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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)

profilejagadish
2013-08-17 21:33:15 
realy much helpfull for d competative exams.....thanx
like 0 dislike 0 reply
profileNeha Kumari
2013-08-13 15:01:58 
IS there any method to find cubes?? Pls reply soon on my e-mail Id nehakumari18@neha@ gmail.com

like 0 dislike 0 reply
profilesrikaja
2013-08-01 11:44:41 
954862..pls solve 
like 0 dislike 0 reply
profilekiya
2013-09-07 10:50:28 
98 is the answer

954 and 862
first group = 862 => so the RHS digit is 8 
now 954 - 9 cube(729)
so the LHS digit is 9


like 0 dislike 0
profileAswath
2013-08-12 10:21:53 
it is not a perfect cube
like 0 dislike 0
profileKiran
2013-08-03 00:53:19 
Only prefect cubes can be solved in this way.
like 0 dislike 0
profileraghu babu
2013-07-18 09:49:59 
how can we identify whether the number has perfect cube root or not???

with your method, if i want to find cube root of 3913,
step 1: 3,913
step 2: for 913, right most digit for cube root must be 7.
step 3: for 3, 3-(1)^3=2. so, left most digit of cube root has to be 1.
hence cube root of 3913=17.

but (17)^3=4913,

i'm waiting for your reply.....
like 0 dislike 0 reply
profileKhan
2013-07-18 22:54:05 
3913 is not a perfect cube. This method works only for perfect cubes
like 0 dislike 0
profileHarsha
2013-06-17 20:49:02 
Suppose we need to find cube root of 10648

Step1: 10, 648

Step2: The last digit of 648 is 8
       If the last digit of the perfect cube = 8, the last digit of the cube root = 2
       
step3: Take the group 10 now
2 is maximum cube we can subtract from 10 such that the result >= 0 (because 23 is 8 and 10-8 is 2)
Hence the left neighbor digit of the answer  = 2

Hence answer = 22
like 0 dislike 0 reply
profilejee
2013-06-17 14:59:32 
if have 5 digt number thn how to sprate it
like 1 dislike 0 reply
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