# Root Calculation by Hand

Is it possible to calculate and find the solution of $\; \large{105^{1/5}} \;$ without using a calculator? Could someone show me how to do that, please?

Well, when I use a Casio scientific calculator, I get this answer: $105^{1/5}\approx " 2.536517482 "$. With WolframAlpha, I can an even more accurate result.

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.