Cube root of a perfect cube number (shortcut method)

If there is a number which is perfect cube of a number, then finding its cube root is an easy job. First of all, we should know that cube root will have number of digits equal to number of triplets, whether complete triplet or partial triplet, formed from one end of that number. So if a number has 5 digits then from one end, say right, there will be one complete triplet of 3 digits, and another one partial triplet of 2 digits, so cube root will have two digits. For example cube root of 24389, will have cube root having two digits.

Let us take example of finding cube root of  this number 24389.

Step 1 Take last three digits 389. Consider last digit 9 of this number 389. Multiply this number 9 by itself 9, we get 81,  again multiply 81 by 9. So we get a number whose unit digit number would be 9 as 9 ✖ 9 ❌ 9 = 729. So last digit is 9. Short cut 9 x 9 = 81 ×9 = ...9 as 1 of 81 multiplying to 9 will give 9.

[ Some texts mentions that for the unit digit of cube root of a perfect cube number, the unit digit of the last three digits of that number be corresponded to unit digit of that number whose cube has the last digit equal to this digit. It means if last three digits of the perfect cube number is "a", the get a corresponding unit digit number "x" such that unit digit of x^3 is "a". And therefore, they ask to memorize all unit digits U(a) for every a =1(1)9 like U(1^3)=1, so if a=1, choose x=1,

U(2^3)=8, so if a=8, choose x=2,

U(3^3)=7, so if a=7, choose x=3,

U(4^3)=2, so if a=2, choose x=4,

U(5^3)=5, so if a=5, choose x=5,

U(6^3)=6, so if a=6, choose x=6,

U(7^3)=3, so if a=3, choose x=7,

U(8^3)=2, so if a=2, choose x=8,

U(9^3)=9, so if a=9, choose x=9,

so as to decide which one will the unit digit of the cube root be. But that is not essential to memorize them, because of the special property, these numbers are having. For any unit digit "a" if you cube them, you will get its unit digit the same one, whose cube's unit digit is this number "a". Thus for any "x" = 1(1)9, if we have U(x^3) = a, then U(a^3) = x. Therefore,  for any unit digit "a", choose U(a^3). Furthermore,  we also have 

U(a^3) = U(a^2*a) = U(U(a^2)*a), which reduces the statement as for any unit digit "a", choose U(U(a^2)*a). ]

Step 2 Having decided the unit place of cube root number of perfect cube 24389 by following Step 1, we will decide the tens place of the required number. For that, delete the last three digits of the number 24389 and start guessing the number from the remaining dogits which in this particular case is 24. This 24 is between the cube of 2 and cube of 3. Out of these two boundary numbers 2, 3, take smaller one, which is here "the 2." Take this smaller one 2 as at tens place. Required cube root of 24389 will be 29. That is 29^3 = 24389. This process is applicable only for finding cube roots of perfect cube numbers.

Let us take another perfect cube number 250047.

250047》two parts  first three digits 047, and remaining one 250》consider last digit of first part 7 》get the unit place of its cube that is first digit of 7^3》for that do it  7x7 as 49 and 9 of 49 multiplying by 7 again gives 63, the unit place being 3 from 63, so unit place is 3》now consider second part 250》 to which two numbers cubes, it is sandwiched,  one can go by 1^3=1,2^3=8,3^3=27,.... 6^3= 216, 7^3= 343...》Observed that it is between 6 and 7》Take smallest one 6》Unit place is 3, Tens place / second place is 6》number is 63 or 63^3= 250047.

Examples More :::

Ex.1) 13824 =y^3, find y.

13824》824, 13》824->4->4x4x4->16x4->6x4->24->4》unit place 4》》13->[2^3, 3^3]-> min(2,3)=2》tens place 2》number is 24》24^3=13824.

Ex.2) 262144=y^3, find y.

262144》144, 262》144>>>4》unit place 4》》262 lies in (6^3, 7^3)》min(6,7)=6》tens place 6》y=64

Ex.3) 493039 = y^3, find y.

493039》039, 493》039>>>81x9>>>9》unit place 9》》493 ~ (7^3, 8^3)-->min(7,8)=7<-- tens place》y=79.