__Creative thoughts-10__Power of a number as the difference of two squares

Any power of a number can be expressed as the difference between two squares ,

where the sum and difference of its roots give higher and lower powers of that number,

so that the product of them is exactly equal to the initial power of the number. This can be

applied effectively to any power of any number.

Cubes

1x1x1 = 1 = 1x1 – 0x0 ; 1 + 0 = 1 = 1x1 ; 1-0 = 1

2x2x2 =8 = 3x3 – 1x1 ; 3 + 1 = 4 = 2x2 ; 3-1 = 2

3x3x3 = 27 = 6x6 – 3x3 ; 6 + 3 = 9 =3x3 ; 6 – 3 = 3

4x4x4 = 64 = 10x10 – 6x6 ; 10 + 6 = 16 = 4x4 ; 10-6 = 4

5x5x5 = 125 = 15x15 – 10x10 ; 15 +10 = 25 = 5x5 ; 15-10= 5

It is found that all the squares involved in these relations are related to triangular numbers

( as they can be represented by triangles). The first few triangular numbers are 1,3,6,10,

15,21,28,36,45,……… The n th triangular number is the sum of n natural numbers from

1 and it is equal to n(n+1)/2. The cube of a number N is found to be the difference

between two successive triangular numbers ,whose sum gives square of the cube

root and the difference ,the cube root itself.

NxNxN = [N(N+1)/2] x [N(N+1)/2] - [N(N-1)/2]x[N(N-1)/2]

If T(n) denotes the n th triangular number ,then

T(n) + T (n-1) = n x n

T(n) – T(n-1) = n

4 th power of a number

The fourth power of a number N is equal to the difference of two squares, the sum

of its roots is equal to cube of that number and the difference of its roots is equal to

the number Itself.

2x2x2x2 = 16 = 5x5 – 3x3 ; 5+3 = 8 = 2x2x2 ; 5-3 = 2

3x3x3x3 = 81 = 15x15 – 12x12 ; 15 + 12 = 27 = 3x3x3 ; 15-12 = 3

4x4x4x4 = 256 = 34x34 – 30x30 ; 34 +30 = 64 = 4x4x4 ; 34-30 = 4

5x5x5x5 = 625 = 65x65-60x60 ; 65+60 = 125 =5x5x5 ;; 65-60 = 5

If N is any number

NxNxNxN = (N/2)[(NxN+1)(NxN+1)] – (N/2)[(NxN-1)(NxN-1)]

The smaller square root number in thes relations is (N-1) times the sum of N numbers

from 1 in the natural series and the greater square root number is obtained by adding N

with it.

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