- Ramanraj K
Introduction
Exponentiation is repeated multiplication of the base, and corresponds to symmetrical replication of the base along the axes in n-dimensional space. If b is the base, then its power or exponent n corresponds to dimensional space n extended by symmetrical replication of the base b times along the axes. This addition of exponents in nDspace could be expressed by the following exponential identity:bn + (b-1)bn = b(n+1)
Proof:
bn + (b-1)bn
= bn + b(n+1) - bn
= b(n+1)
Illustration of symmetrical replication in n-dimensional space with powers of 2
The figures are rendered with x3dom and can be rotated with pointer.
2 power 1 in 1-D Space
One dimensional space is defined as a stright line along x-axis with a single vertex O in the centre. 2 power 1 would be 2 units along the x-axis.
2 power 2 in 2-D Space
Two dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O. The co-ordinates for 2 power 2 are 2 units along x-axis and y-axis, making a square with 4 square units.
22 = 2 x 2 = 4
2 power 3 in 3-D Space
Three dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O, and a third straight line z perpendicular to both x and y passing through O. The co-ordinates for 2 power 3 are 2 units along x-axis, y-axis and z-axis, making a cube with 8 subcube units.
23 = 2 x 2 x 2 = 8
2 power 4 in 4-D Space
Four dimensional space is the 3-D space defined by lines x, y and z as above whose vertex O1 lies on a straight line w with vertex O. Therefore, in 4-D space at least two vertices co-exist and the distance between the vertices O and O1 is defined by the 4th dimension w. The lines x and w coincide. The co-ordinates for 2 power 4 are 2 units along x-axis, y-axis, z-axis and w-axis making two cubes with 8 subcube units, and a total of 16 subcube units.
24 = 2 x 2 x 2 x 2 = 16
It may be noted that the exponential identity for addition of exponents in n dimensional space is
bn + (b-1)bn = b(n+1).
24 involves addition of two exponents as follows:
23 + (2-1)23 = 2(3+1)
23 + 23 = 24
This addition of 23 and 23 results in symmetrical replication of cubes (b-1) times extending to the next dimension resulting in 24 corresponding to the fourth dimension. In the fourth dimension along the w-axis, 23 with 8 subcube units is replicated the base times. When seen from the third dimension, it would be 16 subcube units.
2 power 5 in 5-D Space
Five dimensional space is the 3-D space defined above whose vertex lies in a 2-D plane. Therefore, in 5-D space at least two vertices exist and the distance between them is defined by the 4th and 5th dimensions. The co-ordinates for 2 power 5 are 2 units along x-axis, y-axis, z-axis, w-axis and v-axis making four cubes with 32 sub cube units.
25 = 2 x 2 x 2 x 2 x 2 = 32
The addition of 24 and 24 again results in symmetrical replication base times along v-axis extending to the next dimension resulting in 25 corresponding to the fifth dimension. In the fifth dimension along the v-axis, 24 is replicated the base times resulting in a cuboid with 4 two cube units. When seen from the third dimension, it would be 32 subcube units.
2 power 6 in 6-D Space
Six dimensional space is the 3-D space whose vertex lies in another 3-D space. In 6-D space at least two vertices exist and the distance between the two is defined by the 4th, 5th and 6th dimensions. The co-ordinates for 2 power 6 are 2 units along x-axis, y-axis, z-axis, w-axis, v-axis and u-axis making eight cubes with 64 sub cube units.
26 = 2 x 2 x 2 x 2 x 2 x 2 = 64
The addition of 25 and 25 again results in symmetrical replication base times along v-axis extending to the next dimension resulting in 26 corresponding to the sixth dimension. In the sixth dimension along the u-axis, 25 is replicated the base times resulting in a cube with 8 subcube units when viewed from the sixth dimension and 64 subcube units from 3D point of view.
This may be repeated n power times in n dimensional space.
Illustration of symmetrical replication in n-dimensional space with powers of 3
3 power 3 in 3-D Space
The co-ordinates for 3 power 3 are 3 units along x-axis, y-axis and z-axis, making a cube with 27 subcube units.
33 = 3 x 3 x 3 = 27
3 power 4 in 4-D Space
The co-ordinates for 3 power 4 are 9 units along x-axis and 3 units along y-axis and z-axis in 3-D space. In 4-D space, the 3x3x3 cube in 3-D space is symmetrically replicated three times along w-axis resulting in a cuboid with 81 subcube units.
34 = 3 x 3 x 3 x 3 = 81
The exponential addition for 34 using the identity bn + (b-1)bn = b(n+1) is as follows:
33 + (3 - 1)33
=33 + (2)33
=27 + (2)27
=27 + 54
=81
=34
3 power 5 in 5-D Space
The co-ordinates for 3 power 5 are 9 units along x-axis and 9 units along y-axis and 3 units along z-axis in 3-D space. The three 3x3x3 cubes in 4-D are symmetrically replicated three times along v-axis resulting in a cuboid with 81 subcube units.
35 = 3 x 3 x 3 x 3 x 3= 243
The exponential addition for 35 using the identity bn + (b-1)bn = b(n+1) is as follows:
34 + (3 - 1)34
=34 + (2)34
=81 + (2)81
=81 + 162
=243
=35
3 power 6 in 6-D Space
The co-ordinates for 3 power 6 are 9 units along x-axis, y-axis and z-axis in 3-D space. In 6-D space, the 35 cuboid in 5-D space is symmetrically replicated three times along u-axis resulting in a cube with 729 subcube units.
36 = 3 x 3 x 3 x 3 x 3 x 3 = 729
The exponential addition for 36 using the identity bn + (b-1)bn = b(n+1) is as follows:
35 + (3 - 1)35
=35 + (2)35
= 243 + (2)243
= 243 + 486
= 729
=36
Repeated substitution of powers in exponential addition
The identity bn + (b-1)bn = b(n+1) may be used to add or subtract numbers that have different exponents or different bases, by repeated substitution of bases and powers. This may be illustrated with the equation 33 + 63 = 35
The exponential identities for 34 and 35 are as follows:
33 + 2(33) = 34
34 + 2(34) = 35
Therefore,
(33 + 2(33)) + 2(34) = 35
33 + 2(33) + 2(34) = 35
33 + 33( 2 + 6 ) = 35
33 + 33( 8 ) = 35
33 + 33( 23 ) = 35
Thus,
33 + 63 = 35