Proof that x^0=1

Lesson on why x^0=1.
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[Inspirator_AG]

*x⁰ = 1 Proof*





*TL;DR: Using 0 as an exponent is an empty product. Empty products always output 1, regardless of the input.*



Using a capital pi to define integer exponents in both directions (±), x⁰ always results in an empty product (upper bound less than lower bound), which evaluates to 1.



Here are basic formulas for integer exponents (positive/negative integers) using capital Pi.
bⁿ = Πₖ₌₁ⁿ b
b⁻ⁿ = Πₖ₌₁ⁿ (1÷b)

When n is 0, this results in an empty product for both equations.
b⁰ -= Πₖ₌₁⁰ b- = 1
b⁰ -= Πₖ₌₁⁰ (1÷b)- = 1

*This even holds true when b and n are both 0.* (Undefined values are overridden by the empty product.)
*0⁰ **-= Πₖ₌₁⁰ 0-** = 1*
*0⁰ **-= Πₖ₌₁⁰ (1÷0)-** = 1*



The same thing happens with factorials of natural numbers and 0!.
n! = Πₖ₌₁ⁿ k
0! -= Πₖ₌₁⁰ k- = 1



_(X-posted under multiple vids)_