Cube root

Extracting cube roots by hand

Start by grouping the digits under the radical into groups of three as you normally would. This seems natural to do, even if it is a little more complicated than the square root.

${\sqrt[{3}]{10,460,353,203}}$

The first step here is to find the largest digit whose cube does not exceed the first group of digits under the radical, cube it, subtract to find the remainder, and bring down the next three digits.

${\begin{array}{rrr|rrrr}&&&2\\\hline &&{\sqrt[{3}]{}}&10&460&353&203\\{\underline {2}}^{3}&=&8&-8&\downarrow \\\hline &&&2&460\end{array}}$

The next step is based on the identity

$(10x+y)^{3}=1000x^{3}+300x^{2}y+30xy^{2}+y^{3}$

if we let $x=2$, the number written so far above the radical, and y be the next digit to calculate. The term $1000x^{3}=8000$ has already been subtracted when we bring down the next group of three digits, so the next digit to write above the radical will be the largest y such that $300\times 2^{2}y+30\times 2y^{2}+y^{3}\leq 2460$. Here we have found $y=1$, and for the next step set $x=21$ to find the next digit of the root after that.

${\begin{array}{rrr|rrrr}&&&2&1&\_\\\hline &&{\sqrt[{3}]{}}&10&460&353&203\\{\underline {2}}^{3}&=&8&-8&\downarrow \\\hline 300\times 2^{2}\times {\underline {1}}&&&2&460\\{}+30\times 2\times {\underline {1}}^{2}+{\underline {1}}^{3}&=&1261&-1&261&\downarrow \\\hline 300\times 21^{2}\times \_&&&1&199&353\\{}+30\times 21\times \_^{2}+\_^{3}&=&\end{array}}$

${\begin{array}{rrr|rrrr}&&&2&1&8&7\\\hline &&{\sqrt[{3}]{}}&10&460&353&203\\{\underline {2}}^{3}&=&8&-8&\downarrow \\\hline 300\times 2^{2}\times {\underline {1}}&&&2&460\\{}+30\times 2\times {\underline {1}}^{2}+{\underline {1}}^{3}&=&1261&-1&261&\downarrow \\\hline 300\times 21^{2}\times {\underline {8}}&&&1&199&353\\{}+30\times 21\times {\underline {8}}^{2}+{\underline {8}}^{3}&=&1099232&-1&099&232&\downarrow \\\hline 300\times 218^{2}\times {\underline {7}}&&&&100&121&203\\{}+30\times 218\times {\underline {7}}^{2}+{\underline {7}}^{3}&=&100121203&&-100&121&203\\\hline &&&&&&0\end{array}}$

An exact root has been reached when (and if) the residue is zero. Typically the number whose root is being taken is called the solvend, the equations to the left of the vertical line are resolvent equations, and at every step, to the right of the vertical line, the resolvent is subtracted from the previous resultant to yield a residue, as each digit of the root is calculated as the greatest digit whose resolvent does not exceed the resultant. The new resultant is the residue with three more digits of the solvend brought down.

Knowing the value of π, and the exact density of lead, gunners would have used cube roots for calculating the precise diameter of spherical lead shot sold by weight among other matters, and gunnery was considered an important branch of applied mathematics, to calculate the trajectories of artillery shells using geometry, square and cube roots, trigonometry, calculus and so forth, long before the days of electronic calculators and computers.