已知ax立方=by立方=cz立方,1/x+1/y+1/z=1,求证：立方根ax²+by²+cz²=立方根a+立方根b+立方根c

已知ax立方=by立方=cz立方,1/x+1/y+1/z=1,求证：立方根ax²+by²+cz²=立方根a+立方根b+立方根c
已知ax立方=by立方=cz立方,1/x+1/y+1/z=1,求证：立方根ax²+by²+cz²=立方根a+立方根b+立方根c

已知ax立方=by立方=cz立方,1/x+1/y+1/z=1,求证：立方根ax²+by²+cz²=立方根a+立方根b+立方根c
证：
ax^3=by^3=cz^3
xa^(1/3)=yb^(1/3)=zc^(1/3)
y/x=(a/b)^(1/3) y/z=(c/b)^(1/3)
1/x+1/y+1/z=1
y/x+y/y+y/z=y
(a/b)^(1/3)+(b/b)^(1/3)+(c/b)^(1/3)=y
a^(1/3)+b^(1/3)+c^(1/3)=yb^(1/3)
ax²+by²+cz²=ax^3/x+by^3/y+cz^3/z=by^3(1/x+1/y+1/z)=by^3
(ax²+by²+cz²)^(1/3)=yb^(1/3)
因此(ax²+by²+cz²)^(1/3)=a^(1/3)+b^(1/3)+c^(1/3)