Exponent calculation confused about the negative sign

Question

In the below example why did he use -1/12?

For me I just take the twelfth square root i.e. 1/12.

How did the - sign show?

Answer 1

They should have taken both sides to the power of $\frac{1}{12}$, not $-\frac{1}{12}$.

More importantly, it looks like they just made a typo as $(1.1)^{-\frac{1}{12}}=.992\neq 1.00797$.

The correct step to take after $$(1+i_m)^{12}=(1+.1)$$ is to take both sides to the power of $\frac{1}{12}$.

This is because $x^{\frac{1}{12}}$ is the inverse of $x^{12}$, which implies that $$\left(x^{12}\right)^{\frac{1}{12}}=x$$ After applying this transformation, the LHS, which has the variable we want to solve for ($i_m$), is linear in $i_m$. Hence, it is much easier to solve for it.