Cubes of 1 & 2-Digit Numbers

Cubes of Single and 2-Digit Numbers

 13 = 1  113 = 1331
 23 = 8  123 = 1728
 33 = 27  133 = 2197
 43 = 64  143 = 2744
 53 = 125  153 = 3375
 63 = 216  163 = 4096
 73 = 343  173 = 4913
 83 = 512  183 = 5832
 93 = 729  193 = 6859
 103 = 1000  203 = 8000

speedy calculation of cube roots (of perfect cubes).

  1. The lowest cubes (i.e. the cubes of the fist nine natural numbers) are 1, 8, 27, 64, 125, 216, 343, 512 and 729.
  2. They all have their own distinct endings; with no possibility of over-lapping (as in the case of squares).
  3. The last digit of the cube root of an exact cube is obvious:
    • 1³ = 1    > If the last digit of the perfect cube = 1, the last digit of the cube root = 1
    • 2³ = 8    > If the last digit of the perfect cube = 8, the last digit of the cube root = 2
    • 3³ = 27  > If the last digit of the perfect cube = 7, the last digit of the cube root = 3
    • 4³ = 64  > If the last digit of the perfect cube = 4, the last digit of the cube root = 4
    • 5³ = 125 > If the last digit of the perfect cube = 5, the last digit of the cube root = 5
    • 6³ = 216 > If the last digit of the perfect cube = 6, the last digit of the cube root = 6
    • 7³ = 343 > If the last digit of the perfect cube = 3, the last digit of the cube root = 7
    • 8³ = 512 > If the last digit of the perfect cube = 2, the last digit of the cube root = 8
    • 9³ = 729 > If the last digit of the perfect cube = 9, the last digit of the cube root = 9
  4. In other words,
    • 1, 4, 5, 6, 9 and 0 repeat themselves as last digit of cube.
    • Cube of 2, 3, 7 and 8 have complements from 10 (e.g. 10’s complement of 3 is 7 i.e. 3+7=10) as last digit.
  5. Also consider, that
    • 8’s cube ends with 2 and 2’s cube ends with 8
    • 7’s cube ends with 3 and 3’s cube ends with 7

If we observe the properties of numbers, Mathematics becomes very interesting subject and fun to learn. Following same, let’s now see how we can actually find the cube roots of perfect cubes very fast.

Example 1:  Find Cube Root of 13824

Step 1:
Identify the last three digits and make groups of three digits from right side. That is 13824 can be written as
13  ,   824

Step 2: 
Take the last group which is 824.  The last digit of 824 is 4.
Remember point 3, If the last digit of the perfect cube = 4, the last digit of the cube root = 4
Hence the right most digit of the cube root  = 4

Step 3:
Take the next group which is 13.
From point 3, we see that 13 lies between 8 and 27 which are cubes of 2 and 3 respectively. So we will take the cube root of the smaller number i.e. 8 which is 2.
So 2 is the tens digit of the answer.

We are done and the answer is ’24’

Example 2:  Find Cube Root of 185193

Step 1:
185193 can be written as
185  ,   193

Step 2:
Take the last group which is 193.  The last digit of 193 is 3.
Remember point 3, If the last digit of the perfect cube = 3, the last digit of the cube root = 7
Hence the right most digit of the cube root  = 7

Step 3:
Take the next group which is 185.
From point 3, we see that 185 lies between 125 and 216 which are cubes of 5 and 6 respectively. So we will take the cube root of the smaller number i.e. 125 which is 5.
So 5 is the tens digit of the answer.

So, the answer = 57.

Try some of the perfect cubes like 287496, 658503, 46656.