how can it be proved using field axioms that
$\frac{1}{\sqrt[3]{100}}=\frac{\sqrt[3]{10}}{10}$
I have the next sketch proof:
First I applied the definition of quotient. Then I used that $1=(\sqrt[3]{100^2})(\sqrt[3]{100^2})^{-1}$. And then use commutativity together with $(a\cdot b)^{-1}=a^{-1}\cdot b^{-1}$
But how do I justify that $\sqrt[3]{100^2}\sqrt[3]{100}=100$?
Thanks a lot.