Rút gọn biểu thức \(M=\left(x^{\frac{1}{2}}-y^{\frac{1}{2}}\right)^2\left(1-2\sqrt{\frac{y}{x}}+\frac{y}{x}\right)^{-1}\). \(M=x\) \(M=\frac{1}{x}\) \(M=-x\) \(M=\sqrt{x}\) Hướng dẫn giải: \(M=\left(x^{\frac{1}{2}}-y^{\frac{1}{2}}\right)^2\left(1-2\sqrt{\frac{y}{x}}+\frac{y}{x}\right)^{-1}\) \(=\left(\sqrt{x}-\sqrt{y}\right)^2\left[\left(1-\sqrt{\frac{y}{x}}\right)^2\right]^{-1}\) \(=\left(\sqrt{x}-\sqrt{y}\right)^2\left(\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}}\right)^{-2}\) \(=\frac{\left(\sqrt{x}-\sqrt{y}\right)^{2-2}}{\left(\sqrt{x}\right)^{-2}}=\left(\sqrt{x}-\sqrt{y}\right)^0.x=x\)
Cho \(a>0,b>0,a\ne b\), đơn giản biểu thức sau: \(N=\frac{a^{\frac{1}{3}}b^{-\frac{1}{3}}-a^{-\frac{1}{3}}b^{\frac{1}{3}}}{\sqrt[3]{a^2}-\sqrt[3]{b^2}}\) \(N=\sqrt[3]{a}\) \(N=\sqrt[3]{b}\) \(N=\frac{1}{\sqrt[3]{ab}}\) \(N=\sqrt[3]{ab}\) Hướng dẫn giải: \(N=\frac{a^{\frac{1}{3}}b^{-\frac{1}{3}}-a^{-\frac{1}{3}}b^{\frac{1}{3}}}{\sqrt[3]{a^2}-\sqrt[3]{b^2}}=\frac{a^{-\frac{1}{3}}b^{-\frac{1}{3}}\left(a^{\frac{2}{3}}-b^{\frac{2}{3}}\right)}{a^{\frac{2}{3}}-b^{\frac{2}{3}}}\) \(=a^{-\frac{1}{3}}b^{-\frac{1}{3}}=\frac{1}{\sqrt[3]{ab}}\)
Với a, b là các số thực dương. Rút gọn biểu thức: \(P=\frac{a^{\frac{1}{2}}.\sqrt[3]{b}+b^{\frac{1}{2}}.\sqrt[3]{a}}{\sqrt[6]{a}+\sqrt[6]{b}}\) \(P=\sqrt[6]{ab}\) \(P=\sqrt[3]{ab}\) \(P=\sqrt[6]{a}+\sqrt[6]{b}\) \(P=\frac{1}{\sqrt[6]{a}+\sqrt[6]{b}}\) Hướng dẫn giải: \(P=\frac{a^{\frac{1}{2}}.\sqrt[3]{b}+b^{\frac{1}{2}}.\sqrt[3]{a}}{\sqrt[6]{a}+\sqrt[6]{b}}=\frac{a^{\frac{1}{2}}.b^{\frac{1}{3}}+b^{\frac{1}{2}}.a^{\frac{1}{3}}}{a^{\frac{1}{6}}+b^{\frac{1}{6}}}\) \(=\frac{a^{\frac{1}{3}}.b^{\frac{1}{3}}\left(a^{\frac{1}{2}-\frac{1}{3}}+b^{\frac{1}{2}-\frac{1}{3}}\right)}{a^{\frac{1}{6}}+b^{\frac{1}{6}}}=\frac{a^{\frac{1}{3}}.b^{\frac{1}{3}}\left(a^{\frac{1}{6}}+b^{\frac{1}{6}}\right)}{a^{\frac{1}{6}}+b^{\frac{1}{6}}}\) \(=a^{\frac{1}{3}}.b^{\frac{1}{3}}=\sqrt[3]{ab}\)
Rút gọn biểu thức sau với a là số dương: \(P=\frac{\left(a^{2\sqrt{2}}-1\right)\left(a^{3\sqrt{2}}-a^{2\sqrt{2}}+a^{\sqrt{2}}\right)}{a^{4\sqrt{2}}+a^{\sqrt{2}}}\) \(p=a^{\sqrt{2}}\) \(p=a^{\sqrt{2}}-1\) \(p=a^{\sqrt{2}}+1\) \(p=a^{2\sqrt{2}}-1\) Hướng dẫn giải: Đặt \(b=a^{\sqrt{2}}\) thì: \(P=\frac{\left(b^2-1\right)\left(b^3-b^2+b\right)}{b^4+b}=\frac{\left(b-1\right)\left(b+1\right)b\left(b^2-b+1\right)}{b\left(b+1\right)\left(b^2-b+1\right)}\) \(=b-1=a^{\sqrt{2}}-1\)
Rút gọn biểu thức: \(Q=\frac{2:4^{-2}+\left(3^{-2}\right)^3\left(\frac{1}{9}\right)^{-3}}{5^{-3}.25^2+\left(0,7\right)^0.\left(\frac{1}{2}\right)^{-3}}\) \(Q=\frac{1}{8}\) \(Q=\frac{33}{5}\) \(Q=\frac{33}{13}\) \(Q=\frac{32}{13}\) Hướng dẫn giải: \(Q=\frac{2:4^{-2}+\left(3^{-2}\right)^3\left(\frac{1}{9}\right)^{-3}}{5^{-3}.25^2+\left(0,7\right)^0.\left(\frac{1}{2}\right)^{-3}}\) \(=\frac{2.4^2+\left(\frac{1}{9}\right)^3\left(\frac{1}{9}\right)^{-3}}{\frac{1}{5^3}.5^4+1.2^3}=\frac{32+1}{5+8}=\frac{33}{13}\)
Rút gọn biểu thức \(P = {x^{\frac{1}{3}}}.\sqrt[6]{x},x > 0 \). \(P = {x^2}\) \( P = \sqrt x \) \( P = {x^{\frac{1}{8}}} \) \(P = {x^{\frac{2}{9}}}\) Hướng dẫn giải: \({x^{\frac{1}{3}}}.\sqrt[6]{x} = {x^{\frac{1}{3}}}.{x^{\frac{1}{6}}} = {x^{\frac{1}{3} + \frac{1}{6}}} = \sqrt x \)
