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

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.

**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 $=17$

**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 $=93$

first group having last digit is 1.

hence last digit is 1.

now next group is 1.

subtract 1-1 because cube of 1 is 1. because result should be equal to 0 or > 0.

So ur left digit is 1.

∴ ur answer is 11.

^{2}* 3

^{2}* 5

^{2 }* 7

^{2}) = 1470

find the cube root of 1953125 & 2628072

for 1953125

Solution:

first form it as given rule like

1953, 125

As we know if the last digit of the perfect cube=5, the last digit of the cube root =5

and as we have another rule like

1953-x

^{3}>0

or. 1953- (12)

^{3}>0 (it's true)

and if 1953-13

^{3 }>0 (it's false)

So, the prefect cube root of 1953125 = 125

Similarly for 2628072

like,2628,072

As we know if the last digit of the perfect cube = 2, the last digit of the cube root = 8

So for 072 has 8

and for 2628 we have

2628-13

^{3}>0 (it's true)

2628-14

^{3 }>0 (it's false)

So, the prefect cube root of 2628072 = 138

This way we can solve the problem of 7 digits cube problem

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