Is there any manual method to calculate cubic root as there is for sq root ?

Question

Please enlighten me if you know of the method

Answer 1

"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."
-Doctor Rob

Answer 2

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Answer 3

There is a nice numerical method called Newton-Raphson approximation. It is a stepwise method, by which you can arrive at a more and more accurate approximation in every step.

If you want to compute the cubic root of a non-negative number a, start with an arbitrary (but preferably, reasonable) guess x0.
(You can do the same for negative numbers by computing the cubic root of the positive number and then multiplying it by -1.)

Then a more accurate guess x1 is achieved by substituting into the formula

x1 = (2*x0^3 + a) / (3*x0^2).

In general, the formula (a so-called recursion) to compute the next approximation from the previous one is:

x(n+1) = (2* x_(n)^3 + a) / (3*x(n)^2).

For example, suppose you would like to compute the cubic root of a=10. Let's start with a guess x0=2.

Then you get x1 = (2*2^3 + 10) / (3*2^2) = 26/12 = 2.1666...

By applying the above formula, you get x2 = 3277/1521 = 2.1545

You can do this up to any desired accuracy.

For example, in just 4 approximations, you arrive at x4=2.154434690, which is identical to the actual answer at least up to 9 digits as you may verify it.

The method can be applied to finding roots of several other equations as well. You can read more about it at the following link:

http://en.wikipedia.org/wiki/Newton%27s_method

Answer 4

The Newton-Raphson is just what I was going to mention, although I might suggest a slightly easier way of calculating the iterations. Instead of using

x_[n+1] = (2x_n³ + a)/(3x_n²)

You might try

x_[n+1] = (2x_n + a/x_n²) / 3

which eliminates the need to cube x_n.

The Newton-Raphson method doesn't give you the exact number a digit at a time like the "long-division-like" method does, but it does converge on the answer faster and with a lot less work. In general, the formula for deriving the kth root of a number a is:

x_[n+1] = ((k-1)x_n + a/x_n^(k-1)) / k

In fact, you can create and use an appropriate Newton-Raphson formula for deriving numerical solutions for all sorts of equations, not just for calculating roots.

Good luck!

Answer 5

best manual method is u first take out the factors, then from there see which ones r coming thrice, and thus can calculate cube root.

Answer 6

I hear no method to calculate manual

Answer 7

http://mathforum.org/library/drmath/view/52605.html

(I already answered this for you)