What is a perfect cube ?

A perfect cube number,  that you can write as a product of three equal factors of natural numbers.
perfect cube K=a*a*a=a³ ( a stand for integers.)

On the other hand a cube number can be found out by multiplying an integer by itself three times.

a*a*a=a³=K.

How to find out if a number is perfect cube or not ?

Before trying to jumping in for the formula to calculate cube root, we should need to know that formula can’t be used with every number but only with perfect cube of natural numbers. For example we can use this method with 5832 because it’s cube of 18, but we can’t use it with 1739 because it’s not a perfect cube.

Factorization method to find perfect cube

how to find cube root in 5 seconds

Now the classic method we learned in our school to find out if a number is a perfect cube or not is factorization. However  factorization is a cumbersome process and requires lot of time and effort. In a exam where you have to solve 100 questions in 60 minutes you can’t spend 30-90 seconds doing factorization. But some times in the case of prime number perfect cubes it is next to impossible to find the perfect cube. In order to save time we use digital sum method to determine if a number is perfect cube.

Digital sum method to verify if a number is perfect cube

Digital sum is nothing sum of all digits of a number, sum of digits of thus obtained numbers will be called digital root. note 9 and zero are considered the same numbers in digital soot. Before we begin, let us recall the cubes of first 10 natural numbers and obtain their digital sum. The following table shows digital sum and digital root of first 9 natural numbers-

As you can see that the digital root of a perfect cube is always 1, 8 or 9. So, we found our first condition

Check 1- all perfect cubes must have digital root equal to 1, 8 or 9 .

Moreover, the digital root of any number’s cube can be determined by the remainder the number gives when divided by 3:

Check 2:

• If the cube has digital root 9, it’s cube root will be divisible by 3
• If the cube has digital root 1, the cube root will have remainder of 1 when divided by 3.
• If the cube has digital root 2, the cube root will have remainder of 2 when divided by 3.

With this understanding, if we have to check whether a given number is a perfect cube, just obtain the digital root of the given number. If the digital root is not 1, 8 or 9  we can be very sure that the number is not a perfect cube. However this necessary condition is not sufficient at all, if a number has a digital root of 1, 8 or 0, that does not mean, that the given number must be a perfect cube.

Determining the Cube Root Fast

Cube Root of numbers up to 6 digits

Step 1- Ignore the last three digits of the number and compare the remaining number to the cube of first 10 numbers, choose the perfect cube which is just lower (or equal) to remaining number. This is the first digit of your cube root.

Now for the last digit of the cube root check the last digit of the cube. For example, if the last digit of the number called out is 3, then the last digit of the cube root is 7.

Important rules for finding cube root fast –

1. if last digit of the perfect cube is 0, 1,4,5,6 or 9 the last digit of cube root will also be same.
2. If last digit of perfect cube is 7 last digit of cube root will be 3, if last digit is 3 cube root will be 7.
3. If last digit of perfect cube is 8 last digit of cube root will be 2 and vice-versa.

Please see the examples in following table

We can use the same logic by using nested intervals and find the cube of larger numbers. Make group of 3 digits and apply similar logic-

Finding cube root of a 9 digit number

  x =5,177,717^1/3
take the first digit- The number x must be between 100 and 200 (100³=1,000,000 and 200³=8,000,000).
now take 5177, as we know cube root of 17 is 4913 and 18 is 5832 so x will be somewhere between 170-180. And 7 is on the ones place. so x will have 3 at unit digit.

Hence 173 comes into consideration. x =173.

Cube root of an eleven digit number

x =(23,295,638,016)^1/3
take the first digit- The number x must be between 2000 and 3000 (2000³=8,000,000,000 and 3000³=27,000,000,000).
now take 23,295 as we know cube root of 23,295 will be around 28.5, as 28³=21962 and 29³=24969 (you don’t need to calculate it, once you start practicing our guessing method you will find it very easy)

And 6 is on the ones place. so x will have 6 at unit digit.

Hence x =(23,295,638,016)^1/3

I hope this article will help you in finding the cube roots in 5 to 10 seconds. You may ask the question in comment section if you were not able to understand any part of the article.