Rút gọn biểu thức sau: \(P=\left[\frac{\sqrt[3]{a^{2} b}-\sqrt[3]{a b^{2}}}{\sqrt[3]{a^{2}}-2 \sqrt[3]{a b}+\sqrt[3]{b^{2}}}-\frac{a+b}{\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}}\right](\sqrt[6]{a}+\sqrt[6]{b})^{-1}+\sqrt[6]{a}\), với \(a\gt 0, b>0, a \neq b\)
Giải thích:
Ta có:
\(\begin{array}{l}P=\left[\frac{\sqrt[3]{a^{2} b}-\sqrt[3]{a b^{2}}}{\sqrt[3]{a^{2}}-2 \sqrt[3]{a b}+\sqrt[3]{b^{2}}}-\frac{a+b}{\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}}\right](\sqrt[6]{a}+\sqrt[6]{b})^{-1}+\sqrt[6]{a} \\=\left[\frac{\sqrt[3]{a b}(\sqrt[3]{a}-\sqrt[3]{b})}{(\sqrt[3]{a}-\sqrt[3]{b})^{2}}-\frac{(\sqrt[3]{a}+\sqrt[3]{b})\left(\sqrt[3]{a^{2}}-\sqrt[3]{a b}+\sqrt[3]{b^{2}}\right)}{(\sqrt[3]{a}-\sqrt[3]{b})(\sqrt[3]{a}+\sqrt[3]{b})}\right](\sqrt[6]{a}+\sqrt[6]{b})^{-1}+\sqrt[6]{a} \\=\left[\frac{\sqrt[3]{a b}}{\sqrt[3]{a}-\sqrt[3]{b}}-\frac{\sqrt[3]{a^{2}}-\sqrt[3]{a b}+\sqrt[3]{b^{2}}}{\sqrt[3]{a}-\sqrt[3]{b}}\right] \frac{1}{\sqrt[6]{a}+\sqrt[6]{b}}+\sqrt[6]{a} \\=-\frac{(\sqrt[3]{a}-\sqrt[3]{b})^{2}}{\sqrt[3]{a}-\sqrt[3]{b}} \frac{1}{(\sqrt[6]{a}+\sqrt[6]{b})}+\sqrt[6]{a} \\=\frac{\sqrt[3]{b}-\sqrt[3]{a}}{\sqrt[6]{a}+\sqrt[6]{b}}+\sqrt[6]{a}=\sqrt[6]{b}-\sqrt[6]{a}+\sqrt[6]{a}=\sqrt[6]{b} \\\end{array}\)Câu hỏi này nằm trong: