Bài 1: Lũy thừa

\(P=\sqrt[3]{6+\sqrt{\frac{847}{27}}}+\sqrt[3]{6+\sqrt{\frac{847}{27}}}\)

Ta áp dụng hằng đẳng thức : 

\(\left(a+b\right)^3=a^3+b^3+3ab\left(a+b\right)\)

\(\Rightarrow P^3=6+\sqrt{\frac{847}{27}}+6-\sqrt{\frac{847}{27}}+3\sqrt[3]{6+\sqrt{\frac{847}{27}}}.\sqrt[3]{6-\sqrt{\frac{847}{27}}}\left(3\sqrt[3]{6+\sqrt{\frac{847}{27}}}.\sqrt[3]{6-\sqrt{\frac{847}{27}}}\right)\)

\(\Leftrightarrow P^3=12+3.\sqrt[3]{36-\frac{847}{27}}.P=12+5P\)

\(\Leftrightarrow P^3-5P-12=0\)

\(\Leftrightarrow\left(P-3\right)\left(P^2+3P+4\right)=0\)

\(\Leftrightarrow P=3\) hoặc \(P^3+3P+4=0\) vô nghiệm

Vậy \(P=3\)

\(=\left(\sqrt{a}-\sqrt{b}\right)^2:\left(\sqrt{b}-\frac{b}{\sqrt{a}}\right)^2=\left(\sqrt{a}-\sqrt{b}\right)^2:\left[\frac{\sqrt{b}}{\sqrt{a}}\left(\sqrt{a}-\sqrt{b}\right)\right]^2\)

\(=\left(\sqrt{a}-\sqrt{b}\right)^2:\left[\frac{\sqrt{b}}{\sqrt{a}}\left(\sqrt{a}-\sqrt{b}\right)\right]^2\)

 \(=\left(\sqrt{a}-\sqrt{b}\right)^2.\frac{a}{b\left(\sqrt{a}-\sqrt{b}\right)^2}=\frac{a}{b}\)

\(M=\frac{\left(a^{\frac{1}{3}}+b^{\frac{1}{3}}\right)^2}{\sqrt[3]{ab}}:\left(2+\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\right)=\frac{\left(a^{\frac{1}{3}}+b^{\frac{1}{3}}\right)^2}{\sqrt[3]{ab}}:\frac{2\sqrt[3]{ab}+\left(\sqrt[3]{a}\right)^2+\left(\sqrt[3]{a}\right)^2}{\sqrt[3]{ab}}\)

    \(=\frac{\left(\sqrt[3]{a}+\sqrt[3]{b}\right)^2}{\sqrt[3]{ab}}-\frac{\sqrt[3]{ab}}{\left(\sqrt[3]{a}+\sqrt[3]{b}\right)^2}=1\)

\(=\left[\frac{\left(a^{\frac{1}{2}}-b^{\frac{1}{2}}\right)\left(a+a^{\frac{1}{2}}b^{\frac{1}{2}}+b\right)}{a^{\frac{1}{2}}-b^{\frac{1}{2}}}+a^{\frac{1}{2}}b^{\frac{1}{2}}\right]\left[\frac{a^{\frac{1}{2}}-b^{\frac{1}{2}}}{\left(a^{\frac{1}{2}}-b^{\frac{1}{2}}\right)\left(a^{\frac{1}{2}}+b^{\frac{1}{2}}\right)}\right]^2\)

\(=\frac{a+2a^{\frac{1}{2}}b^{\frac{1}{2}}+b}{\left(a^{\frac{1}{2}}+b^{\frac{1}{2}}\right)^2}=\frac{\left(a^{\frac{1}{2}}+b^{\frac{1}{2}}\right)^2}{\left(a^{\frac{1}{2}}+b^{\frac{1}{2}}\right)^2}=1\)

\(=\frac{a\sqrt{ab}+ab-ab}{a+\sqrt{ab}}.\frac{a-b}{\sqrt[4]{ab}-\sqrt{b}}.\frac{1}{\sqrt{b}+\sqrt[4]{ab}}\)

\(=\frac{a\sqrt{ab}}{a+\sqrt{ab}}.\frac{a-b}{\sqrt{ab}-b}=\frac{a\sqrt{ab}}{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}.\frac{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)}{\sqrt{b}\left(\sqrt{a}-\sqrt{b}\right)}=a\)

\(I=\frac{a^{\frac{4}{3}}-8a^{\frac{2}{3}}b}{a^{\frac{2}{3}}+2\sqrt[3]{ab}+4b^{\frac{2}{3}}}\left(1-2\sqrt[3]{\frac{b}{a}}\right)^{-1}-a^{\frac{2}{3}}=\frac{a^{\frac{1}{3}}\left(a-8b\right)}{a^{\frac{2}{3}}+2a^{\frac{1}{3}}.b^{\frac{1}{3}}+4b^{\frac{2}{3}}}\left(\frac{\sqrt[3]{a}-2\sqrt[3]{b}}{\sqrt[3]{a}}\right)^{-1}-a^{\frac{2}{3}}\)

  \(=\frac{\sqrt[3]{a}\left[\left(\sqrt[3]{a}\right)^3-\left(2\sqrt[3]{b}\right)^3\right]}{a^{\frac{2}{3}}+2\sqrt[3]{ab}+4b^{\frac{2}{3}}}.\frac{\sqrt[3]{a}}{\sqrt[3]{a}-2\sqrt[3]{b}}-a^{\frac{2}{3}}\)

  \(=\frac{\left(\sqrt[3]{a}\right)^2\left(\sqrt[3]{a}-2\sqrt[3]{b}\right)\left[\left(\sqrt[3]{a}\right)^2+2\sqrt[3]{ab}+\left(2\sqrt[3]{b}\right)^2\right]}{\left(\sqrt[3]{a}-a\sqrt[3]{b}\right)\left[\left(\sqrt[3]{a}\right)^2+2\sqrt[3]{ab}+\left(2\sqrt[3]{b}\right)^2\right]}-a^{\frac{2}{3}}=a^{\frac{2}{3}}-a^{\frac{2}{3}}=0\)

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