Mental calculation of the cubic root of a 1, 2 or 3digit number
Mental calculation of the cubic root of a 1, 2 or 3digit number.
Cubic root of a 3digit number
This is very simple to do and all you need is to memorize or calculate on the fly what a 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 
Example:
7^{3}=343. Reverse lookup in table 1 you get the correct answer to be 7.
Cubic root of a 6digit number
Cubic root of a 6digit number is generated by raise a 2digit number to the power of 3. The range of digits are between 46 digits. You can use the knowledge to do another cool party trick that is easy to perform. Ask one in the audience to choose a random 2digit 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 2digit number was raised to the power of 3. A 2digit number with digits a and b. (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+10a3b^{2}+b^{3}
The lowest possible 2digit 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 can be seen by looking at the table below and show 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
N 
N^{3} 
N^{3} 
Mapping 
0 
0^{3} 
0 
Same 
1^{} 
1^{3} 
1 
Same 
2 
2^{3} 
8 
108 
3 
3^{3} 
27 
107 
4 
4^{3} 
64 
Same 
5 
5^{3} 
125 
Same 
6 
6^{3} 
216 
Same 
7 
7^{3} 
343 
103 
8 
8^{3} 
512 
102 
9 
9^{3} 
729 
Same 
We also notice by looking at the thousands that it consists of the 1000a3 plus something more (100a^{2}3b+10a3b^{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 3^{rd} 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 10digit.
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 2digit whole number. This is not entirely accurate. The method works just as well for any 2digit decimal number e.g. 7.9^{3}=493.039
Notice that there is no thousand parts and therefore the number must be less than 1000. 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 2digit 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 9digit number
Cubic root of a 9 digits number is generated by raising a 3digit number to the power of 3 (79 digits result). It gets a little bit complicated to find the 3digit number giving the 9digits 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 9digit number.
5^{th} root of a 5digit number
The fifth root of a 5digit number. First, we look at a single digit number to the power of 5 can generate up to 5digits numbers (15 digits).
Table 3 single digit number raised to the power of 5
N 
N^{5} 
N^{5} 
0 
0^{5} 
0 
1^{} 
1^{5} 
1 
2 
2^{5} 
32 
3 
3^{5} 
243 
4 
4^{5} 
1,024 
5 
5^{5} 
3,125 
6 
6^{5} 
7,776 
7 
7^{5} 
16,807 
8 
8^{5} 
32,768 
9 
9^{5} 
59,049 
We notice that a single digit raised to the power of 5 has the same digit as the last digit making it very easy a simple to predict the fifth root of that number. Simply look at the last digit and you have the number right away.
5^{th} root of a 10digit number
For 2digit number raised to the power of you can get upto a 10digit number (6to 10 digits). Using the same technic as for cubic numbers you also have an easy way to remember it.
Table 4 double digit raised to the power of 5.
N 
N^{5} 
N^{5} 
Easy remember 
0 
0^{5} 
0 
0 
10^{} 
10^{5} 
100,000 
100K 
20 
20^{5} 
3,200,000 
3M 
30 
30^{5} 
24,300,000 
24M 
40 
40^{5} 
102,400,000 
100M 
50 
50^{5} 
312,500,000 
300M 
60 
60^{5} 
777,600,000 
777M 
70 
70^{5} 
1,680,700,000 
1.6B 
80 
80^{5} 
3,276,800,000 
3B or 3.2B 
90 
90^{5} 
5,904,900,000 
5.9B 
An example is better to show how it works:
57^{5}=601,692,057. Since the closed matching lower number is 300M we get the 50 range, so we can immediately say that the first digit is 5 and by just looking at the last digit 7 we also know from table 3 that the second digit is 7 so the answer is 57
Another example:
34^{5}=45,435,424. Since the closed matching lower number is 24M for 30 range we can immediately say that the first digit is 3 and by just looking at the last digit 4 we also know from table 3 that the second digit is 4 so the answer is 34.
Have fun
