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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)

profilenaveen tewari
2014-09-08 10:18:36 

its good technique but explain this? how we calculate cube root of 3048625?

 

like 0 dislike 0 reply
profilekrish
2014-08-18 06:27:56 
But sir how to find the cube root of 3048625?
While following the method 3  048  625
unit digit (of cube root) will be 5
cube smaller than  048 is 27 ie 3^3
so ten's digit will be 3 
Hundred's digit will be 1
so by rule we get the cube root is 135
but 135^3 = 2460375
Please clear my confusion 
like 0 dislike 0 reply
profileMurali Pappala
2014-11-11 18:44:15 
3048625
Last group = 625
Hence first (unit's) digit of cube root is 5
Next group is 3048
15*15*15=3375
14*14*14=2744
3048-14*14*14= some positive value
therefore Answer is 145
like 0 dislike 0
profileamit
2014-08-02 11:21:09 

sir its only 6 digits methods means calculated maximum 6 digit and send me 50243409 solved equtions

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profilehyan
2014-08-08 09:42:02 
50243409
ANS- 50  243  409
calculate the nearest cube root of 50=3(smaller than 50) 
then calculate the cube root of 243=6( smaller than 243)
then calculate the cube of 9(UNIT PLACE OF THE NUMBER)=729
SO THE ANS IS-369
like 0 dislike 0
profileSoubhagya
2014-11-05 14:58:48 
@Hyan, While your method has worked for 50243409 ,but the same is not working for 8615125 could you explain plz
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profilekava ram
2014-07-10 18:55:07 
Wt abt non perfect numbers
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profileEROL CURMAK
2014-07-09 20:30:51 
Wonder how is it with no exact cube roots. Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja had that extracting too in his book vedic Mathematics, but I have problems, need help
Thanks
like 0 dislike 0 reply
profilepriya
2014-04-27 22:50:00 
how can we find the cube root for non perfect cube

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profilePritam Kumar Kar
2014-04-26 09:35:37 
How to find cube root for numbers greater that 6 digit like for 1061208
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