Creative thoughts-11

__Fifth power of a number__The fifth power of a number 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 square of the

number.

2x2x2x2x2 = 32 = 6x6 – 2x2 ; 6 + 2 = 8 = 2x2x2 ; and 6 – 2 = 4 = 2x2

3x3x3x3x3 = 243 = 18x18 – 9x9 ; 18 + 9 = 27 = 3x3x3 and 18-9 = 9 = 3x3

4x4x4x4x4= 1024 = 40x40 – 24x24 ; 40 + 24 = 64 = 4x4x4 and 40 – 24 = 16 = 4x4

5x5x5x5x5 = 3125 = 75x75-50x50 ; 75+50 = 125 = 5x5x5 and 75- 50 = 25 = 5x5

In general, it can be shown as ,

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

(N-1)/2][(NxN)(N-1)/2]

The greater square root number in this relation is N times the sum of N numbers from 1

In the natural series and the smaller square root number is obtained by subtracting

NxN from it.

__Sixth power of a number__The sixth power of a number N is equal to the difference of two squares ,the sum of its

roots is equal to fourth power of that number and the difference of its roots is equal to

the square of that number.

2x2x2x2x2x2 = 64 = 10x10 – 6x 6 ; 10 + 6 = 16 = 2x2x2x2 and 10 – 6 = 4 = 2x2

3x3x3x3x3x3 = 729 = 45x45 – 36x36 ; 45+36 = 8l = 3x3x3x3 and 45-36 = 9 = 3x3

4x4x4x4x4x4 = 4096 = 136x136 – 120x120 ;136+120=256=4x4x4x4 and

136-120 =16 = 4x4 .

5x5x5x5x5x5 = 15625 = 325x325 – 300x300 ; 325+300=625 =5x5x5x5 and

325-300 = 25=5x5 .

The general form of this type of relation is

NxNxNxNxNxN =(NxN)(NxNxNxN) = [(NxN +1)(NxN)/2][(NxN +1)(NxN)/2]

- [(NxN-1)(NxN)/2]{(NxN-1)(NxN)/2]

If NxNxNxNxNxN is split into N and NxNxNxNxN ,then

NxNxNxNxNxN = [(N+1)(NxN) √N/2][(N+1)(NxN)√ N/2]

– [(N-1)(NxN)√ N/2][(N-1)(NxN)√ N/2]

If N is an even square number ,then NxNxNxNxNxN can be expressed as a

difference of two squares in another way also e.g., when N = 4,

4x4x4x4x4x4 = 80x80 – 48x48

This idea can be extended to any power of a number .e.g.,

NxNxNxNxNxNxN = [(N+1)NxNxNx/2][(N+1)NxNxN/2]

- [(N-1)NxNxN/2]{(N-1)NxNxN/2]

NxNxNxNxNxNxNxN= [(NxN+1)NxNxN/2][(NxN+1)NxNxN/2]

-[(NxN-1)NxNxN/2][(NxN-1)NxNxN/2]

NxNxNxNxNxNxNxNxN= [(N+1)NxNxNxN/2][(N+1)NxNxNxN/2]

-[(N-1)NxNxNxNxN/2][(N-1)NxNxNxN/2]

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