Cube tricks and tips for competitive exams (11 - 99)

Cube of a natural number is the product obtained by multiplying the number three times. In these a long time is taken by the students due to long tradition method of multiplication. Now I am going give you a tricky method which will help you get the cube of a natural number in quick time.

But as usual practice is the much needed factor of this trick otherwise the motive of giving this trick will be worth less and most importantly you have to essentially learn the cubes from 1 to 10 which is much needed for this trick and a special technique of addition is also to be understood.

Try to understand the concept of Carry Forward Addition with help of demonstrations shown in this topic. Let’s now proceed with the chapter:-

The trick for the range (11 to 99) contains nothing but an algebraic formula with proper and fixed sequence, which should not be disturbed.

Formula :- (a + b)³ = a³ +3a²b + 3ab² + b³ 

                    Where tenth place digit of these numbers is taken as “a”

                              & unit place digit of these numbers is taken as “b”

Important thing is that the sequence of this formula should not be changed.

Cube to be learnt first:-

1³  = 1 × 1 × 1 = 1.

2³ = 2 × 2 × 2 = 8.

3³ = 3 × 3 × 3 = 27.

4³ =4 × 4 × 4 = 64.

5³ =5 × 5 × 5 = 125.

6³ =6 × 6 × 6 = 216.

7³ =7 × 7 × 7 = 343.

8³ =8 × 8 × 8 = 512.

9³ =9 × 9 × 9 = 729.

10³ =10 × 10 × 10 = 1000.

Now I am giving the tricky method for calculation of Cube of Natural Numbers

Method :-

Step 1^st :- Set the given number in form of (a + b)³.

Step 2^nd :- Now expand in the sequence a³ +3a²b + 3ab² + b^3, don’t disturb or change the sequence.

Step 3^rd :- Now dissolve the expansion.

Step 4^th :- Now apply the Carry Forward Addition to get the required Cube.

Demonstration of above method (Try to Understand the procedure of Carryu Forward Addition):-

11³ →   Step 1^st    Let a =1 and b =1 and set it as (1 + 1)³

               Step 2^nd   Expand it as → 1³ + 3 ×( 1² × 1) + 3 × (1 × 1²) + 1³

               Step 3^rd   1 + 3 + 3 + 1

               Step 4^th    Arrange it like this by applying carry forward addition → 1331

Get the required answer 11³ = 1331

12³ →   Step 1^st    Let a =1 and b =2 and set it as (1 + 2)³

               Step 2^nd   Expand it as → 1³ + 3 ×( 1² × 2) + 3 × (1 × 2²) + 2³

               Step 3^rd   1 + 6 + 12 + 8

            Step 4^th    Arrange it like this by applying carry forward addition → 1(6+1)28 = 1728

Get the required answer 12³ = 1728.

13³ →   Step 1^st    Let a =1 and b =3 and set it as (1 + 3)³

               Step 2^nd   Expand it as → 1³ + 3 ×( 1² × 3) + 3 × (1 × 3²) + 3³

               Step 3^rd   1 + 9 + 27 + 27

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 1 + 9 + (27 + 2)7 = 1 + (9 + 2)97 = (1 + 1)197 = 2197.

Get the required answer 13³ = 2197.

14³ →   Step 1^st    Let a =1 and b =4 and set it as (1 + 4)³

               Step 2^nd   Expand it as → 1³ + 3 ×( 1² × 4) + 3 × (1 × 4²) + 4³

               Step 3^rd   1 + 12 + 48 + 64

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 1 +12 + (48 + 6)4 = 1 + (12 + 5)44 = (1 + 1)744 = 2744.

Get the required answer 14³ = 2744.

15³ →   Step 1^st    Let a =1 and b =5 and set it as (1 + 5)³

               Step 2^nd   Expand it as → 1³ + 3 ×( 1² × 5) + 3 × (1 × 5²) + 5³

               Step 3^rd   1 + 15 + 75 + 125

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 1 + 15 + (75 +12)5 = 1 + (15 + 8)75 = (1 + 2)375 = 3375.

