Let's look at some examples of negative exponents, and find a pattern!
(Pay close attention to the bolded ones, and see if you can find what happens)
So you see the pattern is that you take the reciprocal of the base number and then raise it to the corresponding power.
Now that we have noticed the pattern think about 2-5 then the answer must be 1/25! so 2-5=1/25. Similarly 4-7=1/47. This is the definition for negative exponents!
This is all about the pattern! If the negative exponent is in the bottom it moves to the top, if it is in the top then it moves to the bottom! so, a-n=1/an similarly, 1/a-n=an
Let's look at some exponents, look to see if you see the pattern!
Each time the exponent lowers, so does the amount it is equal to. If you divide by the base number each time it will lower the number. Lets think about the 10's. 100,000 divided by ten, is obviously 10,000 and so on with the pattern. So if your wondering how anything to the zero power could be 1, look at the previous solution, it is ten so 10 divided by 10 is one and so on. It is like this with anything, look through the 2's, it has the same pattern. Every whole number has this pattern.
Here is another helpful video of Khans explaining negative and zero exponents!
Source: Khan Academy