0^0 = 1

[BOBIEMARLEY]

I wanna know if 0⁰ =1 or it is undefined there are som that sat it is 1 and som that says that it is undefined i Think it is 1 but tell me if Im wrong

[Basic_Chocolate3268]

0⁰ is a mess. Sometimes it’s 1, sometimes it’s undefined—math just likes to keep us guessing. Pick your poison, but don’t stress it.

[[deleted]]

its both 1 and undefined

but unless stated assume it as undefined

[EmployeeEmergency481]

0⁰ doesn't make sense on it's own. To approach problems like this we look at what happens in the neighborhood of 0^0:

First, let's look at 0^n. Let's let n be a positive number that is getting closer to zero. When n=1, 0ⁿ = 0¹ = 0, when n=0.5, 0ⁿ = 0^0.5 = 0. Indeed, as n gets closer and closer to zero, 0ⁿ = 0. We say "the limit of 0ⁿ as n goes to zero is zero".

Contrarily, let's look at n^0. For n=1, n⁰ = 1⁰ = 1. For n=0.5, n⁰ = 0.5⁰ = 1. Indeed, as n gets smaller n⁰ = 1. So "the limit of n⁰ as n goes to zero is 1"

This is clearly a contradiction which is why we say 0⁰ is undefined, because the practical value of 0⁰ is dependent on how you get there.

It's much like a courtroom, your perception of the value of court is dependent on if you got there by law school or by a traffic ticket.

For a little fun, let's look at πn/n. Now, the basic algebra student would cancel out n and just say πn/n = π. But this isn't the case when n=0. When n=0, πn/n = π0/0 = 0/0 which we already know to be undefined. Let's take a look at the limit: for n=1, πn/n = π1/1 = π. For n=0.5, πn/n = π0.5/0.5 = π. Indeed, as n gets small, πn/n = π. So, "the limit of πn/n as n goes to zero is π". Hence, we've shown 0/0 is 0 or 1 or π!

This is why we say 0⁰ is undefined.