Say you were asked to calculate a high-order root by hand. For example, the 13th root of 230,120,000 to 4 significant digits.
How would you do it? Would you do a manual Newton-Raphson iteration? Or perhaps a more crude sequence of repeated guessing? Would you manually calculate a Taylor series? What would be the fastest approach?
One could use logarithms for such a problem .
For example ,lets take ,as you said, 230120000$^{1/13}$
Set Z = 230120000$^{1/13}$
log$_{10}$(Z) = $\frac{1}{13}$.log$_{10}$(230120000)
=$\frac{1}{13}$.(5+3.361954)
=$\frac{8.3619954}{13}$
log$_{10}$(Z)=0.64322723076
Z = 10$^{0.64322723076}$
Z = 4.397717
Therefore the 13$^{th}$ root of 230,120,000 is 4.397717
Virtually any root is calculable along with many other calculations if one has a log book handy.
$Note$ : I'm using base 10 but you can use any base with respect to the number whose root you want to find .
