How to approximate rational exponents?

Question

On a recent exam, in an integer-type question in Chemistry, I calculated my final answer to be 10^0.6, I thought I had calculated the answer incorrectly as there was no way that would be the answer. I just left it since there were no calculators allowed. Turns out, putting it into a calculator later, it's about 3.99 or 4 (That was the answer). I asked my Maths teacher but he didn't have a way (Except memorizing the anti-log table), How do you go about estimating numbers like these?

Answer 1

There is no general recipe for doing it in your head. But in this case you want $$ 10^{0.6}=10^{3/5}=(10^3)^{1/5}=\sqrt[5]{1000}. $$ Since $4^5=1024$, the number $10^{0.6}$ is very slightly below $4$.