# Cube roots of perfect cubes Using Vedic Mathematics

It may take two-three minutes to find out cube root of a perfect cube by using conventional method. However we can find out cube roots of perfect cubes very fast, say in one-two seconds using Vedic Mathematics.

We need to remember some interesting properties of numbers to do these quick mental calculations which are given below.

### Points to remember for speedy calculation of cube roots of perfect cubes

**1. To calculate cube root of any perfect cube quickly, we need to remember the cubes of 1 to 10 which is given below.**

1³ | = | 1 |

2³ | = | 8 |

3³ | = | 27 |

4³ | = | 64 |

5³ | = | 125 |

6³ | = | 216 |

7³ | = | 343 |

8³ | = | 512 |

9³ | = | 729 |

10³ | = | 1000 |

**2. From the above cubes of 1 to 10, we need to remember an interesting property. **

1³ | = | 1 | => | If the last digit of the perfect cube = 1, the last digit of the cube root = 1 |

2³ | = | 8 | => | If the last digit of the perfect cube = 8, the last digit of the cube root = 2 |

3³ | = | 27 | => | If the last digit of the perfect cube = 7, the last digit of the cube root = 3 |

4³ | = | 64 | => | If the last digit of the perfect cube = 4, the last digit of the cube root = 4 |

5³ | = | 125 | => | If the last digit of the perfect cube =5, the last digit of the cube root = 5 |

6³ | = | 216 | => | If the last digit of the perfect cube = 6, the last digit of the cube root = 6 |

7³ | = | 343 | => | If the last digit of the perfect cube = 3, the last digit of the cube root = 7 |

8³ | = | 512 | => | If the last digit of the perfect cube = 2, the last digit of the cube root = 8 |

9³ | = | 729 | => | If the last digit of the perfect cube = 9, the last digit of the cube root = 9 |

10³ | = | 1000 | => | If the last digit of the perfect cube = 0, the last digit of the cube root = 0 |

It’s very easy to remember the relations given above because

1 | -> | 1 | (Same numbers) |

8 | -> | 2 | (10’s complement of 8 is 2 and 8+2 = 10) |

7 | -> | 3 | (10’s complement of 7 is 3 and 7+3 = 10) |

4 | -> | 4 | (Same numbers) |

5 | -> | 5 | (Same numbers) |

6 | -> | 6 | (Same numbers) |

3 | -> | 7 | (10’s complement of 2 is 7 and 3+7 = 10) |

2 | -> | 8 | (10’s complement of 2 is 8 and 2+8 = 10) |

9 | -> | 9 | (Same numbers) |

0 | -> | 0 | (Same numbers) |

Also See

8 -> 2 and 2 -> 8

7 -> 3 and 3-> 7

If we observe the properties of numbers, Mathematics will be a very interesting subject and easy to learn.Now let’s see how we can actually find out the cube roots of perfect cubes very fast.

## Example 1: Find Cube Root of 4913

**Step 1: **

Identify the last three digits and make groups of three three digits from right side. That is 4913 can be written as

4, 913

**Step 2**

Take the last group which is 913. The last digit of 913 is 3.

Remember point 2, If the last digit of the perfect cube = 3, the last digit of the cube root = 7

__Hence the right most digit of the cube root = 7__

**Step 3**

Take the next group which is 4 .

Find out which maximum cube we can subtract from 4 such that the result >= 0.

We can subtract 1³ = 1 from 4 because 4 - 1 = 3 (If we subtract 2³ = 8 from 4, 4 – 8 = -4 which is < 0)

__Hence the left neighbor digit of the answer = 1. __

__That is , the answer = 17__

## Example 2: Find Cube Root of 804357

**Step 1: **

Identify the last three digits and make groups of three three digits from right side. That is 804357 can be written as

804, 357

**Step 2**

Take the last group which is 357. The last digit of 357 is 7.

Remember point 2, If the last digit of the perfect cube = 7, the last digit of the cube root = 3

__Hence the right most digit of the cube root = 3__

**Step 3**

Take the next group which is 804 .
Find out which maximum cube we can subtract from 4 such that the result >= 0.
We can subtract 9³ = 729 from 804 because 804 - 729 = 75 (If we subtract 10³ = 1000 from 729 , 729 – 1000 = -271 which is < 0)

__Hence the left neighbor digit of the answer = 9 __

__That is , the answer = 93__

Comments(89)

^{2}* 3

^{2}* 5

^{2 }* 7

^{2}) = 1470

find the cube root of 1953125 & 2628072

for 1953125

Solution:

first form it as given rule like

1953, 125

As we know if the last digit of the perfect cube=5, the last digit of the cube root =5

and as we have another rule like

1953-x

^{3}>0

or. 1953- (12)

^{3}>0 (it's true)

and if 1953-13

^{3 }>0 (it's false)

So, the prefect cube root of 1953125 = 125

Similarly for 2628072

like,2628,072

As we know if the last digit of the perfect cube = 2, the last digit of the cube root = 8

So for 072 has 8

and for 2628 we have

2628-13

^{3}>0 (it's true)

2628-14

^{3 }>0 (it's false)

So, the prefect cube root of 2628072 = 138

This way we can solve the problem of 7 digits cube problem

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