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Practice mental Cubic root calculation

Mental calculation of the cubic root of a 1, 2 or 3-digit number

Cubic root of a 3-digit number
This is very simple to do and all you need is to memorize or calculate on the fly what single digit number raised to the power of 3 give the cubic root of that number. The table 1, below can be used and is easy to memorize. All it takes is some practice.
Table 1 Single digit cubic root:

N

N^{3}

N^{3}

0

0^{3}

0

1

1^{3}

1

2

2^{3}

8

3

3^{3}

27

4

4^{3}

64

5

5^{3}

125

6

6^{3}

216

7

7^{3}

343

8

8^{3}

512

9

9^{3}

729

Cubic root of a 6-digit number
Here is another cool party trick that is easy to perform. Ask one in the audience to choose a random 2-digit whole number x and raise it to the power of x^{3}. Ask for the resulting number after the operation and you can right away tell which 2-digit number was raised to the power of 3. A 2-digit number ab can also be written as 10a+b where a and b are single digit number. Raise it to the power of 3 you get:
1000a^{3}+100a^{2}3b+10a^{3}b^{2}+b^{3}
The lowest possible 2-digit number is 10^{3}=1,000 and the highest possible value is 99^{3}=970,299. By looking at the number on the left of thousand separator you can establish what single digit number raised to the power of 3 is less or equal that number to determine the first digit. You can also see that b^{3} will represent the last digit since the other component is multiplied by 10, 100 and 1000 and does not contribute to the last digit. The reason why this is working is by looking at the table below that any single digit raised to the power of 3 all end in a unique digit from 0 to 9 where the unique single digit is underscored.
Table 2 double digit cubic root

We also notice by looking at the thousands that it consists of the 1000a^{3} plus something more (100a^{2}3b+10a^{3}b^{2}+b^{3}). This information indicates that if we find the nearest number less or equal than the 1000a^{3} then we can establish the digit a.
An example is better to show how it works. 57^{3}=185,193. Now taking the thousand only (185) and find closet smaller number from the table which is 125 and 5^{3}. We now know the cubic root of the number start with 5. Next look at the last digit of the last 3 digits (193). Which is 3 and match it up with the last digit in the 3rd Column that end with the number 3. The number is 343 and represent 7^{3} and now you know that the last digit must be 7 and therefore the correct answer is 57. With some practice is goes lightning fast. A way to remember the mapping for the last digit is to say that 1, 4, 5, 6 & 9 end in the same digit and the other ends in 10-digit.

34^{3}=39,304. The thousand part is 39 and the closest n^{3} less is 3^{3}=27. The first digit is then 3. Looking at the last digit 4 you know by the table that 4^{3} ends in 4 and therefore the last digit is 4 and the result is 34 which is the answer.

I mention in the start that the number needs to be a 2-digit whole number. This is not entirely accurate. The method works just as well for any 2-digit decimal number e.g. 7.9^{3}=493.039
Notice that there is no thousand parts and therefore the number must be less than 10. Separating the number before and after the fraction sign (instead of before and after the thousand separator) you get 493 which closest n^{3} number less than 493 is 7^{3}=343. The first digit is then 7. Looking at the last digit 9 and the table above 9^{3} ends in 9 and therefore the last digit is 9 and the result is then 7.9.
You can even do 2-digit fraction number e.g. 0.13^{3}=0.002197. Again, separating the number into two parts 002 and 197 you quickly see that the first digit is 1 and the second digit is 3 (last digit is 7 and the corresponding number from the table which last digit is 7 is 3) and since the result is less than 1 all digits must be fraction digit and the result is 0.13

Cubic root of a 9-digit number
Cubic root of a 9 digits number is generated by raising a 3-digit number to the power of 3.? It gets a little bit complicated to find the 3-digit number giving the 9-digits number and require more practice to handle it perfectly. Instead of describing the method below I will instead refer the reader to this web site at http://thinkinghard.com/blog/CubeRoots.html where the Author Phillip Dorrell provide an elegant and comprehensive description of how to handle mental calculation of the cubic root of a 9-digit number.

Have fun

Corrections:

21-Jun-2017

Vs 1.0

Initial release of practice mental cubic root calculator