Root Calculation by Hand

I go back to the days BC (before calculators). We did have electricity, but you had to rub a cat's fur to get it.

We also had slide rules, from which a $2$ to $3$ place answer could be found quickly, with no battery to go dead in the middle of an exam. Engineering students wore theirs in a belt holster. Unfortunately, slide rules were expensive, roughly the equivalent of two meals at a very good restaurant. For higher precision work, everyone had a book of tables.

My largish book of tables has the entry $021189$ beside $105$. This means that $\log(105)=2.021189$ (these are logarithms to the base $10$, and of course the user supplies the $2$). Divide by $5$, which is trivial to do in one's head (multiply by $2$, shift the decimal point). We get $0.4042378$.

Now use the tables backwards. The log entry for $2536$ is $404149$, and the entry for $2537$ is $414320$. Note that our target $0.4042378$ is about halfway between these. We conclude that $(105)^{1/5}$ is about $2.5365$.

The table also has entries for "proportional parts," to make interpolation faster. As for using the table backwards, that is not hard. Each page of the $27$ page logarithms section has in a header the range of numbers, and the range of logarithms. The page I used for reverse lookup is headed "Logs $.398\dots$ to $.409\dots$."

There are other parts of the book of tables that deal with logarithms, $81$ pages of logs of trigonometric functions (necessary for navigation, also for astronomy, where one really wants good accuracy). And of course there are natural logarithms, only $17$ pages of these. And exponential and hyperbolic functions, plus a few odds and ends.

You can try using binomial theorem for real exponents.

You can write this as

$$ ((2.5)^5 + (105 - (2.5)^5))^{1/5} = 2.5 \left(1 + \frac{105 - (2.5)^5}{2.5^5}\right)^{1/5} = \frac{5}{2} \left(1 + \frac{47}{625}\right)^{1/5}$$

Taking first three terms of the binomial series

$$(1+x)^r = 1 + rx + \frac{r(r-1)x^2}{2!} + \frac{r(r-1)(r-2)x^3}{3!} + \dots$$

using $r = \frac{1}{5}$ and $x = \frac{47}{625}$ gives us

$$ \frac{5}{2} \left(1 + \frac{47}{5*625} - \frac{4 * 47^2}{2*5^2*625^2}\right) = \frac{4954041}{1953125} \approx 2.5365$$

If you need a better approximation, you can include more terms.

All this can be done by hand using integer arithmetic, but is tedious.