Ответ:
[tex]\sqrt[3]{2}+\sqrt[3]{3}[/tex]
Пошаговое объяснение:
[tex]\frac{2^{\frac{7}{3}}-2^{\frac{8}{3}}\cdot 3^{\frac{2}{3}}+2\sqrt[3]{3^{4}}}{2^{\frac{5}{3}}-\sqrt[3]{48}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}}}:\sqrt[3]{2}=[/tex]
[tex]\frac{2^{1+\frac{4}{3}}-2^{2+\frac{2}{3}}\cdot 3^{\frac{2}{3}}+2\cdot3^{\frac{4}{3}}}{2^{\frac{5}{3}}-48^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}}}\cdot\frac{1}{2^{\frac{1}{3}}}=[/tex]
[tex]\frac{2\cdot2^{\frac{4}{3}}-2^2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+2\cdot3^{\frac{4}{3}}}{(2^{\frac{5}{3}}-(2^4\cdot3)^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}})\cdot2^{\frac{1}{3}}}=[/tex]
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знаменатель
[tex](2^{\frac{5}{3}}-(2^4\cdot3)^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}})\cdot2^{\frac{1}{3}}=[/tex]
[tex](2^{\frac{5}{3}}-2^{\frac{4}{3}}\cdot3^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}})\cdot2^{\frac{1}{3}}=[/tex]
[tex]2^2-2^{\frac{5}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{4}{3}}\cdot3^{\frac{2}{3}}+3\cdot2=[/tex]
[tex]2^{1+\frac{3}{3}}-2^{1+\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{1+\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\cdot2=[/tex]
[tex]2\cdot \left(2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\right)[/tex]
[tex]\frac{2\cdot2^{\frac{4}{3}}-2^2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+2\cdot3^{\frac{4}{3}}}{2\cdot \left(2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\right)}=[/tex]
[tex]\frac{2\cdot \left(2^{\frac{4}{3}}-2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+3^{\frac{4}{3}}\right) }{2\cdot \left(2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\right)}=[/tex]
[tex]\frac{2^{\frac{4}{3}}-2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+3^{\frac{4}{3}} }{2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3^{\frac{3}{3}}}=[/tex]
подставляем
[tex]a=2^{\frac{1}{3}}[/tex]
[tex]b=3^{\frac{1}{3}}[/tex]
[tex]\frac{a^4-2a^2b^2+b^4}{a^3-a^2b-ab^2+b^3}=\frac{(a^2-b^2)^2}{a^2(a-b)-b^2(a-b)}=[/tex]
[tex]\frac{a^4-2a^2b^2+b^4}{a^3-a^2b-ab^2+b^3}=\frac{(a^2-b^2)^2}{(a-b)(a^2-b^2)}=[/tex]
[tex]\frac{a^2-b^2}{a-b}=\frac{(a-b)(a+b)}{a-b}=a+b=2^{\frac{1}{3}}+3^{\frac{1}{3}}=\sqrt[3]{2}+\sqrt[3]{3}[/tex]
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Answers & Comments
Ответ:
[tex]\sqrt[3]{2}+\sqrt[3]{3}[/tex]
Пошаговое объяснение:
[tex]\frac{2^{\frac{7}{3}}-2^{\frac{8}{3}}\cdot 3^{\frac{2}{3}}+2\sqrt[3]{3^{4}}}{2^{\frac{5}{3}}-\sqrt[3]{48}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}}}:\sqrt[3]{2}=[/tex]
[tex]\frac{2^{1+\frac{4}{3}}-2^{2+\frac{2}{3}}\cdot 3^{\frac{2}{3}}+2\cdot3^{\frac{4}{3}}}{2^{\frac{5}{3}}-48^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}}}\cdot\frac{1}{2^{\frac{1}{3}}}=[/tex]
[tex]\frac{2\cdot2^{\frac{4}{3}}-2^2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+2\cdot3^{\frac{4}{3}}}{(2^{\frac{5}{3}}-(2^4\cdot3)^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}})\cdot2^{\frac{1}{3}}}=[/tex]
----------------
знаменатель
[tex](2^{\frac{5}{3}}-(2^4\cdot3)^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}})\cdot2^{\frac{1}{3}}=[/tex]
[tex](2^{\frac{5}{3}}-2^{\frac{4}{3}}\cdot3^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}})\cdot2^{\frac{1}{3}}=[/tex]
[tex]2^2-2^{\frac{5}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{4}{3}}\cdot3^{\frac{2}{3}}+3\cdot2=[/tex]
[tex]2^{1+\frac{3}{3}}-2^{1+\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{1+\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\cdot2=[/tex]
[tex]2\cdot \left(2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\right)[/tex]
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[tex]\frac{2\cdot2^{\frac{4}{3}}-2^2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+2\cdot3^{\frac{4}{3}}}{2\cdot \left(2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\right)}=[/tex]
[tex]\frac{2\cdot \left(2^{\frac{4}{3}}-2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+3^{\frac{4}{3}}\right) }{2\cdot \left(2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\right)}=[/tex]
[tex]\frac{2^{\frac{4}{3}}-2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+3^{\frac{4}{3}} }{2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3^{\frac{3}{3}}}=[/tex]
подставляем
[tex]a=2^{\frac{1}{3}}[/tex]
[tex]b=3^{\frac{1}{3}}[/tex]
[tex]\frac{a^4-2a^2b^2+b^4}{a^3-a^2b-ab^2+b^3}=\frac{(a^2-b^2)^2}{a^2(a-b)-b^2(a-b)}=[/tex]
[tex]\frac{a^4-2a^2b^2+b^4}{a^3-a^2b-ab^2+b^3}=\frac{(a^2-b^2)^2}{(a-b)(a^2-b^2)}=[/tex]
[tex]\frac{a^2-b^2}{a-b}=\frac{(a-b)(a+b)}{a-b}=a+b=2^{\frac{1}{3}}+3^{\frac{1}{3}}=\sqrt[3]{2}+\sqrt[3]{3}[/tex]