Easy Method of Finding Cube Root of a Perfect Cube

H.D. Motiramani

A cube is a number obtained by multiplying the same number thrice. For example cube of 3 will be 3×3×3=27. Likewise cube of 7 is 7×7×7=49×7=343. Because the number 3 has a cube 27, the cube root of 27 is 3. Similarly cube root of 343 is 7.

Let us draw a table of cube of 1 to 10 and try to observe the result closely.  On close observation, we find that the last digit in the resultant cube for the underlined numbers is same as the number itself. This observation will be very helpful in cube root derivation. So, if we are told that 729 is a perfect cube number, by looking at the last digit 9, we should be able to say that the cube root of this number is 9. Last digit of a perfect cube number is little different for the number 2, 3 and 7, 8.  In the case of cube of these numbers, we make an observation that the cube of number 2 is 8 but the cube of 8 ends with 2. Likewise the cube of number 3 is 27 ending with 7 but the cube root of 343 (ending with 3) is 7.

 Number Cube 1 1 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729 10 1000

In the above table of cube of the numbers 1 to 10, cube of a number 1 to 5 and the number 10 is very easy to arrive at as we can do it without pen and paper. For the cube of a number 6 to 9, let us try to remember the numbers 216,343, 512 and 729. Let us cram or adopt some memory technique to remember this. Once we remember the above result of cubes of 1 to 10, our job becomes very easy.

We can now easily find cube root of even a 4 to 6 digit number of a perfect cube. What we do is break the number in two parts with right side having 3 digits. For example number 2744 to be broken as 2/744, 438976 to be seen as 438/976. The job is now very easy. In the case of number 438976 the last three digits that is the right hand part ends with 6. We know that the cube of 6 ends with 6 (See above table of cubes from 1 to 10) so the right hand part of the answer is 6. Now comes the left hand part. The three digits on left hand side are 438. We will observe that this number lies between 343 (cube of 7 and  512 cube of 8). We have to choose lower number 7 whose cube is 343. So the answer of cube root of 438976 becomes 76. Likewise for the cube root of 2744 right hand three digits has 4 in the end therefore right hand side answer is 4. For left hand side answer we see from above table that the number 4 lies between 1 and 8 which are the cubes of 1 and cube of 2. Since we have to pick up a lower of the two, we take 1 as the left hand number for our answer. The cube root of 4744 is therefore 16. We therefore get the answer simply by observation. Few more examples will make the procedure very clear.

 Cube root of Break into 2 parts RHS number LHS number Answer 79507 79   507 3 79 is between cube of 4 and 5 so select 4 43 551368 551  368 2 551 is between cube of 8 and 9 so select 8 82

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