how do you work out the cube root of a number?

Question

for example what is the cube root of 234? do you know? does anyone know?

Answer 1

You can memorize the cubes of the first few numbers, e.g.
1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000.

For any other numbers, use a calculator.

Or you can perhaps estimate:
6³ = 216 and 7³ = 343 => the cube root of 234 is between 6 and 7.

But if you REALLY REALLY have time and want to do cube roots by hand, use this method:

To find a cube root by the "longhand" method, we proceed very much as we do to find a square root by hand. Example: 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+8²=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+8²=5824 | 49 000
46 592
----------
691200+1440·__+__²=?????? | 2 408 000

Step 4:
4 . 8 3
3/-----------
\/ 113.000 000
64
-------
4800+120·8+8²=5824 | 49 000
46 592
----------
691200+1440·3+3²=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,
4.83³ = 112.6786, and 4.84³ = 113.3799, so the answer is correct.

Answer 2

About Cubic Root of Numbers

As given in some of the earlier answers there is a 'long division' method similar to extraction of square root that is available for cubic roots also.
I read about this method in Leelavathy of Bhaskara ( I presume that, this will be oldest text dealing with that method)

2. The above method being laborious , we can extract the roots by a simple method as follows to the desired degree of accuracy

1. Find nearest exact cube - In this case 6 {6^3= 216)

2. Find the difference 234-216=18
3. Divide it by 3 times square of the previous exact cube

18/(3*6*6) = 0.167

4. Now an approximate answer is 6+0.167=6.167 ( Exact answer 6.162240...) Therefore it is correct up two decimals
5. Now find cube of the result = (61/6)^3 = 254.5046296

6. Deduct 254 & divide it again by thrice the square of 6

Answer = 0.004672496
7. Deduct from previous result
Answer = 6.16666667-0.004672496=6.16199417
Now result is almost ok

Note : When nearest cube is greater , reverse the step deduct & then add - For Example Cubic Root of 124 is approximately 5- (1/3*25) = 4 .98666...
( Actual being 4.9866309)

Answer 3

There is a way to find square roots longhand that makes long division look simple. There's a way to do cube roots longhand as well, EXTREMELY complex in comparison. I've never done it. In the days BC, Before Calculators, you'd look up the log of 234 in a table [average students carried 4 place tables, math nerds carried 5 place tables], divide it by 3, and look up the antilog. Now you just ask the calculator for the cube root and get the answer to 10 decimal places. If there's no obvious cube root button, find 234^(1/3) = 6.162240148, which is more than accurate enough to build starships.

Answer 4

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Answer 5

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Answer 6

A SQUARE root x^2 always gives a positive result regardless if x is actually a negative or positive number, whereas a CUBE root x^3 only gives a positive number if x is actually positive.

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Answer 7

There is no easy and quick method like for square root. However, you can have an iterative series calculation that will converge to the answer in a few steps. See more in this wikipedia page (link) in the "cube root on a standard calculator" section.

Answer 8

cube root = 616

If square root is twice cube root is thrice
Ex:
2x2x2 = 8
Therefore the cube root of 8 is 2.
3x3x3 = 27
Therefore the cube root of 27 is 3.

So on and so forth....

TC! GOD BLESS!

Answer 9

6.162240148


There aren't any particularly easy methods for doing it in your head. So usually, you'll just use a calculator.

234^(1/3)

Or most scientific calculators actually have a cube-root button (often a secondary function to x³, or the sqrt button)

Answer 10

Square Root Long Hand