When doing financial, actuarial or probability calculations, often a person is faced with computing exponentiations.
For example, let's say the chance of a particular ship sinking is 1% per year. Then to compute the chance of the ship sinking over the course of a 30 year lifetime is 1 - 99% × 99% ... × 99% (30 times) or $1-0.99^{30}$. So, we have the problem of computing $\frac1{99}$ raised to the 30th power. Is there a fast way to do this kind of calculation manually in the head or using a pencil and paper?
I am not sure about anything precise, but I can give the following approximate rule which turns out to be rather useful: if $n \geq 10$, $(1-\frac{1}{n})^n \approx \frac{1}{e} \approx 0.37$.
So in your example, you want to approximate now $0.37^{3/10} \approx 0.37^{1/3}$. Now, $0.7^3 =0.343$, $0.8^3=0.512$ so, since the cube function grows faster as the argument increases (eg: a linear function has constant growth, so if the cube was linear, you would approximate $0.37 \approx 0.715^3$ but it is convex, albeit not too strongly), you are likely to have some sort of $0.73^3 \approx 0.37$.
So your quantity can be guessed at $0.73$. The actual value is $\sim 0.739$...
Of course, the hundredth-precision is a stroke of luck, but the tenth-precision is not.