October 2022 Problem of the Month

Zero Power

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Solution to the Problem:

n⁰ = n^{x - x}
= (n^x) (n^-x)
= (n^x) (n^x)^-1
= (n^x) (1/((n^x) ))
= 1

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Extra Credit:
I accepted two answers for this.
0⁰ = undefined or 0⁰ = 1.

Davit Banana gives his explanation:
If n = 0,
0⁰ = 0^{x - x}
= (0^x) (0^-x)
= (0^x) (0^x)^-1
= (0^x) (1/((0^x) ))
= (0^x) / (0^x)
= 0 / 0
= undefined since division by zero is undefined.

However, Colin Bowey pointed out that Euler gives the value as 1:
0⁰ = (a - a) ^(n-n) = (a-a)ⁿ / (a-a)ⁿ = 1 (Euler)
https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero

Some authors define 0⁰ as 1 because it simplifies many theorem statements in mathematics.
We define 0! = 1 for the same reason.   It makes everything work right.

K. Sengupta sent in arguments for both answers:
We observe that a^0 is always 1, and accordingly 0^0 is 1
However, we also observe that 0^a is zero, whenever a>0, giving 0^0 as 0
0 is of course a real number. In view of the foregoing, it follows that a real number exponent of a real number has two distinct values, viz. 0 and 1
This is a contradiction.
Therefore, there is no agreement upon the value of 0^0, and the quantity is deemed as INDETERMINATE.

Counter argument
-------------------------------
In higher mathematics, by defining 0^0 as 1 allows some formulas to be expressed in a facile manner. Also,
(i) The interpretation of b^0 as an empty product assigns the value 1.
(ii) The combinatorial interpretation of b^0 is the number of 0-tuples of elements from a b-tuple of elements from a b-element set; there is only one 0-tuple.
(iii) The set theoretic interpretation of b^0 is the number of functions from the empty set to a b-element set; there is exactly one such function, namely the empty function.
Ref: WIKIPEDIA
So, if asked for a definitive single value of 0^0, the answer would be 1.

Correctly solved by: