Với \(a,b\) là hai số thực dương. Biểu thức \(P=\frac{a^{\frac{1}{2}}.\sqrt[3]{b}+b^{\frac{1}{2}}.\sqrt[3]{a}}{\sqrt[6]{a}+\sqrt[6]{b}}\) được rút gọn bằng
\(\sqrt[6]{ab}\). \(\sqrt[3]{ab}\). \(\sqrt[6]{a}+\sqrt[6]{b}\). \(\frac{1}{\sqrt[6]{a}+\sqrt[6]{b}}\). Hướng dẫn giải:\(P=\frac{a^{\frac{1}{2}}.\sqrt[3]{b}+b^{\frac{1}{2}}.\sqrt[3]{a}}{\sqrt[6]{a}+\sqrt[6]{b}}=\frac{a^{\frac{1}{2}}.b^{\frac{1}{3}}+b^{\frac{1}{2}}.a^{\frac{1}{3}}}{a^{\frac{1}{6}}+b^{\frac{1}{6}}}\)
\(=\frac{a^{\frac{1}{3}}.b^{\frac{1}{3}}\left(a^{\frac{1}{2}-\frac{1}{3}}+b^{\frac{1}{2}-\frac{1}{3}}\right)}{a^{\frac{1}{6}}+b^{\frac{1}{6}}}=\frac{a^{\frac{1}{3}}.b^{\frac{1}{3}}\left(a^{\frac{1}{6}}+b^{\frac{1}{6}}\right)}{a^{\frac{1}{6}}+b^{\frac{1}{6}}}\)
\(=a^{\frac{1}{3}}.b^{\frac{1}{3}}=\sqrt[3]{ab}\).