Get the required answer 15³ = 3375.

16³ →   Step 1^st    Let a =1 and b =6 and set it as (1 + 6)³

               Step 2^nd   Expand it as → 1³ + 3 ×( 1² × 6) + 3 × (1 × 6²) + 6³

               Step 3^rd   1 + 18 + 108 + 216

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 1 + 18 + (108 + 21)6 = 1 + (18 + 12)96 = (1 + 3)096 = 4096.

Get the required answer 16³ = 4096.

17³ →   Step 1^st    Let a =1 and b =7 and set it as (1 + 7)³

               Step 2^nd   Expand it as → 1³ + 3 ×( 1² × 7) + 3 × (1 × 7²) + 7³

               Step 3^rd   1 + 21 + 147 + 343

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 1 + 21 + (147 + 34)3 = 1 + (21 + 18)13 = (1 + 3)913 = 4913.

Get the required answer 17³ = 4913.

18³ →   Step 1^st    Let a =1 and b =8 and set it as (1 + 8)³

               Step 2^nd   Expand it as → 1³ + 3 ×( 1² × 8) + 3 × (1 × 8²) + 8³

               Step 3^rd   1 + 24 + 192 + 512

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 1 + 24 + (192 + 51)2 = 1 + (24 + 24)32 = (1 + 4)832 = 5832.

Get the required answer 18³ = 5832.

19³ →   Step 1^st    Let a =1 and b =9 and set it as (1 + 9)³

               Step 2^nd   Expand it as → 1³ + 3 ×( 1² × 9) + 3 × (1 × 9²) + 9³

               Step 3^rd   1 + 27 + 243 + 729

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 1 + 27 + (243 + 72)9 = 1 + (27 + 31)59 = (1 + 5)859 = 6859.

Get the required answer 19³ = 6859.

21³ →   Step 1^st    Let a =2 and b =1 and set it as (2 + 1)³

               Step 2^nd   Expand it as → 2³ + 3 ×( 2² × 1) + 3 × (2 × 1²) + 1³

               Step 3^rd   8 + 12 + 6 + 1

            Step 4^th    Arrange it like this by applying carry forward addition →(8 +1)261 =9261.

Get the required answer 21³ = 9261.

22³ →   Step 1^st    Let a =2 and b =2 and set it as (2 + 2)³

               Step 2^nd   Expand it as → 2³ + 3 ×( 2² × 2) + 3 × (2 × 2²) + 2³

               Step 3^rd   8 + 24 + 24 + 8

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 8 + (24 + 2)48 = (8 + 2)648 = 10648.

Get the required answer 22³ = 10648.

23³ →   Step 1^st    Let a =2 and b =3 and set it as (2 + 3)³

               Step 2^nd   Expand it as → 2³ + 3 ×( 2² × 3) + 3 × (2 × 3²) + 3³

               Step 3^rd   8 + 36 + 54 + 27

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 8 + 36 + (54 + 2)7 = 8 + (36 + 5)67 = (8 + 4)167 = 12167.

Get the required answer 23³ = 12167.

24³ →   Step 1^st    Let a =2 and b =4 and set it as (2 + 4)³

               Step 2^nd   Expand it as → 2³ + 3 ×( 2² × 4) + 3 × (2 × 4²) + 4³

               Step 3^rd   8 + 48 + 96 + 64

               Step 4^th    Arrange it like this by applying carry forward addition →

                                 8 + 48 + (96 + 6)4 = 8 + (48 + 10)24 = (8 + 5)824 = 13824.

Get the required answer 24³ = 13824.

To be continued like this upto 99 only (this trick is valid for 11 to 99).

First of you should practice this trick several times and try to omit the unnecessary steps during the calculations and feel free write for your queries and doubts.