Why is a number to the 0th power always 1?

Answer 1

(Note that 0 to the power 0 is not defined. Any other number to the power 0 are always equal to 1.)

If it is defined that way, the following rule is still valid with powers of zero: xa times xb = xa+b. If you use a zero exponent, you have xa times x0 = xa. Now, which number can you replace for x0 in this equation? - 1, of course.

Another way of looking at it is to consider the following sequence:

103 = 1000

102 = 100

101 = 10

100 = ???

In the left column, the exponent reduces by one every time, in the right column, the result gets reduced by a factor of 10. So, what number should you replace for the question marks? - If you continue the sequence, you get 10/10 = 1. By the same token, just by continuing the sequence, it should be obvious that 10 to the power -1 would be 1/10.