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This can be seen clearly in the case of positive integer exponents, from the definition of powers as repeated multiplications. Here is an example (I will use the symbol "^" for powers):
10^2 x 10^3
= (10 x 10) x (10 x 10 x 10)
= 10^5
As you can see, in this example - and in any similar example you can invent - when you write out the factors, you get as many factors as there are factors in the individual parts. In the above example, the first part has two factors, the second part has three factors, so if you combine them, you have five (2 + 3) factors.
For zero, negative, or non-integer exponents, the rule still applies; in this case, it is simply defined this way for consistency.
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