Question

Nếu ax³ = by³ = cz³ và \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1\)

Chứng minh rằng : \(\sqrt[3]{ax^2+by^2+cz^2}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}\)

Answer 1

\(ax^3=by^3=cz^3\Rightarrow\dfrac{ax^2}{\dfrac{1}{x}}=\dfrac{by^2}{\dfrac{1}{y}}=\dfrac{cz^2}{\dfrac{1}{z}}=\dfrac{ax^2+by^2+cz^2}{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}=ax^2+by^2+cz^2\)

=> \(\sqrt[3]{ax^2+by^2+cz^2}=\sqrt[3]{ax^3}=\sqrt[3]{by^3}=\sqrt[3]{cz^3}\)

\(=\dfrac{\sqrt[3]{a}}{\dfrac{1}{x}}=\dfrac{\sqrt[3]{b}}{\dfrac{1}{y}}=\dfrac{\sqrt[3]{c}}{\dfrac{1}{z}}=\dfrac{\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}}{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}.\)

Vay \(\sqrt[3]{ax^2+by^2+cz^2}=\)\(\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}.\)