Bài 103812

Question

Cho

$ax^3 = by^3 = cz^3$ và

$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$
Chứng minh :

$\sqrt[3] {ax ^2+ by^2 + cz^2} = \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c}$

Tiểu Bắc · Answer 1 · 5/9/2012 4:11:37 PM

Đặt :

$E = \sqrt[3]{{{\rm{a}}{{\rm{x}}^2} + b{y^2} + c{{\rm{z}}^2}}} = \sqrt[{3}]{{\frac{{{\rm{a}}{{\rm{x}}^3}}}{x} + \frac{{b{y^3}}}{y} + \frac{{c{{\rm{z}}^3}}}{z}}}$

$= \sqrt[3]{{\frac{{{\rm{a}}{{\rm{x}}^3}}}{x} + \frac{{{\rm{a}}{{\rm{x}}^3}}}{y} + \frac{{{\rm{a}}{{\rm{x}}^3}}}{z}}} = \sqrt[3]{{{\rm{a}}{{\rm{x}}^3}\left( {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right)}} = x\sqrt[3]{a}$

$\Rightarrow\frac{\sqrt[3]{a}}{\sqrt[3]{ax^2+by^2+cz^2}}=\frac{1}{x}$