Cubing - How to find the Cube of numbers quickly

CUBE

In this post i will explain different cubing methods and tricks to find cube of  2-digit and 3-digit numbers without calculator. Cube of any number 'n' is its third power n3. A positive integer 'n' will be a perfect cube if one can arrange n cubes into a large cube. For example 3 x 3 Rubik's cube have 27 cubes inside that large cube. So 27 is a perfect cube.

Cube of Two Digit Numbers

Formula: (AB)3 = (A)3 | 3A2B | 3AB2 | (B)3
After writing in terms of given formula, keep the unit digit of the rightmost number and transfer rest of the digits to the left and add them. 
Example: Find the cube of 12?
Solution: Here A = 1 ; B = 2
  ⇒ (12)3 = (1)3 | 3(1)2(2) | 3(1)(2)2 | (2)3
                = 1 | 6 |⃔12 | 8
                = 1728

 Second Method 
Example: Find the cube of 28?
Solution:  Step 1:  Find the cube of Tens Digit number which is '2' here => 23 = 8.

 Step 2:  Find the ratio by dividing the units digit by tens digit.
         ⇒ Ratio = 8/2 = 4

 Step 3:  Multiply the cube of Tens digit with this ratio and form a G.P (geometric progression) of 4 terms.
⇒ First term = 8
     Second term = 8 x 4 = 32
     Third term = 32 x 4 = 128
     Fourth term = 128 x 4 = 512
Write these four terms in straight line.

 Step 4:  Double the value of middle two terms and write just below the middle terms.

 Step 5:  Add the terms. Keep the unit digit and transfer rest to the left hand number.

Cube of Three Digit Numbers

Formula: (ABC)3 = (AB)3 | 3(AB)2C | 3(AB)(C)2 | (C)3
After writing in terms of given formula, keep the unit digit of the rightmost number and transfer rest of the digits to the left and add them. 
Example: Find the cube of 123?
Solution: Here AB = 12 ; C =3
⇒ (123)3 = (12)3| 3(12)2(3) | 3(12)(3)2 |33
                = 1728 | 1296 | 324 |⃔27
                = 1728 | 1296 |⃔326 | 7
                = 1728 |⃔1328 | 6 | 7
                = 1860 | 8 | 6 | 7
                = 1860867

We can also use this Formula:
Formula: (A + B + C)3= (A)3 + (B)3 + (C)3 + 3(A + B)(B + C)(C + A)
For Cube of 123: A = 100; B = 20 ; C =3
⇒ (123)3 = (100)3 + (20)3 + (3)3 + 3(100 + 20)(20 + 3)(3 + 100)
                = 1000000 + 8000 + 27 + 3(120x23x103)
                = 1860867 

NOTE: Difference between the cube of consecutive integers is: 3(n − 1)n + 1 or 3(n + 1)n + 1.

We can also find Cube using normal multiplication but that will waste a lot of time in examination and every second is important. Its better to remember these formulas to find the cubes up-to 99. Examiner will rarely ask to find cube of three digit numbers. But if he asks then it will waste almost 4 to 5 minutes using normal multiplication method. So try to adopt these formulas of cubing to save time and score high. Good Luck  👍