Cube Root of 2
The cube root of 2, denoted as ^{3}√2, is the value which gives the original number when multiplied by itself thrice. This definition is applicable to all the cube roots of natural numbers. Now, since 2 is not a perfect cube, therefore we cannot find it using the prime factorisation method.
Cube root of 2, ^{3}√2 = 1.2599 |
We will try to find the value of ^{3}√2, without using a calculator and using the approximation method.
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Calculation of Cube Root of 2
As we already know, 2 is not a perfect cube, therefore we cannot use the general methods, to find its cube root. Let us find it by using approximation or we can also say, as an estimation method.
Assume, ^{3}√2 = X
So, here X should be equal to a number, which after getting multiplied by itself three times gives the result as 2.
But if we see, 2 is nearly equal to the cube of 1, i.e. 1^{3} = 1
So let us just say,
X = p × p × q
Take p =1 and q = 2
X = 1 × 1 × 2 = 2
This is possible only when the average of the three factors (1,1,2) is roughly equal to the cube root of 2.
Hence,
^{3}√2 = (1+1+2)/3 = 4/3 = 1.3
Again let us say p = 1.3 and q = 1.18
So,
1.3×1.3×1.18 = 1.99 ≈ 2
Taking the average of three factors, 1.3,1.3 and 1.18, we get;
^{3}√2 = (1.3+1.3+1.18)/3 = 3.78/3 = 1.26
Hence,
^{3}√2 = 1.26
Therefore, we get the value of the cube root of 2 equal to 1.26 which is approximately equal to its actual value, i.e.,1.2599210.
Cube Root of Non Perfect Cubes
To find the cube root of perfect cubes is easy, but difficult to find for the non-perfect cubes.
For example, cube root of 27 is equal to 3.
Since, 27 = 3 x 3 x 3 = 3^{3}
Therefore,
^{3}√27 = ^{3}√(3^{3}) = 3
But for non-perfect cubes we cannot find their cubic roots this easily. Thus, here is a table for the cube roots from 1 to 20 consisting of only non-perfect cubes.
Number |
Cube Root (^{3}√) |
2 |
1.260 |
3 |
1.442 |
4 |
1.587 |
5 |
1.710 |
6 |
1.817 |
7 |
1.913 |
9 |
2.080 |
10 |
2.154 |
11 |
2.224 |
12 |
2.289 |
13 |
2.351 |
14 |
2.410 |
15 |
2.466 |
16 |
2.520 |
17 |
2.571 |
18 |
2.621 |
19 |
2.668 |
20 |
2.714 |
From the table, we can see there are only two numbers eliminated from the table between 1 and 20, which are 1 and 8. Apart from that, all the natural numbers are non-perfect cubes.