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# Solving the cube root of 19,683 mentally

## Solving the cube root of 19,683 mentally

By Nigamanth Srivatsan

Written on 3rd February, 2023

## Um, what?

That’s right, I will show you the step by step working of how to calculate the cube root of any 5-7 digit perfect cube mentally.

But, before we dive into calculating complex cube roots, we will need to understand the basics.

## The Basics

This information is very necessary for solving complex cube roots.

`1^3` = 1

`2^3` = 8

`3^3` = 27

`4^3` = 64

`5^3` = 125

`6^3` = 216

`7^3` = 343

`8^3` = 512

`9^3` = 729

`10^3` = 1000

And I could go on like this forever, but this is all we need to know to calculate complex cube roots.
Notice how, the cubes of 1, 4, 5, 6, and 9 always end with the number itself.

This is the first property we will use to calculate complex cube roots. If we have a number, we will look at it’s unit’s digit and find the unit’s digit of the cube root.

## Say what?

A lot of people are surprised when they hear this.

If the last digit of any perfect cube is:

1: then the unit’s digit of its cube root is a 1.

2: then the unit’s digit of its cube root is a 8

3: then the unit’s digit of its cube root is a 7

4: then the unit’s digit of its cube root is a 4

5: then the unit’s digit of its cube root is a 5

6: then the unit’s digit of its cube root is a 6

7: then the unit’s digit of its cube root is a 3

8: then the unit’s digit of its cube root is a 2

9: then the unit’s digit of its cube root is a 9.

So now, we already have the first unit digit of our cube root.

If you remember, our number was `19,683`, and therefore the cube root of this number is _7.

## The Next Part

For the next step, if our number is 19,683 then we need to only check the thousands of the number.
Here, for us, the number 19,683 has 19 thousands. What cube root is closest to 19?

2^3 = 8; but 3^3 = 27. Therefore the one right before 19 is 2^3, and as such 2 will be the other number.

Meaning that, the cube root of 19,683 is 27.

## Examples

For practicing this trick, we can use the following examples:

1. Find the cube root of 175,616

2. Find the cube root of 653,503

3. Find the cube root of 1,259,712

The first 2 examples are fairly basic, however the 3rd one can take you by surprise.

The third one has 1,259 thousands, therefore closest to 10^3. The answer is 108 for the third one

Additionally, if you memorize 10^3 – 20^3 then you can find the cube root of numbers that exceed the 1 million mark too!

This trick will forever work, no matter how big a number is (it just has to be a perfect cube).

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