Why is 1 to the power 0 is 1?

Answer 1

For the same reason that any other number (except zero) to the power 0 is 1. This is related to the fact that 1 is the neutral element of multiplication. It can be understood if you see this sequence - I'll use 10 to different powers, but any other number (except zero) will do just as well:

103 = 1000

102 = 100

101= 10

100 = ???

What is the logical continuation of this sequence? As you can see, on the left, the exponents get reduced one at a time; on the right, each number is one-tenth of the previous number in the sequence. Therefore, the logical continuation is 1.

Observe:

35 * 32 = 3*3*3*3*3*3*3 = 37, which is the same as 35 * 32 = 35+2 = 37

7 * 73 = 7*7*7*7 = 74, or 7 * 73 = 71+3 = 74

411 * 40 = (by above observation) 411+0 = 411, it means that 40 played the role of 1.

24/22 = (2*2*2*2)/(2*2) = 2*2 = 22, which is the same as 24/22 = 24-2 = 22

89/86 = (8*8*8*8*8*8*8*8*8)/(8*8*8*8*8*8) = 8*8*8 = 83, or 89/86 = 89-6 = 83

53/53= (5*5*5)/(5*5*5) = 1 , or (by above observation) 53-3 = 50

Thus, it is true that 50 = 1 (things = to the same thing = each other).

Now we can say that 10 = 1.

We can represent 1 such as,

1 = integer/integer, except 0/0 (it's undefined, cannot be both zero and 1)

1 = 5/5, 10 = (5/5)0, 1 = 50/50

1= 1/1 also 10 = (1/1)0, 1 = 10/10

1 cannot be represent by 0/0, or (0/0)0, therefore it cannot be also represented by 00/00.

Thus, 00 is not 1.

Therefore, we make a definition, that any nonzero real number raised to the zero power is 1.