FUN WITH NUMBERS 2

THE POWER OF 2 AND 6
A.K.RAMAN

Let us consider the following numbers which have perfect real and whole cube roots. Having observed the existence of common difference which is 6 in the case numbers raised to the power of three, let us try to find the relation between the common difference and the cube roots.
CUBES 1 8 27 64 125 216 343 512 729
CUBE ROOTS 1 2 3 4 5 6 7 8 9
CD3 6 6 6 6 6 6 6 6 6

Neglecting the first number 1, let us do the following exercise with other numbers in the first row of the above table.
Divide the number by the common difference 6. For example take the case of number 8. If you divide 8 by 6, the reminder comes to be 2 which is the cube root of 8.
Divide the number 27 by 6 and again the reminder (3) is its cube root.
Divide the next number 64 by 6 and the reminder 4 is the cube root of 64.
Same is the case for the number 125. 125 divided by 6 gives a reminder of 5 which is the cube root of 125.
The next number is slightly different. If you divide 216 by 6 the reminder is 0. Since this is the first number in the above table which gives a reminder of 6, let us add a subscript number 1 under it. Now let us make a formula as follows and try further.
CUBES 1 8 27 64 125 216
CD3 6 6 6 6 6 6
REMINDER 0 2 3 4 5 0
SUBSCRIPT 0 0 0 0 0 1
CUBE ROOT 1 2 3 4 5 6

Cube root = Common difference multiplied by subscript plus the reminder.
Let us apply the formula the number 216.
Cube root of 216= 6*1+0 (CD3*Subscript + Reminder)
= 6
In fact the cube root for 216 is 6.
Let us do the same exercise for the next number 343. If 343 is divided by 6 it yields a reminder of 1. Then as the above formula the cube root= 6*1+1
=7
The formula holds good for the remaining numbers also.
Let us consider some more numbers also.
CUBES 343 512 729 1000 1331 1728
CD3 6 6 6 6 6 6
REMINDER 1 2 3 4 5 0
SUBSCRIPT 1 1 1 1 1 2
CUBE ROOT 7 8 9 10 11 12

CUBES 2197 2744 3375 4096 4913 5832
CD3 6 6 6 6 6 6
REMINDER 1 2 3 4 5 0
SUBSCRIPT 2 2 2 2 2 3
CUBE ROOT 13 14 15 16 17 18

CUBES 6859 8000 9261 10648 121617 13824
CD3 6 6 6 6 6 6
REMINDER 1 2 3 4 5 0
SUBSCRIPT 3 3 3 3 3 4
CUBE ROOT 19 20 21 22 23 24

THE ABOVE FORMULA HOLDS GOOD ONLY IN CASE OF NUMBERS THAT HAVE REAL WHOLE NUMBERS AS THEIR CUBE ROOT. ONE CAN NOT WORK OUT CUBE ROOT OF ANY NUMBER WITH THE ABVOE FORMULA. THE SAME FORMULA HOLDS GOOD IN FINDING SQUARE ROOTS OF NUMBERS WHICH HAVE REAL WHOLE NUMBERS AS THEIR SQUARE ROOTS, BUT FOR SQUARE ROOT CD2 IS TO BE USED IN PLACE OF CD3.


Now let us try to find the fourth root of numbers which have real whole numbers as their fourth root.
X 1 2 3 4 5 6
X power 4 1 16 81 256 625 1296
CD4 24 24 24 24 24 24
Reminder 1 16 9 16 25 0
Subscript 0 0 0 0 0 1
If we divide the numbers in row 2 by CD4 and apply the above formula to find their fourth root (CD4* subscript + reminder) you can observe that the formula does not hold good.
But let us make a table considering CD2 (Because we are trying to find the fourth root we are considering CD2 since 4 is an even number)
X 1 2 3 4 5 6
X power 4 1 16 81 256 625 1296
CD2 2 2 2 2 2 2
Reminder 1 0 1 0 1 0
Subscript 0 1 1 2 2 3
Now let us work out the fourth root of 1296 using the formula
Fourth root of 1296= CD2 * subscript + Reminder
= 2*3+0
=6 which is correct.
One can do similar exercise for any number which has a real whole number as its fourth root and observe that the formula holds good. Knowledge of correct subscript is must for this exercise.
Now let us consider the following table.
X 1 2 3 4 5 6
X power 5 1 32 243 1024 3125 7776
CD3 6 6 6 6 6 6
Reminder 1 2 3 4 5 0
Subscript 0 0 0 0 0 1
Let us calculate the fifth root of 7776. Since we are calculating odd number (5) root, we have to use CD3 in the above formula.
Fifth root of 7776 = CD3 * Subscript + Reminder
= 6 * 1 + 0
= 6
For those numbers which has a real whole number as its fifth root, CD3 (6) is to be used in the above formula.
We can generalize by saying
ODD NUMBER ROOT OF A NUMBER FOR WHICH A REAL WHOLE NUMBER EXISTS AS ITS ROOT, THE SAME CAN BE FOUND OUT USING THE FORMULA
ROOT = CD3 * SUBSCRIPT PLUS REMINDER
EVEN NUMBER ROOT OF A NUMBER FOR WHICH A REAL WHOLE NUMBER EXISTS AS ITS ROOT, THE SAME CAN BE FOUND OUT USING THE FORMULA
ROOT = CD2*SUBSCRIPT PLUS REMINDER.
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