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).

$1^3=12^3=83^3=274^3=645^3=1256^3=2167^3=3438^3=5129^3=729{10}^3=1000$

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.

$1⟹1$	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 $3$ 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 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

$4,913$

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 $<0$)

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

$804,357$

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 $<0$)

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

i.e., answer $=93$