Question

Cho các biểu thức: A= \(\frac{x-\sqrt[3]{x}}{x-1}\), B=\(\frac{1}{\sqrt[3]{x}-1}+\frac{1}{\sqrt[3]{x^2}+\sqrt[3]{x}+1}\), với x \(\ne\)1
Rút gọn biểu thức M= A + B

Answer 1

\(ĐKXĐ:x\ne1\)

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

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

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

\(\Leftrightarrow M=A+B=\frac{x-\sqrt[3]{x}}{x-1}+\frac{\sqrt[3]{x^2}+2\sqrt[3]{x}}{x-1}\)

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

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

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

\(\Leftrightarrow M=\frac{\sqrt[3]{x}}{\sqrt[3]{x}-1}\)

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