$ ax^{3} = by^{3} = cz^{3} => \frac{ax^{2}}{\frac{1}{x}} = \frac{by^{2}}{\frac{1}{y}} = \frac{cz^{2}}{\frac{1}{z}}$
= $ \frac{ax^{2} + by^{2} + cz^{2}}{\frac{1}{x} + \frac{1}{y} + \frac{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^{2}}$
= $ \frac{\sqrt[3]{a}}{\frac{1}{x}} = \frac{\sqrt[3]{b}}{\frac{1}{y}} = \frac{\sqrt[3]{c}}{\frac{1}{z}} = \frac{\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c}}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} = \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c}$
Vậy $ \sqrt[3]{ax^{2} + by^{2} + cz^{2}} = \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c}$