Practice mental Cubic root calculation

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

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

The 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 a single digit number raised to the power of 3 gives the cubic root of that number. Table 1, below can be used and is easy to memorize. All it takes is some practice. Table 1 Single-digit cubic root

Example:

7³=343. Reverse lookup in table 1 you get the correct answer to be 7.

The Cubic root of a 6-digit number

The Cubic root of a 6-digit number is generated by raising a 2-digit number to the power of 3. The range of digits is between 4-6 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 2-digit whole number x and raise it to the power of x3. 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 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:

 1000a³+100a²3b+10a3b²+b³

The lowest possible 2-digit number is 10³=1,000 and the highest possible value is 99³=970,299. By looking at the number on the left of the thousand separators you can establish what single-digit number raised to the power of 3 is less or equal to the number to determine the first digit. You can also see that b³ 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 showing 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 1000a3 plus something more (100a²3b+10a3b²+b³). This information indicates that if we find the nearest number less or equal to 1000a³ then we can establish the digit a.

An example is better to show how it works.

57³=185,193. Now take the thousand only (185) and find closet smaller number from the table which is 125 and 5³. We now know the cubic root of the number starts 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 ends with the number 3. The number is 343 and represents 7³ and now you know that the last digit must be 7 and therefore the correct answer is 57. With some practice, it 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 the 10-digit.

34³=39,304. The thousand part is 39 and the closest n³ less is 3³=27. The first digit is then 3. Looking at the last digit 4 you know by the table that 4³ ends in 4 and therefore the last digit is 4 and the result is 34 which is the answer.

I mention at 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. 7.9³=493.039

Notice that there are 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 separators) you get 493 which closest n³ number less than 493 is 7³=343. The first digit is then 7. Looking at the last digit 9 and the table above 9³ ends in 9 therefore the last digit is 9 and the result is then 7.9.

You can even do 2-digit fraction numbers e.g. 0.13³=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 (the last digit is 7, and the corresponding number from the table which the 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

The cubic root of a 9-digit number

The cubic root of a 9 digits number is generated by raising a 3-digit number to the power of 3 (7-9 digits result).? It gets a little bit complicated to find the 3-digit number giving the 9-digits number and requires more practice to handle it perfectly. Instead of describing the method below I will instead refer the reader to this website at http://thinkinghard.com/blog/CubeRoots.html where the Author Phillip Dorrell provides an elegant and comprehensive description of how to handle the mental calculation of the cubic root of a 9-digit number.

5^th root of a 5-digit number

The fifth root of a 5-digit number. First, we look at a single-digit number with the power of 5 that can generate up to 5-digit numbers (1-5 digits).

Table 3 single digit number raised to the power of 5

We notice that a single digit raised to the power of 5 has the same digit as the last digit making it very easy and 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 10-digit number

For a 2-digit number raised to the power of you can get up to a 10-digit number (6-to-10 digits). Using the same techniques as for cubic numbers you also have an easy way to remember them.

Table 4 double-digit raised to the power of 5.

An example is better to show how it works:

57⁵=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:

34⁵=45,435,424. Since the closed matching lower number is 24M for the 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