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Cube Roots of Perfect Cubes Using Vedic Mathematics

Cube roots of perfect cubes can be found out faster using Vedic Mathematics.

Points to remember

1. To calculate cube root of any perfect cube quickly, we need to remember the cubes of $1$ to $10$ (provided below).

$1^3=1\\2^3=8\\3^3=27\\4^3=64\\5^3=125\\6^3=216\\7^3=343\\8^3=512\\9^3=729\\10^3=1000$

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

$1^3=1$ $\Rightarrow$ If last digit of perfect cube $=1,$ last digit of cube root $=1$
$2^3=8$$\Rightarrow$ If last digit of perfect cube $=8,$ last digit of cube root $=2$
$3^3=27$$\Rightarrow$ If last digit of perfect cube$=7,$ last digit of cube root $=3$
$4^3=64$$\Rightarrow$ If last digit of perfect cube$=4,$ last digit of cube root $=4$
$5^3=125$$\Rightarrow$ If last digit of perfect cube $=5,$ last digit of cube root $=5$
$6^3=216$$\Rightarrow$ If last digit of perfect cube$=6,$ last digit of cube root $=6$
$7^3=343$$\Rightarrow$ If last digit of perfect cube$=3,$ last digit of cube root $=7$
$8^3=512$$\Rightarrow$ If last digit of perfect cube$=2,$ last digit of cube root $=8$
$9^3=729$$\Rightarrow$ If last digit of perfect cube$=9,$ last digit of cube root $=9$
$10^3=1000$$\Rightarrow$ If last digit of perfect cube$=0,$ last digit of cube root $=0$

It’s very easy to remember the relations given above as follows.

$1\implies 1$same numbers
$8 \implies 2$$10$'s complement of $8$ is $2$ and $8+2=10$
$7 \implies 3$$10$'s complement of $7$ is $3$ and $7+3=10$
$4 \implies 4$same numbers
$5 \implies 5$same numbers
$6 \implies 6$same numbers
$3 \implies 7$$10$'s complement of $3$ is $7$ and $3+7=10$
$2 \implies 8$$10$'s complement of $2$ is $8$ and $2+8=10$
$9 \implies 9$same numbers
$0 \implies 0$same numbers

Also see
$8 \implies 2$ and $2 \implies 8$
$7 \implies 3$ and $3 \implies 7$

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

Example 1: Find Cube Root of 4913

Step 1

Identify the last three digits and make groups of three three digits from right side. i.e., $4913$ can be written as

$4,\quad 913$

Step 2

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

Remember point 2, If last digit of perfect cube$=3,$ last digit of 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 $\ge 0$

We can subtract $1^3=1$ from $4$ because $4-1=3$ (If we subtract $2^3=8$ from $4,$ $4-8=-4$ which is $\lt 0$)

Hence the left neighbor digit of the answer $=1$

i.e., 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. i.e., $804357$ can be written as

$804,\quad 357$

Step 2

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

Remember point 2, If last digit of perfect cube $=7$ , last digit of 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 $\ge 0$

We can subtract $9^3=729$ from $804$ because $804-729=75$ (If we subtract $10^3=1000$ from $729,$ $729-1000=-271$ which is $\lt 0$)

Hence the left neighbor digit of the answer $=9$

i.e., answer $=93$

Comments(69)

profileSarvajeet Suman
2016-08-04 04:10:58 
Find the cube root of $33076161$ using this method ? Thanks in advance.
like 0 dislike 0 reply
profileJaved Khan
2016-08-15 19:57:39 
Grouping it gives
$33076,\quad161$

Last digit of $161$ is $1$. Therefore, last digit of the cube root is $1$

$33076 - 32^3 \gt 0$
(Note that $33076-33^3 \lt 0$).
So, left side of the answer is $32$

Answer is $321$

For larger numbers, this method may not be that easy and to get the left side $32$, some calculations are involved. However, faster than the conventional way. If one can remember cubes of numbers up to $20,$ certainly more numbers can be covered easily.
like 1 dislike 0
profileNishant yadav
2015-12-24 14:43:06 
How to find cube root of a decimal number?
like 0 dislike 0 reply
profileManoj (Alex)
2015-07-29 09:19:57 
How to find cube root more than 6 digit
like 0 dislike 0 reply
profileAnduna Abdul
2015-07-21 18:43:08 
this method is only applicable for finding cube root upto 6 digit numbers.  But very nice though.
like 0 dislike 0 reply
profilechithiraivel
2015-07-15 05:05:47 
how to find a cube root of a simple number other than perfect cube?
like 0 dislike 0 reply
profileRaywant Bhatia
2015-06-17 08:12:35 
How to find cube root of 10 digit no.
like 0 dislike 0 reply
profileswarup
2015-07-10 18:57:24 
for that u need to know the cube of 10-20 or may be 20-30.
process is all same u have to solve unit digit rule as above then neglect last 3 digits and find the no whose cube value is nearest to remaining 7 digits.

if u cant then do again or reply back.
like 0 dislike 0
profileMd.Arman
2015-05-30 01:52:16 
I have read all the process, it was very interesting but I may like to know how to find the cube root of a decimal and what if the cube root of the number is in decimal?
like 0 dislike 0 reply
profilenivedita
2015-04-21 09:54:56 
how to find cube root of a number which is in decimal?
like 0 dislike 0 reply
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