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The simple way to reason it is this: think backwards. Say I tell you that 210=1024. Given this how would you find 29? You would never want to multiply the whole thing out again. You would probably divide 210 by 2. What's 23? 8. What is 22? divide that last answer by 2 to get 8/2 = 4. What is 21? divide the last answer by 2 to get 4/2 = 2. What is 20? divide the last answer by 2 to get 2/1 = 1.

In higher mathematics, the reason comes from the definition. Mathematicians define a0 = 1 and then they define an+1 as an * a. Let me think of an example. If we define 20=1, then what is 21? 21= 20+1= 20*2 (this is because we defined it this way an*a = an+1). = 1*2 = 2. We can similarly define every exponent in this way by using the one below it. Mathematicians call this a definition by induction.

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Q: Why exponent zero on a number is equal to one?
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