Let $n\in N,n=2k+1,and\text{}\frac{1}{a+b+c}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Show that $\frac{1}{{a}^{n}+{b}^{n}+{c}^{n}}=\frac{1}{{a}^{n}}+\frac{1}{{b}^{n}}+\frac{1}{{c}^{n}}$

I have tried, but I don't get anything. Can you please give me a hint?

Show that $\frac{1}{{a}^{n}+{b}^{n}+{c}^{n}}=\frac{1}{{a}^{n}}+\frac{1}{{b}^{n}}+\frac{1}{{c}^{n}}$

I have tried, but I don't get anything. Can you please give me a hint?