Why is anything to the 0th power 1?

This is actually a defined value.

nm can be defined as a recursive function:

nm = n * nm-1 for n,m > 0

If we follow this definition, we will always come to m = 0, so a value must be defined for n0. The logical choice is n0 = 1, since it will always make our definition hold true.

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The work above is not mine, I'm in no way correcting this work rather enforcing and elaborating it by giving an example to make it easier to understand (hopefully). As above nm can be reworked:nm = n * nm-1

• for an example let's use 23(n=2 and m=3). using the recursive function we have : 23= 2*23-1......23 is 8 so the other side of the equation should also give us 8 which it does because 2*23-1 or 2*22 or 2*4 =8

although it is simple enough to solve 23 without using the recursive function we now understand the logic and see it works (which always helps me) and can now use this function to understand why a # to the 0th power =1.

• So now for an example we'll use: 20(n=2, m=0) using the recursive function we have 20 = 2*20-1 ; so even not knowing 20 we do know that solving the other side of the equation will give us the answer so: 2*20-1 or 2*2-1 or 2*1/2 = 1. So we know that 20=1

this can be applied with any base number to discover that anything to the 0th power is in fact 1..hope that helped

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The above works aren't mine, I'll try to offer the most simple explanation possible

First, let's ask how do we end up with n0?

As the rules of indices defines, nm/nm = nm-m = n0, while the fraction itself, if we do not subtract their indices, can be simplified from nm/nm to 1/1, because we have cancelled out the same nm at both numerator and denominator, and if you know your divisions, 1 divided by 1 is 1, therefore we have come to the conclusion n0 = 1.