Well, do you agree that $\sqrt[7]9$ is a number such that $(\sqrt[7]9)^7=9$?
Now, suppose $$\begin{align*}\sqrt[7]9&=9^x\\\ (\sqrt[7]9)^7&=(9^x)^7 \tag 1\\\ 9^{7x}&=9 \tag 2\\\7x&=1 \tag 3\\\ x&=\dfrac 1 7\end{align*}$$
We have used in going from $(1)$ to $(2)$ that,
$$(x^a)^b=x^{ab}$$
In going from $(2)$ to $(3)$ we use the fact that, $$x^a=x^b \implies a=b$$