Indices and the units place

 The units place

I have described patterns relating to the colour codes of resistors, and the reduced mass of a two- body rotating system. Today I want to come closer home and discuss the units place of a given number knowing that it is a number that is the product of raising one number t0 a certain power. For the purposes of this article I'll stay committed to the denary system. However, Ill be open to exploring other concepts in Maths and Science if anyone requests me to do so. 

To start with I'd like define two terms of an indice, and the term indice itself

• Indice: Repetitive multiplication of a number
• Base: Number that is being repetitively multiplied 
• Power: The number of times the number is repetitively multiplied. 

Numbers and the units place of their results 

Correlation of  the Base units place place to the results unit place 

Correlation of the Indice and the units place of the result

The table

Observations

n begins from 1 for the purposes of this section

If looked at from Units place of Base

Units place number	Pattern
0	0 in all cases
1	1 in all cases
2	With the exception of being 1 when indice number is 0, it cycles through 2,4,8,6 repetitively
3	Cycles through 1,3,7,9 repetitively
4	With the exception of being 1 when indice number is 0, it alternates between 4 and 6 where 4 is for odd indices and 6 is for even indices
5	With the exception of being 1 when indice number is 0, it is 5 in all cases
6	With the exception of being 1 when indice number is 0, it is 6 in all cases
7	Cycles between 1,7,9,3 repetitively
8	With the exception of being 1 when indice number is 0, it cycles through 8,4,2,6 repetitively
9	Alternates between 1 and 9 repetitively, where 1 applies to even indices and 9 to odd indices

So to conclude:

• ·         0 and 1 stay as they are regardless of indice
• ·         With the exception of being 1 when the indice is 0, 5 and 6 stay the same
• ·         With the exception of being 1 when the indice is 0, the units place of the results of raising 2 and 8 cycles through differing permutations of 2,4,6,8 for each base unit
• ·         The result for 3 and 7 cycles between 1,3,7,9 respectively albeit in differing patterns for each number
• ·         4 while it is 1 when the indice is 0 always alternates between 4 and 6 while 9 always alternates between 1 and 9

If looked at from the indice number 

Indice number	Pattern for raising bases that don’t have a unit 1
0	1 in all cases
4n-3	Is the units place of the units base its being raised to
4n-2	Progresses as 1,4,9,6, then becomes 6 before reversing the former pattern to start again
4n-1	No specific rule, but there is a pattern that is repetitive and happens through out
4n	Alternates between 1 and 6 till the unit of the base is 5 in which case it becomes 5 before the former pattern reverses and becomes an alternation of 6 and 1 till the units number of the base is 9

Conclusion

There is a repetitive pattern that undermines what the units place is going to be when a number is raised to a power. The pattern for indice numbers repeat every quartet while the pattern for the units place of the base numbers repeat every duplet, or every every quartet or just stay the same through out with just 1 exception in some cases