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i want a method to get cube root of a real number without using computer or log table. i want a method that i can use anywhere even in a train or plane only using pen and paper .we can find square root of a number using a division method.we can find square root of any rational or irrational number by using this simple division method . I want the same to find cube root ,too.But how? Please provide the answer.

2006-12-16 21:43:50 · 9 answers · asked by raj 1 in Science & Mathematics Mathematics

9 answers

though there isn't a method exactly like the one for finding the square root the cube root can be found without the calculator.suppose you want to find the sq.rt. of 129.5^3 is 125 and 6^3 is 216
since 129 is very close t0 125 try 5.1.5.1^3 is 132.xxx it must be >5 but <5.1.try 5.06 it is 129.5,very close
this method can be very effective with practice

2006-12-16 23:10:06 · answer #1 · answered by raj 7 · 0 0

Well, if you are acquainted with calculus, there is a simple method. Only problem is that you will have to know cubes of some natural numbers close to the value you are calculating.

Here is the method,

Any function f(x) can be written as

f(x)=f(t) + (x-t)f '(t)/1!+ (x-t)^2(f ''(t))/2! ...................Using Taylor's series.

Now if you have to find the cube root of a number say 740, you can use the series. The closest perfect cube to 740 is 729 (9^3=729).

In the above equation x=740, t=729 and f ' indicates differential.

Now f(x)=(x)^(1/3)

=>f '(x)=1/3(x^(-2/3))

Substitute values in equation, using first 2 terms
=>f(740)=f(729) + (740-729)f '(729)
=>f(740)=9 + 11*[1/3(729^(-2/3))]
=>f(740)=9 + 11/3 * 1/81
=>f(740)=9 + 0.045
=>f(740)=9.045

Correct to 2 decimal places!
The beauty of the method is that f(x) can be anything sin, cos, log, tan..........But you will have to know some standard values. Also the accuracy of the answer depends on the closeness of x to t and also the number of terms used in the series.


For other numbers like keep dividing by the number which is the smallest factor and then make triplets of the factors. Write each number in each triplet once and multiply.

eg: 512
clearly, divisible by 2
512/2=256..........first factor is 2
also 256 is divisible by2
256/2=128..........second factor is 2
also,128/2=64....third is 2
64/2=32....4th is 2

in this way you get 9 2's

=>(2*2*2) * (2*2*2) * (2*2*2)
=2*2*2=8

Therefore cube root of 512 is 8

If you want to find square root

f '(x)=1/2(x^(-1/2))
f ''(x)=-1/4(x^(-3/2))

2006-12-17 01:07:52 · answer #2 · answered by Anonymous · 0 0

though there isn't a method exactly like the one for finding the square root the cube root can be found without the calculator.suppose you want to find the sq.rt. of 129.5^3 is 125 and 6^3 is 216
since 129 is very close t0 125 try 5.1.5.1^3 is 132.xxx it must be >5 but <5.1.try 5.06 it is 129.5,very close
this method can be very effective with practice

2006-12-17 16:08:04 · answer #3 · answered by arpita 5 · 0 0

Say you want to find the cubed root of a. Start with x0=good first guess. Then iterate the formula x(n+1)=[2*(xn)^3+a]/[3(xn)^2].

For instance, cubed root of two is about 1.259921049894 873164767211. Start with guess x0=1 (cubed root of 2 is *about* 1), then calculate

x1=(2*1^3+2)/(3*1^2)=4/3=1.33333 (closer)
x2=(2*[4/3]^3+2)/(3*[4/3]^2)=91/72=1.263888 (closer)
x3=(2*[91/72]^3+2)/(3*[91/72]^2)=1126819/894348=1.259933 (closer still)

The number of accurate digits in the decimal expansion will approximately double with each step.

2006-12-17 02:02:48 · answer #4 · answered by a_math_guy 5 · 0 0

Left alone, on a desert island, without any means than a pencil and a sheet of paper, I would proceed as such: Iteratively! That is by trial and error OK, imagine that you have to calculate the cube root of 25 2x2x2=8 3x3x3=27 so it is between 2 and 3, probably rather towards 3 I try 2.9 2.9x2.9x2.9=29x29x29 x 1/1000 = 24389/1000 =24,389 So it is between 2.9 and 3.0, rather towards 2.9 So I try with 2.94 and so on It works!

2016-03-13 07:52:03 · answer #5 · answered by Anonymous · 0 0

a slide rule (same as a log chart ) instant awnser ,a calculator or a computer just guesses at it then splits the difference add it or subtracts it from the root base spools the numbers agane and so on till it works out ( most times it finds in 4 trys go figure)

2006-12-17 01:23:41 · answer #6 · answered by Bob B 3 · 0 0

There could be some ways but i never heard any method to find the cube root! You can estimate. for example (10)^(1/3) is between 3 and four. so choose the midpoint which is 3.5 and find out whether 3.5 is less than cube root of 10 or greater than it. (like 3.5*3.5*3.5 < 10 or not) if it is less than it, (3.5 < cube root of 10 < 4), this time choose 3.75 and calculate for 3.75 and so on ...

you can search it at google for better and easier solutions!!

2006-12-16 22:13:00 · answer #7 · answered by ___ 4 · 0 0

Either use trial and error, or use the Windows calculator in your computer, or invest in a pocket calculator having a y^1/x key. Cost about £3, or $5.
I do not know how people manage to live without such calculators.

2006-12-17 06:17:06 · answer #8 · answered by Anonymous · 0 1

yes of course you can! without even bringing something provided that you know what is the number that is to be converted. Use your MIND...

2006-12-16 22:09:22 · answer #9 · answered by CriZzie ツ 3 · 0 3

square

2006-12-16 21:45:41 · answer #10 · answered by chi 4 · 0 2

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