Is there any method for calculating cube root of any integer number ?

Question

Method may be similar to the process used for calculating square
root of any integer.

Answer 1

You can use the Newton-Raphson Method to calculate the cube root of a number. It can also be used to calculate the square root of a number.

Let's assume you want to find the cube root of 9.

x^3 = 9

The first thing that you want to do is turn it into a function that equals zero. The Newton-Raphson Method calculates the root values of a function; that means where it crosses the x-axis.

f(x) = x^3 - 9 = 0

We find the derivative the f(x).

f'(x) = 3x^2

The equation for the Newton-Raphson Method is

x2 = x1 - f(x1)/f'(x1)

So substituting the equations we found we get

x2 = x1 - [ (x1)^3-9 / 3(x1)^2 ]

x1 is an approximation of the cube root of a number. We want the cube root of 9 so a safe guess would be 2.

x2 = 2 - [ ((2)^3-9) / (3(2)^2) ] = 2.08333...

The cube root of 9 is 2.08008382305 which is pretty close. The wonderful thing about the Newton-Raphson Method is that it is a recursive function which means you can use the value you got for x2 again to get an even better approximation.

x3 = x2 - [ ((x2)^3 - 9) / (3(x2)^2) ]
x3 = 2.0800888...

x3 is even closer than x2 to the cube root of 9.

Solving for x4, we get

x4 = 2.08008382306

which is really, really close. One more iteration and I most precise value my calculator gets by taking the cube root of 9.

Take note that you can use this equation to solve for the cube root of any number, "b".

x2 = x1 - [ [(x1)^3 - b] / [3(x1)^2] ]

This method works for the cube root of a number because the line only crosses the x-axis in one spot. This method solves for the closest root to your guess. When using this method to solve for the square root of a number, you can get a positive or negative answer depending on your starting approximation. For polynomial equations of the 4th degree and higher, there are even more roots that you can get depending on your initial starting value.

Answer 2

To find the Cube root of a number there four way to do this :

1) look up this website it explain in detail how to extract a cube root of a number:

http://www.nist.gov/dads/html/cuberoot.h...

2) take the logarithm of a number and divide it by 3
{ for example the cube root of 5)

log 5 = 0.69897000 ( I've rounded it to 9 decimal places)

0.69897000 / 3 = 0.23299

Anti log( 0.23299) = 1.709975941

Cube root of 5 = 1.709975941 { accurate to 8 decimal places)

Proof :
1.709975941 * 1.709975941 * 1.709975941 = 4.99999995 or 5

Close enough for grovernment work

The other is the Newton- Raplhin Method:

Gn= ((N/(Gn^2)+(2*Gn)/3


Gn = guess number or new guess number
N = number to be cube rooted

Do this for 5 time to 10 times or until the number no longer changes

for example the cube root of 9 , Gn =2


Run # 1:

Gn=( 9/(2^2) +(2*2))/3

Gn= (9/8 + 4)/3

Gn = (1.125+4)/3

Gn = 5.125/3

Gn = 1.7083333333333 ( round to 12 decimal places )

End of run # 1

Run #2

Gn = ((9/(1.7083333333333 ^2))+2*1.7083333333333 ))/3

Gn= ((3.083878643665) + 2*1.708333333)/3

Gn =(3.083878643665 + 3.416666666)/3

Gn = 6.500545310331/3

Gn = 2.166848436777

end of run # 2

Gn =( (9/2.166848436777^2)+ (2*2.166848436777)))/3

Gn = (1.916838127781) + (2*2.166848436777)

Gn = (1.916838127781+ 4.333696873554)/3

Gn = 6.250535001335/3

Gn = 2.083511667118

end of run # 2

Run # 3 :

Gn = ((9/ 2.083511667118^2) + (2*2.083511667118))/3

Gn = (2.073245044358 + 4.167023334236)/3

Gn = 6.240268378594/3

Gn = 2.080089459532

End of run # 4

Gn= ((9/2.080089459532^2)+ (2*2.080089459532))/3

Gn = (2.080072550138+ (2* 2.080089459532))/3

Gn=( 2.080072550138 + 4.160178919064)/3

Gn = 6.240251469202/3

Gn = 2.080083823067

End Of run 4

Run # 5 :

Gn = ((9/2.080083823067^2) + (2*2.080083823067))/3

Gn = (2.080083823022+ 4.160167646135)/3

Gn = 6.240251469157/3

Gn = 2.080083823052

End of run 5

Cube root of 9 appox equals 2.080083823052

Proof:

2.080083823052^3 = 9.0000000000055713657113417957...

accurate to 11 decimal places

4) Here a another one

the square root of the number than Multiply

Than do the square root twice than multiply it.

Than do the square root four times then multiply it.

Than do the square root eight times than multiply it

Than do the square root sixteen times than multiply it

Than do the square root thirty two times than multiply

If more accurate results are needed than continue this by square root it (64.128,256,512,1024..etc times) and multiply after each square rooting if you like

Square root the number

this will be your cube root

For example the cube root of 7 ( I'm not writing out all the square roots of each step here, but I will right out the results of the steps at the end of of the square rootings here (Okay)


enter 7

Square root of 7

2.645751311 {round to 9 decimal places here}

Multiply it or hit the times key on the Calulator (which I'll use X to represent it)

than square root it twice

1.2753731078

X

3.374320070

do the square root four times

1.078975646

X

3.6408091771458413696541366439634

Do the square root eight times.

1.005060441

X

3.659233275

Do the square root sixteen times

1.000019795

X

3.65930571

Do the square root thirty two times

1.000000000

X

3.659305710

Square root it

The Cube of seven appoximatlly equals

1.912931183

Proof:

1.912931183 * 1.912931183 * 1.912931183 = 7

3.659305710893779489 * 1.912931183 = 7

7.000000002498693585233905487 = 7 ( Accurate to 9 decimal close enough for government work)

So Cube root 7 is 1.912931183

Answer 3

The method used is similar to determining a square root. Its just slighty more complicated. http://mathforum.org/library/drmath/view/52605.html

Answer 4

sophisticated factor. search into the search engines. that will can assist!