Cube Root

Cube root of a number can be found by a very simple method which is the prime factorization method. It is not an easy task to find the cubic root but a complex process. To find it, we need to find a number which when multiplied three times by itself gives the original number. Example: ∛8 = ∛(2 × 2 × 2) = 2. Since 8 is a perfect cube number, it is easy to find the cube root of a number.

Let us consider another example of number 2. Since 2 is not a perfect cube number. It is not easy to find the cubic root of 2. With the help of the long division method, it is possible to find the cube roots for non-perfect cube numbers. The approximate value of the ∛2 is 1.260.

How to Find Cube Root of a Number?

Cube root is the inverse process of calculating the cube of a number. It is denoted by the symbol ‘∛’. Let us see some examples here now.

To find the cube root of a number 64, we want a number which when multiplied thrice with itself shall give 64. We can write,

64 = 4 × 4× 4 = 43

Taking cubic root on both the sides;

or 64 = 43

Therefore, cube-root of 64 is 4.

Please note that we will only consider the positive values cube roots of the natural numbers.

Cubes and Cube Roots From 1 to 15

Number Cube(a3) Cube root∛a
1 1 1.000
2 8 1.260
3 27 1.442
4 64 1.587
5 125 1.710
6 216 1.817
7 343 1.913
8 512 2.000
9 729 2.080
10 1000 2.154
11 1331 2.224
12 1728 2.289
13 2197 2.351
14 2744 2.410
15 3375 2.466

 

Cube Root By Prime Factorisation Method

We can find the cube-root of a number by the method of prime factorisation. Consider the following example for a clear understanding:

2744= 2 × 2× 2 × 7 ×7 × 7= (2 × 7 )3

Therefore the cube root of 2744 = ∛2744 = 2 × 7 = 14

Cube Root of 343

Let us find the cube root of 343 with the help of the prime factorization method.

Dividing 343 by smallest prime factor, till we get the remainder as 1. Follow the below steps;

Cube root of 343

Therefore, 343 = 7 × 7 × 7

And, 343 = 7

Cube Root of 729

Now, let’s find the cubic root of 729.

Cube root of 729

729 = 3 × 3 × 3 × 3 × 3 × 3 = 9 × 9 × 9

Therefore, ∛729 = 9

Illustration 1: Solve: 24389
Solution: Prime factors = 29×29×29 = 293
Therefore ∛24389= 29

Illustration 2: Find 46656 by the method of prime factorization.

Solution: Let us first find the prime factors:

Cube Root by Prime Factorisation Method

∛46656 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 = 23 × 23 × 33 × 33 = (2 × 2 × 3 × 3)3

Therefore, 46656= 36.

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