How to find cube root of a number without using log-tables or computers?

Question

The method to find square-roots of a number is taught in schools. It is based on principle of divisions. How can one find cube-root of a positive number arithmatically, in a similar fashion. Use of log-tables, calculators or computers not allowed.

Answer 1

Try the method at this link.

To find a cube root by the "longhand" method, we proceed very much as
we do to find a square root by hand. I intersperse numbered steps
with an example. We will find the cube root of 113 to two decimal
places.

1. Draw a cube root symbol, or radical, with the number whose root you
are seeking underneath. Start with the decimal point and mark off
digits in both directions in groups of three. Put a decimal point
above the radical, and directly above the other decimal point.

.
3/-----------
\/ 113.000 000

2. Start with the first group of 1, 2, or 3 digits. Find the largest
cube of a single-digit integer less than it. Write the single digit
above the radical, and its cube under the first group. Draw a line
under that cube, and subtract it from the first group.

4.
3/-----------
\/ 113.000 000
64
-------
49

3. Bring down the next group below the last line drawn. This forms
the current remainder. Draw a vertical line to the left of the
resulting number, and to the left of that line put three hundred
times the square of the number above the radical, a plus sign,
thirty times the number above the radical, a multiplication sign,
an underscore character, another plus sign, another underscore
character, the exponent 2, an equals sign, and some blank space for
the answer.

4.
3/-----------
\/ 113.000 000
64
-------
4800+120*_+_^2=???? | 49 000

4. Pick the biggest digit D that would fit into both underscore
places, and give a number such that D times it is less than the
current remainder. Put it above the radical above the last group of
digits brought down, and put it in each of the blanks where the
underscore characters are. Compute the number given by the
expression, and put it after the equals sign. Multiply D times that
number, and put that below the current remainder, draw a horizontal
line below that, and subtract, to give a new current remainder.

4. 8
3/-----------
\/ 113.000 000
64
-------
4800+120*8+8^2=5824 | 49 000
46 592
----------
2 408

5. If the current answer, above the radical, has the desired accuracy,
stop. Otherwise, go back to step 3.

Step 3:
4. 8
3/-----------
\/ 113.000 000
64
-------
4800+120*8+8^2=5824 | 49 000
46 592
----------
691200+1440*_+_^2=?????? | 2 408 000

Step 4:
4 . 8 3
3/-----------
\/ 113.000 000
64
-------
4800+120*8+8^2=5824 | 49 000
46 592
----------
691200+1440*3+3^2=695529 | 2 408 000
2 086 587
---------
321 413

Step 5: Stop.

Thus the cube root of 113 to two decimal places is 4.83. Checking,
4.83^3 = 112.6786, and 4.84^3 = 113.3799, so the answer is correct.

Answer 2

Cubed Root Table

Answer 3

You can find systematic method in your high school algebra/arithmatic books.

It may be on line. There are calculators with cub root keys available.

Answer 4

Stunning, isn't it, the number of people who don't seem to understand what the words "Use of log-tables, calculators or computers not allowed." actually means?

And I'll bet that a lot of them would even claim that English is their native language.


Doug

Answer 5

trial and error is generally the best approach

this site:
http://mathforum.org/library/drmath/view/52606.html

provides an algorythm that makes the trial and error a little more systematic

Answer 6

have u evr seen the button on calculatr ? and then said to yuor self yep that must b it

Answer 7

You split it into a factor tree

...27
I....I....I
3 *3 *3

Answer 8

use calculater !!!!!!! heheheheheheee