Cube Root

Introduction

The cube root is applied to cube numbers to find their initial value, from which they were introduced. The cube root acts as a base for entire mathematics. It is an important topic that needs to be learned properly. In this article we will be learning, what cube root is and different methods to find the cube root. It is necessary to have a clear understanding of this concept to solve higher-level problems later.

Cube root

The cube root of a given number is a value that when multiplied by itself three times gives the given number. Let x be the cube root of y. It can be written in the form x = ∛y. Here ‘∛’ represents the cube root symbol. 

How to find cube root

There are different methods to find the cube-like prime factorization method, estimation method. Let us take a look at all the examples.

Cube root by prime factorization method

The cube root of any given perfect cube can be found by the prime factorization method. It is similar to finding square root by this method. Steps involved in finding the cube root:

1. The prime factors of the given number are written.
2. Prime factors of similar numbers are written in traid.
3. One factor is chosen from each triad.
4. The product of chosen factors is the cube root of a given number. 

Let us consider a few examples to understand:

1. ∛8 

Prime factorization of 8 = 2 x 2 x 2. 

So, ∛ (2 x 2 x 2) = ∛23 = 2.

From the triad 2 is chosen and taken out of cube root.

2.               ∛216

Prime factorization of 216 = 2 x 2 x 2 x 3 x 3 x 3.

So, ∛ (2 x 2 x 2 x 3 x 3 x 3) = ∛23 x 33 = ∛ 63 = 6.

           The factors chosen from the triad are 2 and 3, their product gives the value of 6, which is the cube root.

3.               ∛3375

Prime factorization of 3375 = 3 x 3 x 3 x 5 x 5 x 5.

So, ∛ (3 x 3 x 3 x 5 x 5 x 5) = ∛33 x 53 = ∛153 = 15

The factors chosen from triad are 3 and 5, their product gives the value of 15, which is the cube root.

In the above example, we have seen how to find the cube root of perfect cubes. Moving on, we can look at how to find the cube root of large numbers and non-perfect cubes.

 Estimation method to find the cube root of large numbers

This method is effective to find the cube root of large numbers but before looking at this method it is important to memorize the cube values of 1 to 10. 

Let us understand using a few example problems:

1. Find the cube root of 4913.

Step 1: Consider the last three digits and the unit’s place has the value 3.

Hence the cube root’s unit place value is 7. 

Step 2: Ignore the last three places and consider 4. It lies in between 1 to 8. Hence, we select the lowest number, which is 1.

This gives us the two-digit cube root of 4913.

Hence, ∛4913 = 17.

2.               Find the cube root of 778688.

Step 1: Consider the last three digits and the unit’s place has the value 8.

Hence the cube root’s unit place is 2.

Step 2: Ignore the last three places and consider the remaining, which is 778. It lies in between 729 and 1000. Hence, we select the lowest number which is 9.

This gives us the two-digit cube root of 778688.

Hence, ∛778688 = 92.

 Cube root of imperfect cubes

In previous methods, we learned how to find the cube root of perfect cubes. In this method, we look at some special tricks to estimate the cube root. 

 Find the cube root of 265. 

Step 1: 

As we can see 265 lies between 216 and 343, which is the cube of 6 and 7. Consider the lowest number, which is 6.

Step 2:

Divide it by a square of 6.

265 /36 = 7.36

Step 3:

Take the difference of 7.36 and 6 and multiply it by ⅓.

7.36 – 6 = 1.36.

1.36 / 3 = 0.4

Step 4:

Add the lowest number chosen and the final value in step 3.

So, 6+0.4 = 6.4

 The cube root of 265 is 6.4, ∛265 = 6.4.

Conclusion

The cube root is an important operation in maths, just like any other arithmetic operation. It is necessary to learn how to solve a problem. Problems with roots are important to further advance into higher-level maths.