Question

Помогите решить. Упростить выражение. Ответ: ∛2 + ∛3

Answer 1

Ответ:

$\sqrt[3]{2}+\sqrt[3]{3}$

Пошаговое объяснение:

$\frac{2^{\frac{7}{3}}-2^{\frac{8}{3}}\cdot 3^{\frac{2}{3}}+2\sqrt[3]{3^{4}}}{2^{\frac{5}{3}}-\sqrt[3]{48}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}}}:\sqrt[3]{2}=$

$\frac{2^{1+\frac{4}{3}}-2^{2+\frac{2}{3}}\cdot 3^{\frac{2}{3}}+2\cdot3^{\frac{4}{3}}}{2^{\frac{5}{3}}-48^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}}}\cdot\frac{1}{2^{\frac{1}{3}}}=$

$\frac{2\cdot2^{\frac{4}{3}}-2^2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+2\cdot3^{\frac{4}{3}}}{(2^{\frac{5}{3}}-(2^4\cdot3)^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}})\cdot2^{\frac{1}{3}}}=$

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знаменатель

$(2^{\frac{5}{3}}-(2^4\cdot3)^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}})\cdot2^{\frac{1}{3}}=$

$(2^{\frac{5}{3}}-2^{\frac{4}{3}}\cdot3^{\frac{1}{3}}-2\cdot3^{\frac{2}{3}}+3\cdot2^{\frac{2}{3}})\cdot2^{\frac{1}{3}}=$

$2^2-2^{\frac{5}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{4}{3}}\cdot3^{\frac{2}{3}}+3\cdot2=$

$2^{1+\frac{3}{3}}-2^{1+\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{1+\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\cdot2=$

$2\cdot \left(2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\right)$

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$\frac{2\cdot2^{\frac{4}{3}}-2^2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+2\cdot3^{\frac{4}{3}}}{2\cdot \left(2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\right)}=$

$\frac{2\cdot \left(2^{\frac{4}{3}}-2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+3^{\frac{4}{3}}\right) }{2\cdot \left(2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3\right)}=$

$\frac{2^{\frac{4}{3}}-2\cdot2^{\frac{2}{3}}\cdot 3^{\frac{2}{3}}+3^{\frac{4}{3}} }{2^{\frac{3}{3}}-2^{\frac{2}{3}}\cdot3^{\frac{1}{3}}-2^{\frac{1}{3}}\cdot3^{\frac{2}{3}}+3^{\frac{3}{3}}}=$

подставляем

$a=2^{\frac{1}{3}}$

$b=3^{\frac{1}{3}}$

$\frac{a^4-2a^2b^2+b^4}{a^3-a^2b-ab^2+b^3}=\frac{(a^2-b^2)^2}{a^2(a-b)-b^2(a-b)}=$

$\frac{a^4-2a^2b^2+b^4}{a^3-a^2b-ab^2+b^3}=\frac{(a^2-b^2)^2}{(a-b)(a^2-b^2)}=$

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