Prove: \(\frac{a+b+c}{a^{-1}+b^{-1}+c^{-1}} = abc\)
Proof
\[ = \frac{a+b+c}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}} \]
\[ = \frac{a+b+c}{\frac{bc+ca+ab}{abc}} \]
\[ = (a+b+c)\cdot \frac{abc}{ab+bc+ca} \]
\[ = abc \]
\[ = \frac{a+b+c}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}} \]
\[ = \frac{a+b+c}{\frac{bc+ca+ab}{abc}} \]
\[ = (a+b+c)\cdot \frac{abc}{ab+bc+ca} \]
\[ = abc \]