Question 1190689: What is z to the zero power times z to the negative 10th? Only using positive exponents. Found 2 solutions by math_tutor2020, Alan3354:Answer by math_tutor2020(3816) (Show Source):
You can put this solution on YOUR website!
Any nonzero number to the exponent of 0 will result in 1. as long as
Multiplying by 1 has no effect, so the z^0 won't affect the answer.
The portion then turns into when we make the exponent positive. Take the reciprocal of the base to make the exponent positive.
The general rule is that or you can think of it like
You can put this solution on YOUR website! 0^0 = 1, by convention.
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That means some people made that decision.
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In 1752, Euler in Introductio in analysin infinitorum wrote that a0 = 1[15] and explicitly mentioned that 0^0 = 1.[16] An annotation attributed[17] to Mascheroni in a 1787 edition of Euler's book Institutiones calculi differentialis[18] offered the "justification"
{\displaystyle 0^{0}=(a-a)^{n-n}={\frac {(a-a)^{n}}{(a-a)^{n}}}=1}{\displaystyle 0^{0}=(a-a)^{n-n}={\frac {(a-a)^{n}}{(a-a)^{n}}}=1}
as well as another more involved justification. In the 1830s, Libri[19][17] published several further arguments attempting to justify the claim 0^0 = 1, though these were far from convincing, even by standards of rigor at the time.[20]
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I wouldn't argue with Euler.
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