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If you understand how a^n and b^n actually are made up of three logical parts, these are nice diagrams. Otherwise, go to explanation of the addition of the shells of powers.
If a^n + b^n will equal c^n then every two legs on this graph outward from
the center ADD to the
Assuming a^n < b^n < c^n for the graph, and understanding shells:
a^n + b^n = c^n is three-dimensionally-logical by two approaches―by numbers of shells adding up and by summations of shell values adding up. See that both approaches compute to c or c^n, respectively, by simple addition―as long as progressively related "summaries" of "two dimensional" sets include all of the three directions (by the second computation).
See how easy it can be to know procedures that don't seem so functionally related without handy dandy pictures?
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hypotenuse of their triangle. The Pythagorean Theorem that a^2 + b^2 =
c^2 (for right triangles) is a sub-category of this chart. Fundamentally
the difference is: these charts are indicating both the successful
accumulation of shells as well as the successful accumulation of shell values. Thus
the chart also
works for a^1 + b^1 = c^1. Remember, it works by SIMPLY ADDING. The chart, in
a direct way, is moot
for powers of 3 or higher due to the 
