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.

Answer 2

You can use the "long-division-like" method, but there's an iterative method called Newton-Raphson Iteration that works a lot faster.

Specifically, to find the cube root of any positive number Z: start with a good "initial guess," which we'll call x_0. Then, you'll use the following formula to make successive approximations x_1, x_2, x_3, etc.

x_[n+1] = (2x_n + Z/(x_n)²) / 3

For instance, to find the cube root of 800, we might start with an initial guess of x_0 = 9 (since 9³ = 729). Then we iterate, plugging each approximation back into the formula to get the next one:

x_1 = (2*9 + 800/9²) / 3 = 2258 / 243 ≈ 9.29 21 81 07
x_2 ≈ 9.28 31 86 39
x_3 ≈ 9.28 31 77 67
x_4 ≈ 9.28 31 77 67

Notice that, to at least 8 decimal points, we've now converged on an answer. If you need more accuracy, carry out the iterations until you converge on the answer you need.

Hope that helps!

Answer 3

The Newton-Raphson is merely what i changed into going to point, besides the very undeniable truth that i'd recommend a touch less complicated way of calculating the iterations. instead of using x_[n+a million] = (2x_n³ + a)/(3x_n²) you may attempt x_[n+a million] = (2x_n + a/x_n²) / 3 which eliminates the opt to dice x_n. The Newton-Raphson technique would not provide you the precise form a digit at a time like the "lengthy-branch-like" technique does, even though it does converge on the answer swifter and with plenty a lot less artwork. many times, the formula for deriving the kth root of a range a is: x_[n+a million] = ((ok-a million)x_n + a/x_n^(ok-a million)) / ok in truth, you could create and use an proper Newton-Raphson formula for deriving numerical innovations for all sorts of equations, no longer only for calculating roots. reliable success!

Answer 4

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.