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
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 )
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]
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]