Find the value of $m+n+p$ if $a^{\frac{1}{m}}= b^{\frac{1}{n}} = c^{\frac{1}{p}}$ and $abc = 1$

Question

If $a^{\frac{1}{m}}= b^{\frac{1}{mn}} = c^{\frac{1}{p}}$ and $abc = 1$, then find the value of $m+n+p$.

Answer 1

Let $a^{\frac{1}{m}}=k$.

Hence, $$1=abc=k^m\cdot k^n\cdot k^p=k^{m+n+p}$$ Thus, $m+n+p=0$ or $k=1$.

For $k=1$ we have $a=b=c=1$ and we can not find a value of $m+n+p$.