Cube of a number

Cubing of Numbers:

Example : Find the cube of the number 106.
We proceed as follows:

i) For 106, Base is 100. The surplus is 6. Here we add double of the surplus i.e. 106+12 = 118.
(Recall in squaring, we directly add the surplus)
This makes the left-hand -most part of the answer.
i.e. answer proceeds like 118 / - - - - -

ii) Put down the new surplus i.e. 118-100=18 multiplied by the initial surplus i.e. 6=108.


Since base is 100, we write 108 in carried over form 108 i.e. .

As this is middle portion of the answer, the answer proceeds like
118 / 108 /....

iii) Write down the cube of initial surplus i.e. 63 = 216 as the last portion
i.e. right hand side last portion of the answer.

Since base is 100, write 216 as 216 as 2 is to be carried over. Answer is 118 / 108 / 216
Now proceeding from right to left and adjusting the carried over, we get the
answer
119 / 10 / 16 = 1191016.

let us Consider 106³.

i) The base is 100 and excess is 6. In this context we double the excess and then add.

i.e. 106 + 12 = 118. ( 2 X 6 =12 ) This becomes the left - hand - most portion of the cube.

i.e. 1063 = 118 / - - - -

ii) Multiply the new excess by the initial excess i.e. 18 x 6 = 108 (excess of 118 is 18)

Now this forms the middle portion of the product of course 1 is carried over, 08

in the middle.

i.e. 1063 = 118 / 08 / - - - - -

iii) The last portion of the product is cube of the initial excess.

i.e. 63 = 216.

16 in the last portion and 2 carried over.

i.e. 1063 = 118 / 081 / 16 = 1191016

Example:

Find 1002^3:

 

i) Base = 1000. Excess = 2. Left-hand-most portion of the cube becomes 1002+(2x2)=1006.

ii) New excess x initial excess = 6 x 2 = 12. Thus 012 forms the middle portion of the cube.

iii) Cube of initial excess = 2³ = 8.

So the last portion is 008.

Thus 10023 = 1006 / 012 / 008 = 1006012008.

Find 94³.

i) Base = 100, deficit = -6. Left-hand-most portion of the cube becomes 94+(2x6)

=94-12=82.

ii) New deficit x initial deficit = -(100-82)x(-6)=-18x-6=108

Thus middle potion of the cube = 08 and 1 is carried over.

iii) Cube of initial deficit = (-6)3 = -216

                                 __                    __

Now 943 = 82 / 08 / 16 = 83 / 06 / 16

                         _

                   1    2

                                  = 83 / 05 / (100 – 16)

                                  = 830584.