2-4\right)\sqrt{\dfrac{9}{x^2-4x+4}}\)
\(A=\dfrac{2}{x-1}\sqrt{\dfrac{\left(x-1\right)^2}{4x^2}}=\dfrac{2}{x-1}\left|\dfrac{x-1}{2x}\right|=\dfrac{\left|x-1\right|}{\left(x-1\right)\left|x\right|}\)
\(B=\left(x^2-4\right)\sqrt{\dfrac{9}{x^2-4x+4}}=\dfrac{3\left(x^2-4\right)}{\left|x-2\right|}\)
a) Ta có: \(A=\dfrac{2}{x-1}\cdot\sqrt{\dfrac{x^2-2x+1}{4x^2}}\)
\(=\dfrac{2}{x-1}\cdot\dfrac{x-1}{2x}\)
\(=\dfrac{1}{x}\)
b) Ta có: \(\left(x^2-4\right)\cdot\sqrt{\dfrac{9}{x^2-4x+4}}\)
\(=\dfrac{\left(x-2\right)\left(x+2\right)\cdot3}{\left(x-2\right)^2}\)
\(=\dfrac{3x+6}{x-2}\)