The question is as follows:
Anna has money invested in an account that pays $6$% interest per year. At what rate is the investment increasing each decade? What is the monthly rate of growth?
As for the decade, I calculated by doing $(1.06)^{10} = 1.791$ and from that I subtracted $1$ to get $0.791 \times 100 = 79.1$%.
I am unsure as to what I should do for the monthly rate of growth. Which one of the following calculations should I be doing?
$$r^{12} = 0.06$$ $$(1 + r)^{12} = 1.06$$ $$r^{12} = 1.06$$
Any help will be greatly appreciated?
Either of the latter two calculations you proposed would work, where the rate of growth is $r$ in the second calculation, and $r-1$ in the 3rd.
A third equivalent way is solved $(1.06)^{1/12} = 1+ r_{monthly}$, and it ties in to how you found how much growth you had after 10 years where you solved $(1.06)^{10} = 1+r_{decade}$. (For the decade you had 10 years here you have 1/12 years)
Meanwhile see for yourself why the first calculation you proposed is wrong.