Question

Rút gọn biểu thức

A=\(\frac{1}{x\left(x+2\right)}\)+\(\frac{1}{\left(x+2\right)\left(x+4\right)}\)+\(\frac{1}{\left(x+4\right)\left(x+6\right)}\)+\(\frac{1}{\left(x+6\right)\left(x+8\right)}\)

Answer 1

Answer 2

A = \(\frac{1}{x\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+8\right)}\)

= \(\frac{1}{x}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+4}+\frac{1}{x+4}-\frac{1}{x+6}+\frac{1}{x+6}-\frac{1}{x+8}\)

= \(\frac{1}{x}-\frac{1}{x+8}\)

= \(\frac{x+8}{x\left(x+8\right)}-\frac{x}{x\left(x+8\right)}\)

= \(\frac{x+8-x}{x\left(x+8\right)}\) = \(\frac{8}{x\left(x+8\right)}\)

Answer 3

May lag :(( ko gửi ảnh đc

Answer 4

Mãi mới gửi đc ảnh

Answer 5

\(\frac{1}{x\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+6\right)}\frac{1}{\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{\left(x+4\right)\left(x+6\right)\left(x+8\right)+x\left(x+6\right)\left(x+8\right)+x\left(x+2\right)\left(x+8\right)+x\left(x+2\right)\left(x+4\right)}{x\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{\left(x^2+6x+4x+24\right)\left(x+8\right)+x\left(x+6\right)\left(x+8\right)+x\left(x+2\right)\left(x+8\right)+x\left(x+2\right)\left(x+4\right)}{x\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{\left(x^2+10x+24\right)\left(x+8\right)+x^3+8x^2+6x^2+48x+x^3+8x^2+2x^2+16x+x^3+4x^2+2x^2+8x}{x\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{x^3+8x^2+10x^2+80x+24x+192+x^3+8x^2+6x^2+48x+x^3+8x^2+2x^2+16x+x^3+4x^2+2x^2+8x}{x\left(+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{4x^3+48x^2+176x+192}{x\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{4\left(x^3+12x^2+44x+48\right)}{x\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{4\left(x^3+2x^2+10x^2+20x+24x+48\right)}{x\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{4\left[x^2\left(x+2\right)+10x\left(x+2\right)+24\left(x+2\right)\right]}{x\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{4\left(x+2\right)\left(x^2+10x+24\right)}{x\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{4\left(x^2+6x+4x+24\right)}{x\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{4\left[x\left(x+6\right)+4\left(x+6\right)\right]}{x\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{4\left(x+6\right)\left(x+4\right)}{x\left(x+4\right)\left(x+6\right)\left(x+8\right)}\)

\(\Leftrightarrow\frac{4}{x^2+8x}\)