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### Cube Root by Hand

```
Date: 01/23/98 at 15:48:37
Subject: Cube root

Dear Sir,

Many years ago I learned to calculate the square root of a number
(without calculators or computers, just pencil and paper).

An example will tell you how (square root of 222784)

22.27.84   I   472
16         I ---------------------------
-----       I
62.7     I    87    I    942
60.9     I   X 7    I   X  2
-------- I   -------------------------
188.4 I   609    I   1884
188.4 I
------
0

I suppose this is relatively familiar to you.

Later we learned a comparable but of course more complicated way to
calculate the cube root of a number. I forgot almost completely how.
What I remember is that we had to split the number up in groups of
three figures instead of two (example: 274625  becomes 274.625) and
that we had to take the first group (here 274) and find the
approximate cube root of that number (here 6 since 6x6x6=216 and
7x7x7=343). But then I am lost.

Can you help me further?
Many thanks in anticipation.

Adrian ( Bruges - Belgium )
```

```
Date: 03/03/98 at 16:18:48
From: Doctor Rob
Subject: Re: Cube root

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
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

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.

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```

```
Date: 03/14/98 at 07:24:04
Subject: Cube root

Dear Sirs,

I want to thank you very much for your help. I would like to

1. The "determination" of D seems to me a bit arbitrary: try 9, if 9
is incorrect try 8, if 8 is incorrect, try 7; if 7 is not correct,
try 6, and so on. In a book that I found recently in a shop for
second-hand books (edition 1910!), I found large theoretical
considerations on the square root and the cube root of a number. To
find the cube root of a number up to step 2 they use a similar method
as the one you proposed. To determine D they say (for your given
example): divide 490|00 by 3*4^2 = 3*16 = 48. Result 10, which of
course is too large. Trying 9: still too large, but as we know 8 is
correct. In most cases, however, this way of determination of D gives
the correct D immediately.

2. Another way to control whether D is correct or not is to calculate
(4.8)^3 and to subtract it from the given number 113.000. If the
result is positive then D is correct. Here also (113.000 - 110.592)
= 2.408; however, this result will also be positive if D is taken too
small: nothing is perfect in this world.
```

```
Date: 03/18/98 at 12:36:59
From: Doctor Rob
Subject: Re: Cube root

Usually the next digit D is the quotient gotten by dividing the number
to the left of the vertical line (the divisor), *ignoring D* (or
setting D = 0), into the current remainder. In this case, you would
divide 4920 into 49000, and the quotient digit D is guessed to be 9.
As you progress to later and later stages in the algorithm, this
method is more and more accurate. This problem is exactly analogous to
the one encountered when doing long division: what is the next digit?
You have to estimate the right answer, and adjust if the remainder is
too big or too small.

-Doctor Rob,  The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents
High School Square & Cube Roots

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