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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rrr|rrrr} &&&2\\ \hline &&\sqrt[3]{}&10&460&353&203\\ \underline2^3&=&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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} , the number written so far above the radical, and y be the next digit to calculate. The term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 300\times 2^2y+30\times 2y^2+y^3\le 2460} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rrr|rrrr} &&&2&1\\ \hline &&\sqrt[3]{}&10&460&353&203\\ \underline2^3&=&8&-8&\downarrow\\ \hline 300\times 2^2\times\underline1&&&2&460\\ {}+30\times 2\times\underline1^2+\underline1^3&=&1261&-1&261&\downarrow\\ \hline 300\times 21^2 \times \_&&&1&199&353\\ {}+30\times 21 \times\_^2+\_^3&=& \end{array}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rrr|rrrr} &&&2&1&8&7\\ \hline &&\sqrt[3]{}&10&460&353&203\\ \underline2^3&=&8&-8&\downarrow\\ \hline 300\times 2^2\times\underline1&&&2&460\\ {}+30\times 2\times\underline1^2+\underline1^3&=&1261&-1&261&\downarrow\\ \hline 300\times 21^2 \times \underline8&&&1&199&353\\ {}+30\times 21 \times\underline8^2+\underline8^3&=&1099232&-1&099&232&\downarrow\\ \hline 300\times 218^2\times \underline7&&&&100&121&203\\ {}+30\times 218\times\underline7^2+\underline7^3&=&100121203&&-100&121&203\\ \hline &&&&&&0 \end{array}}

An exact root has been reached when the remainder is zero.