$\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $?

Question

Is it always true that: $\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $ where $m,k \in \mathbb N$ ?

I tried it with a few numbers and it seems to work every time.

Answer 1

It should be \begin{align} \sqrt[\large m]{x+y}\over \sqrt[\large k]{x+y}&=\frac{(x+y)^\frac{1}{m}}{(x+y)^\frac{1}{k}}\\ &=(x+y)^{\large \frac{1}{m}-\frac{1}{k}}\\ &=(x+y)^{\large \frac{k-m}{km}}\\ &=\sqrt[\large km]{(x+y)^{\large k-m}} \end{align}