Question

[Nguyễn Thị Thuỳ Dương]

cho các biểu thức: A=\(\dfrac{x-\sqrt[3]{x}}{x-1}\),B=\(\dfrac{1}{\sqrt[3]{x}-1}+\dfrac{1}{\sqrt[3]{x^2}+\sqrt[3]{x}+1}\);vs x\(\ne\)1

rút gọn biểu thức M=A+B

Answer 1

[Nguyễn Lê Phước Thịnh]

Ta có: M=A+B

\(=\dfrac{x-\sqrt[3]{x}}{x-1}+\dfrac{1}{\sqrt[3]{x}-1}+\dfrac{1}{\sqrt[3]{x^2}+\sqrt[3]{x}+1}\)

\(=\dfrac{x-\sqrt[3]{x}}{\left(\sqrt[3]{x}-1\right)\left(\sqrt[3]{x^2}+\sqrt[3]{x}+1\right)}+\dfrac{\sqrt[3]{x^2}+\sqrt[3]{x}+1+\sqrt[3]{x}-1}{\left(\sqrt[3]{x}-1\right)\left(\sqrt[3]{x^2}+\sqrt[3]{x}+1\right)}\)

\(=\dfrac{x+\sqrt[3]{x}+\sqrt[3]{x^2}}{\left(\sqrt[3]{x}-1\right)\left(\sqrt[3]{x^2}+\sqrt[3]{x}+1\right)}\)

\(=\dfrac{\sqrt[3]{x}}{\sqrt[3]{x}-1}\)