Cube root: Difference between revisions

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explain third step
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:<math>(10x+y)^3=1000x^3+300x^2y+30xy^2+y^3</math>
:<math>(10x+y)^3=1000x^3+300x^2y+30xy^2+y^3</math>


if we let <math>x=2</math>, the number written so far above the radical, and ''y'' be the next digit to calculate. The term <math>1000x^3=8000</math> 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 <math>300\times 2^2y+30\times 2y^2+y^3\le 2460</math>.
if we let <math>x=2</math>, the number written so far above the radical, and ''y'' be the next digit to calculate. The term <math>1000x^3=8000</math> 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 <math>300\times 2^2y+30\times 2y^2+y^3\le 2460</math>. Here we have found <math>y=1</math>, and for the next step set <math>x=21</math> to find the next digit of the root after that.


:<math>\begin{array}{rrr|rrrr}
:<math>\begin{array}{rrr|rrrr}
&&&2&1\\ \hline
&&&2&1&\_\\ \hline
&&\sqrt[3]{}&10&460&353&203\\  
&&\sqrt[3]{}&10&460&353&203\\  
\underline2^3&=&8&-8&\downarrow\\ \hline
\underline2^3&=&8&-8&\downarrow\\ \hline

Revision as of 07:42, 27 December 2024


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.

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.

The next step is based on the identity [1][2]

if we let , the number written so far above the radical, and y be the next digit to calculate. The term 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 . Here we have found , and for the next step set to find the next digit of the root after that.

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

  1. WikiHow: How to Calculate Cube Root by Hand https://www.wikihow.com/Calculate-Cube-Root-by-Hand
  2. Paul E. Black, “cube root” in Dictionary of Algorithms and Data Structures, [online], ed. 6 May 2019. https://www.nist.gov/dads/HTML/cubeRoot.html